An unusual conference on the foundations of geometry was held in 1974 at the University of Toronto. It lasted from July
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English Pages 348 [347] Year 1976
Foundations of geometry Selected proceedings of a cont erence
Foundation s of geometry Selected proceedings of a cont erence
Edited by P. Scherk
University of Toronto Press Toronto and Buffalo
©
University of Toronto Press 1976 Toronto and Buffalo Printed in Canada Reprinted in 2018
LIBRARY OF CONGRESS CATALOGING IN PUBLICATION DATA Main entry under title: Foundations of geometry Conference held at the University of Toronto, July 17Aug. 18, 1974. 1. Geometry  Foundations  Congresses. I. Scherk, Peter. II. Toronto. University. QA681.F74 516 7542127 ISBN 0802022162 ISBN 9781487582401 (paper)
CONTENTS
Preface  vii Contributors  ix Combinatorics of finite planes and other finite geometric structures  A. Barlotti  3 Configurational propositions in projective spaces J. Cofman  16 Unitary planes  Martin Gotzky  54 The LenzBarlotti type III  Christoph Hering  88 Some recent results on incidence groups  H. Karzel  114 Topological Hjelmslev planes  J.W. Lorimer  144 On finite affine planes of rank 3  Heinz Liineburg  147 Every group is the collineation group of some projective plane  E. Mendelsohn  175 Recent advances in finite translation planes  T . G. Ostrom  183 Recent results on preHjelmslev groups  Edzard Salow  206 Finite geometric configurations  J.J. Seidel  215 Distributive quasigroups  K. Strambach  251 Planes with given groups  Jill Yaqub  277
V
PREFACE
In 1974 the University of Toronto sponsored a unique experiment. For more than four weeks, from July 17 to August 18, a group of outstanding workers in the foundations of geometry lived together on the campus of the university, gave and attended lectures, and met informally. Three longer series of sixteen lectures each were given by F. Bachmann (Kiel University, West Germany)
on Hjelmslev
groups, by R. Lingenberg (Karlsruhe, West Germany) on
s
groups, and by J. Tits (College de France, Paris) on generalized polygons.
Two of these are being prepared for publi
cation as separate volumes. In addition, fifteen mathematicians gave not the usual brief accounts of their work but detailed reports which required up to seven hours of lecturing. reports appear in this volume.
Thirteen of these
The other two reports, by
E. Sperner (Hamburg University, West Germany) on affine structures and by J. Tits on quadratic forms and the geometry on quadrics, unfortunately could not be included. The editor wishes to thank the University of Toronto Press and the Blythe Foundation for making publication of the volume possible.
The papers represent an invaluable basis
for further progress in the field, as has already been demonstrated by the appearance of one important paper which received its stimulus from the conference.
vii
CONTRIBUTORS
BARLOTTI, ADRIANO Universita di Bologna, Istituto Matematico, Bologna, Italy COFMAN, JUDITA Fachbereich Mathematik der Johannes Gutenberg Universitat, 6500 Mainz, Postfach 3980, West Germany GOTZKY, MARTIN Mathematisches Seminar der Christian Albrechts Universitat, 2300 Kiel, Neue Universitat, Angerbau Al , West Germany HERING, CHRISTOPH Mathematisches Institut der Universitat Tubingen, 7400 Tubingen 1, Auf der Morgenstelle 10, West Germany KARZEL, HELMUT Institut fur Geometrie der Technischen Universitat, 8000 Munich 2, Arcisstrasse 21, West Germany LORIMER, J . W. Department of Mathematics, University of Toronto, Toronto, Ontario, M5S !Al, Canada LONEBURG, HEINZ Fachbereich Mathematik der Universitat Kaiserslautern, 675 Kaiserslautern, Pfaffenbergstrasse, West Germany MENDELSOHN, ERIC Department of Mathematics, University of Toronto, Toronto, Ontario, M5S !Al OSTROM, T.G . Department of Mathematics, Washington State University, Pullman, Wash . 99163, U. S.A.
ix
SALOW, EDZARD 28 Bremen, Katrepelerstrasse 7, West Germany SEIDEL, J.J. Department of Mathematics, Technological University Eindhoven, PO Box 513, Eindhoven, The Netherlands STRAMBACH, KARL Mathematisches Institut der Universitat ErlangenNiirnberg, 852 Erlangen, Bismarckstrasse l½, West Germany YAQUB, JILL C.D.S. Department of Mathematics, Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210, USA
X
FOUNDATIONS OF GEOMETRY Selected proceedings of a conference
COMBINATORICS OF FINITE PLANES AND OTHER FINITE GEOMETRIC STRUCTURES
A. Barlotti
In the following pages, we survey recent results on some of the topics considered in section 3.2 of P. Dembowski's book Finite Geometries (1968; for other references on this subject, see A. Barlotti, 1974). 1.
SYSTEMS AXIOMATIZING FINITE PLANES AND OTHER FINITE
STRUCTURES Many authors have contributed to the study of systems of axioms for finite projective and affine planes (see Dembowski, 1968, pp. 138139).
Using the notation of Dembow
ski, we state the following results. THEOREM A (A. Basile, 1970).
Let n > 1 be an integer.
Then the following is a complete list of minimal axiom systems, each one of which will ensure that a nondegenerate incidence structure is a finite projective plane of order n: (a)
(l'), (2"), (4"), (5'),
(a')
3
(1"), (2'), (3"), (6');
(b)
(l'),(2 11 ),(4 11 ),(6'),
(b')
(1 11
(c)
(l'),(2 11 ),(3'),(6 11 ) ,
(c')
(1"), (2'), (4'), (5");
(d)
(l'),(2 11 ),(5'),(6 11
(d')
(l 11)
(e)
(1'),(2 11 ),(6),
(e')
(l 1 1 ) , (2'), (5);
(f)
(2'),(3 11 ),(4 11 ),(5'),
( f')
(1
(g)
(1'),(3 11 ),(4 11 ),(5'),
(g')
(2'), (3 11
),
(4"), (6');
(h)
(1 11 ),(2'),(3"),(4'),(6 11
(h' )
( 1' ) , ( 2 11
)
(
),
),
1 )
t
(2'), (3"), (5');
(2 t)
,(3 11
)
t
(5 11
,(4 11
)
t
),
(6
I);
(6');
),
,
3' ) , ( 4 11 )
,
(i)
(2 11
),
(i')
(1"),(3'),(4),(5 11
(j)
(l"), (3), (4'), (6"),
(j')
(2 11 ),(3'),(4),(5");
(k)
(2'), (3"), (5'), (6"),
(k')
(1'),(4"),(5"),(6');
(,Q,)
( l ' ) , (3"), (5'), (6"),
(,Q,')
(2'),(4"),(5"),(6');
(m)
( 3 11
(mt )
(3'), (4"), (5"), (6');
(n)
(2'),(3"),(6)
(n')
(1'),(4"),(5);
(o)
(1'),(3"),(6),
(o')
(2 I)
(p)
(3"),(4'),(6),
(p')
(3'),(4"),(5);
(q)
(l"), (3'), (4'), (5"), (6 11
),
),
) t
(r)
(2
(s)
(1 1 )
(3), (4'), (6 11
(
4
I ) t
(
5
I ) t
(
6 H)
t
t
(4")
t
);
(2"), (3'), (4'), (5"), (6");
(6 11
)
,
(r')
(1'), (5"), (6);
(5)
t
(6 II)
t
(s')
(2'),(5"),(6);
(t)
(4'),(5),(6"),
(t')
(3 I)
(u)
(3'), (5), (6"),
(u')
(4'), (5"), (6).
t
(5")
t
THEOREM B (P. Brutti, 1969; G. Pabst, 1974). be an integer .
(6) ;
Let n > 2
Then the following is a complete list of
minimal axiom systems, each one of which will ensure that
4
;
(q') t
t
5 11 )
(5);
(5)
I) t
(
a nondegenerate incidence structure is a finite affine plane of order n: (a)
(3)
(b')
(5), (7), (9");
(c')
(5),(8),(9);
(d')
(3'), (5"), (7"), (9);
(e)
(3")
(f)
(3'), (5"), (8"), (9);
(g')
(3")
I
(5 I)
I
(8 I)
I
(9") i
(k)
(5")
I
(7")
I
(8 I)
I
(9 I);
(Q,)
(5'),(7
(m)
(3'), (5"), (7'), (8"), (9");
(n)
(3"), (5'), (7"), (8').
(5)
I
I
I
(9) ;
(5 I)
1 ),
I
(7 I)
I
(9") i
(8"), (9");
Clearly additional hypotheses will modify the above systems.
C. Bernasconi (to appear, a) studied the problem
with the additional assumption that the structure is connected, and M. Crismale (to appear) changed (3'), (4'),
(4") to axioms which require that in (3) and (4)
"=" be replaced by "~ 11 or
11
~
bols in Dembowski (1968], v 1 b
1
~
(3"),
n + 1, b
1
~
in average (with the sym
11
~
n + 1, v 1
~
n + 1,
n + 1).
Similar systems of axioms have been studied for other finite structures:
for inversive planes (R. Bumcrot,
private communication) and for projective designs (C. Bernasconi, to appear, b).
5
In connection with the above questions, see P . Biscarini (to appear), N.G. de Bruijn and P. Erdos (1968), R. Magari (1964), and I. Reiman (1963, 1968).
In particular,
we wish to point out that D.A. Drake (1974) has proved that in a finite Hjelmslev plane the parallel axioms cannot be replaced by cardinality assumptions. 2.
REPRESENTATION OF NETS AND PLANES
We can use the representation of nets and planes by sets of mutually orthogonal latin squares to study the properties of the plane (see R. C. Bose and K. R. Nair, 1941; G. Pickert, 1955). For recent results in this field, see A. Barlotti (to appear), J . Denes and A.O. Keedwell (1974), and P. Hohler (1970, 1972). 3.
GALOIS GEOMETRIES
Galois geometries are the study of the sets of points in finite spaces.
In the following, PG(r, q) will denote an
rdimensional projective space of order q .
If r
= 2,
the symbol PG(2, g) will be used only for a desarguesian plane whereas n(q) will denote any projective plane of order q. A (k; n)arc of n(q) is a set of k points of n(q) such that n is the largest number of them which are collinear.
In a given plane, a
there does not exist a
(k; n)arc is complete if
(k'; n)arc which contains it with
6
k' > k. (k; 2)arcs are simply called karcs; i.e. a karc of n(q) is a set of k points of n(q) such that no three of them are collinear. 3.1.
Some properties of karcs
In the last twenty years, karcs have been studied extensively, especially in PG(2, q).
We shall recall here some
of their properties. A line of n(q) is called a secant, a tangent, or an external line to a karc Kif it meets Kin two, one, or zero points.
It is very easy to prove that in n(q) the
number of tangents of a karc passing through a point not on the arc has the same parity ask. A question which in general is still open is determining the maximum value of k for which, in a given n(q), a karc will exist (the packing problem forkarcs). In the general case, we have the following result: THEOREM 3.1.1 (R.C. Bose, 1947). in n(q), then k
$
If there exists a karc
q + 1 when q is odd and k
$
q + 2 when
q is even. The following theorems give the existence of maximal arcs in the desarguesian planes. THEOREM 3.1.2. In PG(2, q), an irreducible conic is a (q
+ 1)~. 7
THEOREM 3.1.3 (B. Quist, 1952). of a
In TT(q), all the tangents
(q + 1)arc are concurrent when q is even. This implies that in TT(q), if q is even, then every
(q + 1)arc is incomplete and can be uniquely completed to form a
(q + 2)arc.
The point at which all the tangents
of a (q + 1)arc meet is called the nucleus of that {q + 1)arc. The question of the existence of {q + 1)arcs in every projective plane TT(q) is still open. THEOREM 3.1.4 (G. Korchmaros, to appear) .
Each Hall plane
of odd order contains a {q + 1)~. A graphic characterization for conics in PG(2, q), q odd, is given by the following theorem. THEOREM 3.1.5 {B. Segre, 1955).
In PG(2, q), if q is odd,
then every (q + 1)arc is an irreducible conic. In PG(2, 2h), we can obtain a
(q + 2)arc by adjoin
ing the nucleus to a conic (see Theorem 3.1.3).
By delet
ing a point other than the nucleus from this (q + 2)arc, we obtain a {q + 1)arc which is not a conic provided q > 5.
For if this arc is a conic, then by the theorem
of Bezout, it cannot have more than four points in common with the original conic.
The question arises as to
whether or not there exist (q + 2)arcs which do not
8
contain a conic.
The following theorem provides the
answer. THEOREM 3.1.6
(B. Segre, 1957).
h
In PG(2, 2 ), there
exist (q + 2)arcs which do not contain q + 1 points forming a conic when h = 4 or h = 5 or h
~
7.
It is not known if the same property holds for h When h
= 6.
= 1, 2, or 3, every (q + 2)arc can be obtained
by adding the nucleus to the q + 1 points of a conic. We state now a theorem which shows how a geometrical property of an arc can determine the nature of the plane
in which the arc can be embedded: THEOREM 3.1.7
(F. Buekenhout, 1966).
If in n(q) there
is a (q + 1 )  ~ K such that every hexagon whose vertices are points of K is pascalian, then n(q) is pappian and K
is a conic. We wish to point out that the introduction of "segments" in the geometry over a field by B. Segre (1973) leads to results of deep interest in the study of conics. For more details on the study of karcs, the reader is referred to A. Barlotti (1965) and B. Segre (1961, 1965, 1967). 3.2.
Some properties of (k; n)arcs
The study of (k; n)arcs, for n > 2, is not as developed
9
as the study of karcs.
The "packing problem" has not
been solved, not even for the case of PG(2, q).
A. Basile
and P. Brutti (1971, 1973) and H.R. Halder (to appear) have obtained interesting results on this problem with the aim of proving the LunelliSce conjecture (see L. Lunelli and M. See, 1964). In the following, we present a brief list of topics with which (k; n)arcs are connected or in which they are of interest. (a) The use of (k; n)arcs in the construction of geometric structures. class of (p
3m
+ 1, p
m
R.C. Bose (1958) used a special + 1)arcs to derive BIBdesigns.
T.G. Ostrom (1962) used arcs to construct BolyaiLobachevsky planes and J.A. Thas (1973, 1974) used (k; n)arcs to construct partial geometries.
The connection between
arcs and certain (finite) Laguerremstructures is also of interest (see H.R. Halder and W. Heise, to appear). Structures in highdimensional finite spaces, corresponding to the karcs of the projective plane over the total matrix algebra of then x n matrices with elements in GF(q), were used by J.A. Thas (1973) to construct special classes of 4gonal configurations.
See also the
constructions due to J. Tits in P. Dembowski (1968). (b) The relationship between {k; n)arcs and other subsets of points of n{q).
A. Bruen and J.C. Fisher
{1973) found an interesting connection between complete
10
karcs and "blocking sets."
(A blocking set Sin n(q) is
a subset of the points of n(q) such that every line of n(q) contains at least one point in Sand at least one point not in S [see, for example, A. Bruen, 1971].) (c) To obtain information on planes, whose existence is not yet known, by studying (k; n)arcs in the planes (with the hypothesis that the planes exist).
As examples:
R. H. F. Denniston (1969) proved that no 6arc is complete in n(l0) and A. Barlotti (1971) obtained some (k; n)arcs in the plane n(l2) with a subplane of order 3. 3.3.
kcaps and the "packing problem"
A set of k distinct points of PG(r, q)
(r
of which are collinear, is called a kcap.
~
3), no three We shall denote
by m(r, q) the maximum number of points in PG(r, q) that belong to a kcap.
The value of m(r, q) is very important
for some applications in coding theory and in statistics. However, it is a very difficult problem to determine m(r, q)
("packing problem") in the general case.
The following are the known values of m(r, q)
(r
~
m(3, q) = q 2 + 1, m(r, 2) = 2r, m(4, 3)
20
m(S, 3) = 56
(G. Pellegrino, 1970), (R. Hill, to appear).
A list of the known bounds for m(r, q) is given in a paper by B. Segre (1967, p.166).
11
3):
For a wide illustration of further problems on caps, see the papers due to appear by G. Tallini and M. Tallini Scafati. REFERENCES Barlotti, A. 1965. Some topics in finite geometrical structures. Lecture Notes, Chapel Hill, N.C. 1971. Alcuni procedimenti per la costruzione di piani grafici non desarguesiani. Conferenze Sem. Mat. Bari 127: 117. 1973. Some classical and modern topics in finite geometrical structures. In: A Survey of Combinatorial Theory, edited by J.N. Srivastava et al., pp. 114. North Holland Co . 1974. Combinatorics of finite geometries. In: Combinatorics, edited by M. Hall Jr. and J.H. van Lint, Part I, pp. 5559. Mathematical Centre Tracts 55, Math. Centrum, Amsterdam. (to appear) Alcune questioni combinatorie nello studio delle strutture geometriche. In: Atti Convengno Teorie Combinatorie, Acc. Lincei, Rome 1973. Basile, A. 1970. Sugli insiemi di proprieta che definiscono un piano grafico finito. Le Matematiche 25: 8495. Basile, A. and P. Brutti. 1971. Alcuni risultati sui {q(n  1) + l; n}archi di un piano proiettivo finito . Ren. Sem. Mat. Univ. Padova 46: 107125. 1973. On the completeness of regular {q(n  1) + m; n}arcs in a finite projective plane. Geom. Dedicata 1: 340343. Bernasconi, C. (to appear) (a) Strutture di incidenza connesse e definizione assiomatica di piani grafici e affini. In: Ann. Univ. di Ferrara. (to appear) (b) Sistemi di assiomi che caratterizzano i disegni proiettivi. Biscarini, P. (to appear) Sets of axioms for finite inversive planes. Bose, R.C. 1947 . Mathematical theory of the symmetrical factorial design. Sankhya 8: 107166 . 1958. On the application of finite projective geometry for deriving a certain series of balanced Kirkman arrangements. The Golden Jubilee Commemoration Volume, Calcutta Math. Soc. 1973. On a representation of Hughes planes. In: Proc. Internat. Conf. on Projective Planes, edited by M.J. Kallaher and T.G. Ostrom, pp. 2757. Wash. State Univ. Press . 12
Bose, R.C . and K.R. Nair. 1941. On complete sets of Latin squares. Sankhya 5: 361382. Bruen, A. 1971. Blocking sets in finite projective planes. SIAM J. Appl. Math. 21: 380392. Bruen, A. and J.C. Fisher. 1972. Arcs and ovals in derivable planes. Math. z. 125: 122128. 1973. Blocking sets, karcs and nets of order ten . Advances in Math. 10: 317320. Brutti, P. 1969. Sistemi di assiomi che definiscono un piano affine di ordine n. Ann. dell'Univ. Ferrara, ser. VII, 14: 109118. Buekenhout, F. 1966. Plans projectifs a ovoides pascaliens. Arch. Math. 17: 8993. Crismale, M. (to appear) Sui sistemi minimi di assiomi atti a definire un piano proiettivo finito. de Bruijn, N.G. and Erdos. 1968. On a combinatorial problem. Indag. Math. 10: 421423. Dembowski, P. 1968. Finite Geometries. SpringerVerlag, BerlinHeidelbergNew York. Denes, J. and A.D. Keedwell. 1974. Latin Squares and Their Applications. Academic Press, New York and London. Denniston, R.H.F. 1969. Nonexistence of a certain projective plane . J. Austral. Math. Soc. 10: 214218. Drake, D.A. 1974. Near affine Hjelmslev planes. J. Comb . Theory 16: 3450 ... Halder, H.R. (to appear) Uber syrnrnetrische (k; n)Kurven in endlichen Ebenen . Halder, H. R. and W. Heise. (to appear) On the existence of finite Laguerremstructures and karcs in finite projective spaces. Hall, M. Jr. 1959. The Theory of Groups. Macmillan, New York. Hill, R. (to appear) On the largest size of cap in s 5 , 3 . Rend. Acc. Naz. Lincei. (to appear) Caps and groups. Hohler, P. 1970. Verallgemeinerung van orthogonalen lateinischen Quadraten auf hohere Dimensionen. Diss. 4522, Eidg. Technischen Hochschule Zurich. 1972. Eigenschaften van vollstandigen Systemen orthogonaler Lateinischer Quadrate, die bestirnrnte affine Ebenen reprasentieren. J. Geom. 2: 161174. Korchmaros, G. (to appear) Ovali nei piani di Hall di ordine dispari. Lunelli, L. and M. See. 1964. Considerazioni aritmetiche e risultati sperimentali sui {k; n}archi. Rend. Ist. Lombardo (A) 98: 352. Magari, R. 1964. Sui sistemi di assiomi "minimali" per una data teoria. Boll. UMI 19: 423435. Menichetti, G. (to appear) qarchi completi nei piani di Hall di ordine q = 2k.
13
Ostrom, T.G. 1962 . Ovals and finite BolyaiLobachevsky planes. Amer. Math. Monthly 69: 899901. Pabst, G. 1974. Private communication. (The result is in the author's master thesis.) See also "Quantitative axioms for affine planes" in Abstract of Communications ICM, Vancouver, 1974, p . 109. Pellegrino, G. 1970. Sul massimo ordine delle calotte in s 4 . Le Matematiche 25: 149157. ,q 1972. Procedimenti geometrici per la costruzione di alcune classi di calotte complete in S 3 . Boll . UMI (4) 5 : 109115. r, Pickert, G. 1955. Projektive Ebenen. SpringerVerlag, BerlinGottingenHeidelberg. Quist, B. 1952 . Some remarks concerning curves of the second degree in a finite plane . Ann. Acad. Sci. Fenn. 134 : 127. Reiman, I. 1963 . Su una proprieta dei piani grafici finiti. Rend. Acc. Naz. Lincei 35: 279281. 1968. Su una proprieta dei due disegni. Rend . Mat. e Appl. 7681. Segre, B. 1955. Ovals in a finite projective plane. Canad . J . Math . 7: 414416. 1957. Sui karchi nei piani finiti di caratteristica due. Rev . Math . Pures Appl . 2: 289300 1961. Lectures on Modern Geometry. Cremonese, Rome . 1965 . Istituzioni di geometria superiore, Vol. I, II, III, Lecture Notes . Istituto Matematico, Universita di Roma. 1967 . Introduction to Galois geometries. Memorie Lincei (8) 7: 133236. 1973. Proprieta elementari relative ai segmenti ed alle coniche sopra un campo qualisiasi ed una congettura di Seppa Ilkka peril caso dei campi di Galois. Ann . Mat . Pura e Appl . ser. IV, XCVI: 289337 . Tallini, G. (to appear) Graphic characterization of algebraic varieties in a Galois space . In: Atti Convegno Teorie Combinatorie, Acc. Lincei, Rome 1973. Tallini Scafati, M. (to appear) The ksets of type (m, n) in Galois spaces S (r ~ 2). In: Atti Convegno r,q Teorie Combinatorie, Acc. Lincei, Rome 1973 . Thas, J.A. 1970. Connection between thendimensional affine space A and the curve C, with equation n,q y = xq, of the affine plane A2 n· Rend. Trieste 2: 146151. ,q 1973 . 4gonal configurations. In: Finite Geometric Structures and Their Applications, C.I.M.E. II ciclo 1972, pp . 249263 . Cremonese, Rome.
14
1973. On 4gonal configurations. Geom. Dedicata 2: 317326. 1973. Construction of partial geometries. Simon Stevin 46: 9598. 1974. Construction of maximal arcs and partial geometries. Geom. Dedicata 3: 6164. (to appear) Some results concerning {(q + 1) (n  l); n}arcs and {(q + 1) (n  1) + l; n}arcs in finite projective planes of order q. (to appear) On 4gonal configurations with parameters r = q 2 + 1 and k = q + 1.
15
CONFIGURATIONAL PROPOSITIONS IN PROJECTIVE SPACES
Judita Cofman
A configurational proposition about points and lines of a projective space Eis any statement about points and lines in E of the following kind: (C)
For any choice of points P 1 choice of lines £ 1 equalities P. i
i
,
£2 ,
P., t J
s
,
••• ,
i
P2
,
••• ,
Pn of E and any
tm of E from certain in
it, certain incidences
P. I t , and negations of some incidences P. J t , i s i s where i, j E {l, 2, •.• , n'}, with n'
~
n, m'
~
s, t
E
{l, 2, •.. , m'}
m, i i j, s i t , certain incidences
of the same type follow. The points P 1
,
••• ,
Pn and the lines £ 1
,
••• ,
tm are called
the variables of (C); the points Pn+l' ••• , Pn' and the lines tm+l'
••• , tm' are the fixed elements of (C).
If
(C) has no fixed elements, we say that (C) is universally valid. Wellknown examples of configurational propositions are the proposition of Desargues and the proposition of Pappus.
Their importance in the theory of projective 16
spaces justifies the study of configurational propositions in general.
It would be desirable to establish hierarchies
between propositions of given classes of projective planes (see, for example, Amitsur [1]); however even special cases of this problem turn out to be extremely difficult.
On
the other hand there are other possibilities for the study of propositions by relating them to different properties of the corresponding spaces.
For instance, R. Baer [3]
has pointed out that (P, g)transitivity of a projective plane n is equivalent to the property that n is (P, g)desarguesian.
This result has a large number of important
consequences for projective planes (see, for instance, problems related to the classification of LenzBarlotti in [7], p. 126).
Or questions of the following kind can
be investigated:
given a configurational proposition (C)
in a restricted form in a projective space E, is it possible to remove some of the restrictions imposed on (C); in particular under which circumstances is (C) universally valid in E? Here I would like to consider some recent results on configurational propositions. be considered:
The following topics will
(1) configurational propositions and cor
relations in projective spaces; propositions in spreads;
(2) configurational pro
(3) configurational propositions
in derivable planes.
17
I.
CONFIGURATIONAL PROPOSITIONS AND CORRELATIONS
I.l.
Projective Planes
A correlation of a projective plane the point setP of
TI
is a 11 mapping of
TI
onto the line set2' of
the incidence relation in
TI.
preserving
TI
A point X (line x) is called
an absolute point (line) of a correlation a of only if it is incident with its image x 0
(x 0
)
TI
if and
under a.
A correlation of order 2 is called a polarity. Although polarities of projective planes have often been investigated, not much is known about correlations of arbitrary power. In this section we shall discuss results of Herzer
(9) on correlations with a large number of absolute points. Correlations with many absolute points were studied by Baer (3) in 1942 . plane
TI
a
Call a correlation
o of
a projective
(P, g)correlation if and only if there is a
nonincident pointline pair (P, g) in
TI
such that x 0 =
PX for any point XI g and £ 0 = £ n g for any line£ IP. It is easy to verify the following:
(a) the points of
g and the lines through Pare the only absolute elements of
o;
(b) for any point Y f P, Y
t
g and any line
h I PY n g, h f PY, g there is at most one (P, g) correlation of
TI
mapping Y onto h.
A plane
TI
is said to be
(P, g)homogeneous if and only if for a given nonincident pointline pair (P, g), any point Y f P, Y
18
t
g and any
line t I PY n g, t f PY, g there is a of
mapping Y onto t .
TI
(P, g)correlation
Baer has proved that a plane
is
TI
pappian if and only if there are two distinct lines m, g in M
such that
TI
#
is (M, g)homogeneous for any point MI m,
TI
m n g.
Consider now a correlation o of a projective plane
TI
with the property that the set of absolute points of~ contains the point set [g] of a line g of subset.
c
correlation TI
as a proper
Then the following is true:
PROPOSITION 1.1.
g of
TI
If the set
of
of
of absolute points of a
contains the point set [g] of a line
TI
as a proper subset, then there exists a unique
line i of
J'f=
such that
TI
[g] u [t].
PROOF STEP 1.
A
=
961
t
For, the image of any line i # g through A
lute lines.
is the point Ao
n
(t n g)
follows that to
=
i
lines i
1 ,
=
g and£~= i
t
1
n
i
g; all lines through A are abso
2
g.
n
c.
Since (t n g)o
n
i
n
g, it
There are at least two distinct
through A distinct from g . 2
=
g are distinct.
The points£~ Hence
At
g.
All
lines through A are absolute . STEP 2.
g 0 = B # A.
Otherwise
c
would be an (A, g)
correlation, contradicting the existence of a point in
/t
[g].
Clearly, B
STEP 3.
t
g.
No point of AB distinct from P
19
=
AB n g is
an absolute point of o. such that Y concurrent.
f
points A,
x,
t
Hence Y0
=
g, P 0
£
r/t 
[g), x 0 I A.
g,
/t" 
ferent from g.
Xe
I
x,
and (AX n g)o I AX n g are
Thus
Xe
= AX.
[g) = [£) for a line£ through P, dif
Since for any point X
£
,It [g) the line
x 0 is incident with A i t follows that Ao £
jt  [ g) .
Take any point z of £
It is easy to see that Zo point.
The
AX n g are three distinct points, hence
three concurrent lines.
X
AB, and Y0 are
Y.
For any point X
their images Ao
STEP 5.
=
The images A0
P.
STEP 4.
For, suppose Y is a point of AB
=
AZ.
=
Ao
1
I X for every
1 I
z
t
£ n g.
Thus z is an absolute
Since every point of£ is absolute and since
there are no absolute points on AB except P, it follows that g, £, and AB are concurrent. DEFINITION.
A correlation o is said to be a
correlation (i
=
(A . , g . )1.
l.
1, 2) if and only if the setv~ of abso
lute points of o consists of the points of gi and Ai 01 = gi for i = 1, 2. Our aim is to investigate projective planes admitting (Ai' gi)correlations. following questions:
(A., g.)correlations? l.
l.
We shall consider the
Which projective planes can admit How is it possible to characterize
pappian projective planes in terms of (A., g . )correla1.
tions?
20
l.
The following definitions are needed: DEFINITION.
An (A . , g . )configuration consists of two 1
1
distinct lines g 1 , g 2 intersecting in a point P and of two distinct points A 1 , A 2 incident with a line h through P
i P i A2 •
where A 1
DEFINITION.
For a given (A . , g.)configuration consider 1
t
any line 2
1
P intersecting gi in Ri' i = 1, 2, and any
point SI 2, S i t n h, R 1
i Si R 2 •
If for all possible
choices of 2 and S the points SA 1 n g
THEOREM 1.1.
A projective plane
SA 2 n g 2
1 ,
admits a
TI
,
and
(A., g . )1
1
correlation if and only if it is (A., g . )pappian. 1
1
PROOF (a) Assume that set of points in lines in mapping
TI
is (A., g.)pappian. 1
o of
ft into ~:
cr is a bijection of
J
'JS
onto
£
J
and the set of
let Y0 be the line
•
It is easy to verify that
i
Moreover, three collinear
are mapped onto concurrent lines of$·
if three points A, B, Care on a line t

the lines A0 (where Ri
Denote the
Consider the following
for any Y
joining YA 1 n g 1 to YA 2 n g 2
points of
JJ
which are not on h by
TI
not through P by~
TI
1
=

,
B0
,
t
P then each of
c 0 is incident with the point A 1 R 2
2 n gi,
i
=
1, 2) since
TI
Namely,
n A2 R1
is (Ai' gi)pappian.
If A, B, Care on a line t IP, consider the intersection
21


Q of A0 and B0
The point Q must belong to h; otherwise
•
the lines through Q would represent the images of the
cr,
points ofJ} on a line not through P under possible.
which is im
But if QI h, this implies that Q
and similarly Q
a is
Thus
tained from

= A0

c0
n
since Q
,
= A0

= A0

B0
n
h.
n
an isomorphism of the affine plane c1f,
ob
by deleting h together with its points onto
TT
the dual of the structure obtained from together with the lines through P.
by deleting P
TT
Obviously,
cr
can be
uniquely extended to an isomorphism o of the corresponding projective planes.
Thus o is a correlation of
If
TT.
T £ g. for i l.
= TAi+l
o is a
Thus TI T0
(i + 1 taken mod 2).
in other words
,
(A., g.)correlation. l.
l.
(b) Leto be a
(A., g.)correlation of l.
l.
From the
TT.
investigations in the proof of Proposition 1.1 it follows
1
that A.
l.
g. and that all lines through A. are absolute l.
l.
for i = 1, 2. set of P.
TT
Denote by h the line A 1 A 2
,
byJ the point
outside h, and by$ the line set of
TT
not through
For any point X £ f i t s image x 0 is the line through
XAi n g i, i
= 1, 2.
The image t O of any line t
point R 1 A 2 n R 2 A 1 where Ri
= t
n gi.
£
$
is the
The correlation o
is uniquely determined on the elements of
?
u
'ii.
As shown
in (a) this implies that o is uniquely determined on
TT.
The points of gi are mapped onto the lines through Ai+i· Thus gf
= Ai+l
for i
=
1, 2,
22
(i +1 taken mod 2).
Hence
=
(g 1 n g 1 ) 0
o
A 1 A 2 and (A 1 A 2 ) 0
interchanges hand P.
g1 n g2•
The points A 1
they form together with g Let i be any line of
=
and g 2 a
1
l.
that
TI
,
Pare collinear;
l.
not through P and let S be any point
TI
,Q,
l.
o maps
A2
,
(A . , g . )configuration.
of i distinct from i n hand from R . Since
In other words,
i = 1, 2.
n gi,
R 1 , S, R 2 onto concurrent lines, it follows
is (A., g.)pappian. l.
l.
This proves Theorem 1.1. REMARK 1.1 . 02 since A .
Ai+l
REMARK 1 . 2.
A (A., g . )correlation is uniquely determined
l.
A (A., g.)correlation l.
=
(i
l.
TI
the plane
1, 2, i + 1 taken mod 2).
l.
by Ai' gi, i = 1, 2 . tion in
cannot be a polarity,
l.
Thus for a given (Ai' gi)configuraTI
can admit at most one (A., g . )l.
l.
correlation. DEFINITION.
A projective plane
TI
is said to be (Al, gl,
g 2 )pappian if it is (Ai' gi)pappian for fixed elements A1
,
g
1 ,
g 2 and any point A 2 on h = A 1 P distinct from A 1
and P. We are now able to prove the following characterization of pappian planes: THEOREM 1 . 2 .
PROOF.
A projective plane is pappian if and only
We have only to show that (A 1
23
,
g1
,
g 2 )pappian
implies that
is pappian.
TI
This will be done in several
steps. STEP 1.
TI
is (A 1 , g 2 )transitive.
For, take two
distinct points R, Son h, different from P and A 1 • Denote by 6R and 6s the (A 1 , R, g 1 g 2 )correlations respectively. collineation of
TI
,
g 2 )  and (A 1 , S, g 1 , 1
Obviously, 6R6S
mapping R onto s.
1
Moreover 6R6S
every line through A 1 and every point on g 2 is a
is a
(A 1 , g 2 )homology mapping R onto S.
•
fixes 1
Thus 6R6S
By fixing Rand
varying S over the set of all points on h distinct from A 1 and P we obtain that STEP 2.
TI
is (A 1
,
TI
is (A 1
,
g 2 )transitive.
h)transitive.
Take a point R on
h distinct from P and Al and a line s I P, s I gl, g 2, h. 1
Denote by s the image of s under the inverse 6R (A 1 , R, g 1 , g 2 )correlation that 6 R
1
6R •
us 6 R is the (A 1 , R, g 1 J:
,
of the
It is easy to verify s)correlation of
TI.
By fixings and varying Rover the points of h distinct from A 1 and P, we get (A 1 , X, g 1 , s)correlations for all points XI h, X/ A 1 , P. plane
TI
Thus, in view of Step 1, the
is (A 1 , s)transitive.
The (A 1 , s)transitivity
and the (A 1 , g 1 )transitivity together imply that
is
TI
(A 1 , h)transitive. STEP 3.
TI
is a translation plane with respect to h.
Namely, the collineation 6 2 interchanges A 1 and A 2 implies that
TI
is also (A 2
transitivity and (A 2
,
,
h)transitive.
(A 1 ,
h)transitivity imply that
24
This
•
h)TI
is a
translation plane with respect to h. STEP 4.
is the dual of a translation plane with
TI
respect to P.
This follows from the fact that
o
inter
changes hand P. STEP 5.
In the ternary ring~ coordinatizing
with
TI
respect to the quadrangle O = (o, o), E = (1, 1), V = {00 ) ,
U = (o) by the method of Hall [8], with V = P, VO= g 1 , VE= g 2 , U =A, and A I

2
= (a) for any element a £ ~ 
{o}
the following relation is satisfied: (*)
(a  ax)z = az 
for all x, z £f/.
(az)x
For, consider the incident pointline pair Q =
(x, y), g = [az, o]
Then Q 0 = ( [ (x, y),
with y = (az)x.
(o) J n OV)
( [ (x, y),
(a) 1 n EV)
= [a  ax, y] and
g 0 = [((az, o]
n OV),
(a)]
[(az, o] n EV,
n
(o)]
= (z, az) (using the rules given in the proof of Theorem 1.2 (b)) . Since 0 6 I g 0 , it follows that az = (a  ax)z + y = (a  ax)z + (az)x, or (*)
(a  ax) z = az STEP 6.
Jl,
(az)x
for all x, z £
is a commutative field.
TI
Jr.. is a
translation plane with respect to hand the dual of a translation plane with respect to V; all a distributive quasifield. is (A, g )transitive and since I
2
25
thus~ is first of
Moreover, the plane TI
TI
is a translation plane
with respect to PA 1 , it follows that transitive.
is also (A 1 , g 1 )
TI
This implies that multiplication is associa
tive in
J1,
any a£
Jl {o}.
In view of Step 5, relation (*) is valid for Take a= l; this implies that xz = zx
for all x, z in.,l •
The ternary ring
Ji
is cornrnutati ve.
In other words, we have proved that REMARK 1.3.
TI
is pappian.
As we have shown in Step 1, if
g 2 )pappian then
TI
TI
is (A 1 , g 2 )desarguesian.
is (A 1 , g 1 , Using the
same arguments as in Step 1, one can prove that if pappian, then
TI
TI
is
is (A, g)desarguesian for all noninci
dent pointline pairs in TI.
This implies that TI is (X, g)
desarguesian for all pointline pairs in TI, i.e. TI is desarguesian.
Thus we have a new proof of the wellknown
statement: a pappian plane is desarguesian. Our next aim is to give a sufficient condition for the existence of a
(A . , g.)correlation in a projective 1
1
plane: THEOREM 1.3. plane V. (*)
TI
Let~ be the ternary rinq 1 of a projective
with respect to the points of reference: 0, E, U,
If TI is (V, UV)transitive and if the relation (a  ax)z = az 
(az)x
is satisfied for all x, z
£Ji,
correlation with A 1 = (o),
A2
then

TI
admits a
(A . , g. )1
1. Throughout the whole section the coordinatization method of Hall (8) is used. 26
1
= (a), g 1 = OV, g 2 = EV for
:,f  {o}.
a fixed element a£ PROOF . and by
Denote by
i:
'J
the set of the points (x, y) in
the set of all lines [m, kl in
mapping 6 on the set
Ju
ij
TI.
TI
Define the
as follows:
( x, y) 6 = [ a  ax, y l and
[az, kJ 6 = (z, az + k).
It remains as an exercise to verify that 6 is an isomorphism of the affine plane ~obtained from its points onto the affine plane
'11
TI
by deleting UV with
which is the dual of
the structure obtained from
TI
by deleting P together with
all lines through P.
6
can be extended uniquely to
a correlation
o of
TI.
Hence
It is easy to check that the points
of VO and VE are absolute points of (a) play the role of A 1 and A 2 • COROLLARY.
o;
the points A 1 and
This proves Theorem 1.3.
There exist nondesarguesian projective planes
admitting (A., g.)correlations. 
PROOF.
1
1
There are known examples of distributive quasi
fields with commutative multiplication which are not fields {see, for example, Dembowski [7), 5.3).
All of
these satisfy relation {*) with a= 1 and their corresponding projective planes are (V, UV)transitive .
Thus any
projective plane coordinatized by a distributive quasifield with commutative multiplication admits a correlation.
27
(A., g . )1
1
REMARK 1.4.
It would be interesting to know which classes
of the Lenz classification for projective planes can admit (A., g.)correlations. i
i
(For the definition of the Lenz
classification see Dembowski [7].)
I shall mention with
out proof the following result of Herzer ([9], Satz 8): THEOREM 1.4.
If a projective plane
TI
is (X, y)transitive
for at least one incidence pointline pair (X, y) and if TI
admits an (A., g.)correlation, then i
i

one of the following classes:

TI
must belong to
Lenz class II, Lenz class
III, or Lenz class V. REMARK 1.5.
It is not known whether there exist planes
of Lenz class III admitting (A., g.)correlations. i
i
Planes
of Lenz class II admitting (A., g . )correlations have been i
i
constructed by Herzer ([9], p. 251).
Distributive quasi
fields with commutative multiplication provide examples for planes of class V with (A., g.)correlations. i
I.2.
i
Desarguesian projective spaces
For an integer n
~
3 and a not necessarily commutative
field K denote thendimensional left vector space over by V(n,K).
Let the projective space isomorphic to the
lattice~ of the subspaces of V(n,K) be denoted by E. A correlation of Eis an antiautomorphism of$.
A sub
space A of Eis called totally isotropic, isotropic, or nonisotropic with respect to a correlation
28
of E if and
only if /1. n /1. 0 ively.
=
/1., or /1. n /1. 0 :f .
Obviously,.,fparallelism is an equi
valence relation in T;t·
Moreover, for any
Q
E
T.,f and any
point X s.;t there is exactly one line through X which is ,:,fparallel to a... We are now able to define two configurational propositions (D) and (P) for regular spreads. DEFINITION.
Let /f.
regular spread
!J7
9, ':c be
three distinct elements of a
and let A., B . , C., i = 1, 2, be points l.
l.
l.
of the projective space E such that Ai s.;f, Bi C.
l.
E
ff.
and
We say that A . , B . , C. satisfy the configurational l.
l.
l.
proposition (D) if and only if A 1 B 1
DEFINITION.
A 2B 2 and A 1 C 1
1I
.,t
A 2C 2
g:, if and only if (D) is true for any
three distinct elements ,,f, choice of Ai
Ed,
DEFINITION.
Let,A,
ff
1I
vt
The configurational proposition (D) is uni
versally valid in
spread
sjJ,
Bi
E
1J,
$,
g'.
E
Take
Let O and P be
of points of L not contained
nX
X ➔ ,Q,P
where ,Q,P is the
JI
parallel to OX through the point P and X the element of ff through
x.
It is easy to verify that
Jonte itself.
Our aim is to show that
extended to a collineation the following:
6OP
cOP
of L.
is a 11 map of
6OP
can be uniquely
For this we observe
(1) Points of jJ on a transversal of any
J)E 9'
are mapped onto points of a transversal in view of
(D) .
(2) Points of 'fi on a line h
mapped onto collinear points.
.S
E
.i E
fP are
This follows from the fact
that h can be embedded into a plane any element of !f'; in view of (1) 34
for any
TT
not contained in
C TT
OP is a plane.
(3)
Co
planar points of I can be embedded into threedimensional spaces not contained in any element of9'; hence coplanar points of I are mapped onto coplanar points.
Thus COP
can be extended to a map c OP of the point setP of I into itself in such a way that c Op is a collineation of L cOP fixes every element of J' .
Clearly,
Denote by b. the
set of all collineations cOX for a fixed point O £~,t and all points X
£;f.
The set b. is a group acting as a
transitive permutation group on the point set of every
jj £!/'. Conversely, if b. is a collineation group of I with the required properties, then for any :iJ transversal g of ~and any
d£
b.,
the points ofJ . LEMMA 2. 2.
Let
gc
11
,l
g.
The group b. is transitive on
Hence (D) is universally valid in J'.
fl, J be two distinct elements of the
regular spread~ and let g be a line intersecting both
A
Define the map og on the point set of .,f;and ,I as C og .R, follows: X = X,g n j for any X £vi and X g = .R, X,g nv"f;
and,B.
for any X through X.
£
2.
Here .R.X ,g denotes the Ifparallel tog
Proposition (P) is universally valid in~ if
and only if for any two distinct elements v4 , 13 of 9' and any three lines a , b , c intersecting A and J3 the following relation is true:
(*) PROOF.
Suppose that (P) holds. 35
If any two of the lines
a, b, c coincide, then (*) is satisfied, since 6 2 = 6 2 = a b 0 C2 = 1.
Otherwise take any point X
£
rJt
and consider the
6aobocoa oaob configuration Al = X, A 2 = X , A3 = X , Bl 0 o obo c o a ob oaoboc X , B 2 = X a, and B3 = X a Because of (P) 6 (6a6b6c) 2 B C = X i t follows that X Thus Al 3 (oaoboc)
= 1.
2
Conversely, by assuming that (*) holds for the above configuration A 1 , A 2 A 1 B 3 I IA 3B 1 , i.e.
,
B2
,
B 3 , we deduce that
(P) is satisfied.
universally valid.
Let.,t,
g a fixed line meeting
Ji}
,
Let y> be a regular spread in which (P) is
LEMMA 2 . 3.
and
A 3 , B1
/t
JJ
and
be two distinct elements of[?,
1J
and h any line of L meeting
Then the set 6 g = {o g • oh induces on the projective space
for all h meeting
tit
A
an abelian col
lineation group transitive on the points of,4. PROOF.
Leth and h' be two arbitrary lines intersecting
d and iJ • ~
= (oh,)
1
According to Lemma 2. 2, oh 6g oh, = ( oh 6g 6h, ) 1
1
og oh= 6h,6goh.
Hence (ogoh) (ogoh,) =
6g(6h6goh,) = og(oh,ogoh) = (ogoh,) (ogoh). 6
g
1
Therefore
induces a commutative permutation group on the point
set of#. Let X and Y be two arbitrary points of.It. The 0 points X g and Y determine a unique line h such that Y o oh = X g hence 6g is transitive on the points ofc/t. From the definition of og and oh it follows that 6g induces a collineation group onvf, which proves the lemma. 36
II.3. Main results We are now able to prove the following theorems: THEOREM 2.1.
A regular spread 9' is desarguesian if and
only i f (D) is universally valid in ~. THEOREM 2. 2.
y
A spread g, is pappian if and only i f
is
regular and (P) is universally valid in!J'. THEOREM 2.3. Then
J'
Let
J' be a spread over a field K with I.Kl> 2.
is a Moufanq spread which is not desarguesian if and
only if .!I' is regular and (D) is not universal in g,. For the proofs of these results some wellknown properties of spreads are needed (see Andre (2) and Bruck and Bose (4),
[SJ):
PROPERTY 1 (BruckBose (5), p. 155).
E.
Let j?be a spread of
Denote by V the vector space over){ from which Eis
obtained .
For any element
ponding subspace of V.
X
£
ff' denote by
X:
the corres
Take an ordered triple;!,
:B,
~ of
distinct elements of!/; there is a unique linear trans+
+

formation a+ a' of ,A, into formation ; + ; + ; ' maps element
X of
1

;;t,
such that the linear trans
into
1/.
The vectors of any
the space corresponding to
expressed in the form ;
Y'£ +
a'
where ;
;r £ Y'  .Jt £
can be
,;,f, ; ' is the
corresponding vector of 1j defined above, and
$Pf is a
unique linear transformation of ~ into itself determined
37
f'.
by Ci:
In particular~= 0 and
:B
of the linear transformations
the properties:
b £ cit and
9'
X
E
Ci:
;
'f _ 
r
'f _
JI
rg, = Cf_ X
I.
The collection
for all X
E
Ef' 
vt has
'J:
i f a,

is nonsingular i f X:/
+
:/ ; , then there exists exactly one element
ar_X
+
such that
+b.
PROPERTY 2 (BruckBose (5], p. 158) .
Let E' be a pro
jective space containing E as a hyperplane and let TT be the translation plane obtained from E', E, and Y as described in II.l.
Then there is a ternary ring of TT iso
morphic to the algebraic structure~(+, ·) defined as follows:
( 1) the elements of _$( +, •) are the vectors of +
(2) "+" is the vector addition; nonzero vector of +
,j;
(3) let e be a fixed
then for any two vectors:,
+
the product a · bis the vector +
a 7+ b +
transformation of Ci: mapping e onto b. the role of the identity.
where
f+
b
b Ed
is the +
Clearly e plays
,I(+, ·) is a quasifield.
PROPERTY 3 (BruckBose [5], p. 159).
If Ci: is closed under
multiplication, then (€, •) is a group with zero isomorphic to
/t( · ) .
PROPERTY 4 (Herzer (11]). ~+, •).
Xis contained in the kernel of
Moreover, J< is contained in the centre of.if(+,·)
if and only if ff is a regular spread.
38
PROPERTY 5 {Andre [2)).
Let
Q
zing a translation plane
TT;
then the multiplicative group
of the kernel of homologies of
TT
Q
be a quasifield coordinati
is isomorphic to the group of all
with a given affine centre and the im
proper line as axis. We shall now indicate the proofs of Theorem 2.1, 2.2, and 2.3. PROOF OF THEOREM 2.1.
Suppose that {D) is universally
valid in a regular spread.9'.
Then, according to Lemma 2.1
the space I admits the collineation group 6.
I
+
+
~
+ + = < {x, y)>c.
6* imply the existence
+
+
➔
+
of a homology mapping onto . Hence
➔
Let
{Vis represented in the form ;it$;/).
TT
is {R, t 00 )transitive, i.e. 39
TT
is desarguesian
(see Pickert [14)). Conversely, if n is desarguesian, then n is (R, i 00 ) Hence I admits a collineation group 6 fixing
transitive.
all elements of !.f and acting transitively on the points of each element in
J'
(because of 6 "X*IJof a
42
derivable projective plane
TI
is an affine hyperplane of a
threedimensional affine space 'T contained in TI 0 TI
is desarguesian.
fixing
TI 0
Hence
TI.
Moreover any Baer collineation of This
pointwise induces a collineation ofr.
gives the following information about the group of all Baer collineations of
TI
fixing
TI 0
IT(TI 0 )
pointwise:
IT(TI 0 )
is a subgroup of the group of all perspectivities of
with axis
TI 0
contained in
'T
and centres on an improper line of .7'not TI 0 •
Our investigations are carried out on the dual planes of derivable planes; the Baer collineations and Baer subplanes belonging to a "dual derivation set" in these planes have the abovementioned properties of III.2.
IT(TI 0 ) .
Main results
Throughout this section plane,
and
TI 0
r:J
TI
denotes a derivable projective
a derivation set in
TI,
and
TI
0
a Baer subplane
belonging toot>. Let us denote by
TI
the plane dual to
of/) i n ; is a set.;c) of lines in
TI
TI.
The dual
through a point L
with the property that for any two distinct lines a, b of
A
TI
not through L for which the intersection of a and b A
belongs to a line ofoiJ there is a Baer subplane ing a, b, and L such that the lines of A
exactly the lines ofdl>. mined by a, b, andJ>.
Obviously
;o
A
TI 0
A
TI 0
contain
through Lare
is uniquely deter
We say that a Baer subplane of; be
43
longs to
A
,;I)
if and only if it contains the lines of
us denote by~ the class of all Baer subplanes of
A
al). TI
Let
belong
A
ing to~.
Take a line g
tobtained from
TI
and consider the affine plane
E ~
by deleting g together with its points.
Since any Baer subplane
A
TI 0
of
A
TI
belonging to ,;3 contains
g, the affine points and lines of tJ!. contained in an affine subplane d{ 0 of t7l obtained from g together with its points in
TI 0 •
A
TI O
A
TI 0
form
by deleting
Denote by~ the class
of all affine Baer subplanes of t:l;obtained in the above way from the subplanes of.$ • The affine points of any plane of $'are points on the A
lines of  {g}.
Let D be an affine point
of Cl on a line k of :f>  {g}, different from h. by Pt, 1 the plane of~ through A, B, D.
Denote
Take an arbitrary
point X on g which is not an improper point of (l 1 • ~ 1
is a Baer subplane of rJ{,, the correspondence
9':
Since Y
XY n h where Y runs over all affine points of (l 1 is a
46
+
oneone mapping of the affine points of t( 1 onto the affine points of h.
In order to verify that is an affine
plane whose lines are blocks of .7'it remains to show that the blocks of Yon the line hare the images of the affine lines of
a
1
points of h.
r•
under
Let M, N be two arbitrary distinct
Denote by Uthe unique point of XM in
by V the unique point of XN in ot,1 •
a
1 ,
and
If U =Mand V = N
then clearly MN is the image of the affine line UV of cl1 under
f·
Suppose U
~
M.
Then U, V, Mis a triangle, and
according to (1) there is a unique plane t1[ 2 of it.
Let
A
7T 2
u,
V the plane
A
7T
2
contains the points X
and L, thus it contains the point N the line UV of ;
r•
tl1
containing
be the plane of jj from which tJ[ 2 was obtained.
Together with M,
7T 2 •
~
XV n ML.
Moreover
is projected from X onto the line MN of
2
Thus the block MN is the image of the block UV under Conversely, by the same argument every affine line of is mapped under
f
onto a block of ✓ on h.
This finishes
the proof of (3).
:r contains
two types of planes:
planes of type 1B
which are the Baer subplanes o f f and planes of type L which are the lines structure
J
J>  {g}.
According to Lenz [ 12] the
is an affine space if the following properties
are satisfied:
(i) between the blocks an equivalence
relation called parallelism is defined;
(ii)
for any
pointblock pair (P, b) there exists a unique block b'
47
incident with P and parallel to b;
(iii) for any four
distinct points, A, B, C, D such that AB is parallel to CD and for any point P € AC, either P s CD or AB and PD have a point in common.
We shall define parallelism as
follows: Two distinct blocks of :rare parallel if and only if
(IP)
they are disjoint sets of points belonging to a common plane of 7'.
Each block is parallel to itself.
In view of (3) all planes of ✓ are affine planes; properties (ii) and (iii)are satisfied .
hence
It remains to prove
that parallelism (~) is an equivalence relation.
Clearly,
(~) is reflexive and symmetric; transitivity has to verified.
Thus we have to show that: If b
(4)
1 ,
b2
,
and b 3 are three blocks of
.r
such that b 1
is parallel to b 2 and b 2 is parallel to b 3 , then b 1 is parallel to b 3 • For the proof of (4) the following cases can be distinguished : CASE 1 . of
Y.
b1, b2
,
b 3 are contained in a common plane
Since the planes of :T'are affine planes,
(4) follows
immediately . CASE 2.
m,
b 1 and b 2 are contained in a plane of type
and b 2 and b 3 are contained in a plane of type L.
note the plane through b
1
and b 2 by
48
Cl1 and the plane
De
through b 2 and b points of ~
2 •
/12 •
by
3
Let 2 be the line carrying the
Take two arbitrary points X, Y of b 1
and an arbitrary point Z of b 3 •
In view of (1) the points
X, Y, Z determine a unique subplane ~ 3 off.
(l 3
The plane
intersects 2 in a block b; through Z having no point In the affine plane 9;2 there is exact
in common with b 2 •
ly one block through Z containing no point of b 2 the block b 3 •
Thus b
3
,
namely
and b; coincide, that is b 1 and
b 3 are parallel lines of
d/ 3 •
Hence b 1 and b 3 are parallel
in./. CASE 3.
b 1 and b 2 are contained in a plane tt' 1 of
type Band b 2 and
b
3
also belong to a plane carrying b 1 respectively. subplane from
,
b2
,
b 3 by 2 1
,
22
,
23
Since any line of~ contains lines of any
r,
there is a line b, of
Pll on
of (1) the blocks b 3 and b 4 are parallel in b, is parallel to b 1 •
23.
In view
J; obviously,
From the investigations in Case 2
it follows that b 1 and b 3 are parallel blocks of.J': This proves (4). Thus (~) is an equivalence relation, which implies: (5)
:r
is an affine space; the affine lines of :J are the
blocks of j'; the affine planes of .J"' are Baer subplanes of
i
and the affine line of 1ft, belonging to£.
Consider an affine plane
£
{g}.
,Z1 of j represented by a line of
Take a point P of .7'in
49
tt 1
and two distinct blocks
b1
,
b 2 in {l{_ 1 through P 1 •
For any other affine plane,~\
of ✓ represented by a line of ~  { g} and for any point P 2 of~ there exist unique blocks b 3 and b 4 through P 2 parallel to b 1 and b 2 respectively. and ,!£ 2 are parallel in J.
Hence the planes
£
1
On the other hand any affine
plane of :T represented by a Baer subplane of ~: in a block, that is in an affine line of
g, intersects
r.
This
implies that (6)
J'"'is a threedimensional affine space. We have completed the proof of the following theorem:
THEOREM 3.1.
Let
jective plane
TI
.J) in
TI.
g
TI
be the dual plane of a derivable pro
and let cl)" be the dual of a derivation set
O[ is the affine plane obtained from
A
TI
by delet
ing a line g s ~ together with its points, then the affine points of~ on the lines of sional affine spacer.
io 
{g} form a threedimen
The affine lines of Jare the
affine lines of those Baer subplanes C(i of dl, which are obtained from the Baer subplanes of deleting g with its points. affine plane of
A
TI
A
belonging to ;lJ after
Any such subplane fZ. is an l

::r'.
As a corollary we obtain the following statement: THEOREM 3.2.
If
jective plane
TI
TI
is the dual plane of a derivable pro
and if
ftJ
is the dual of a derivation set
50
J) of 71 in 71, then any Baer subplane of 71 belonging to ,:J:; A
is desarguesian. Our next aim is to consider Baer collineations of 71 . A Baer collineation of a projective plane is a collineation fixing pointwise a Baer subplane of the plane.
Clearly,
the Baer collineations fixing pointwise a given Baer subplane form a group . Let ; , 3.1.
£,
g,
ct,
J"', and O[ . be defined as in Theorem l.
Denote by IT(Q . ) the group of the Baer collineations l
A
fixing pointwise a Baer subplane ~  belonging to ;!J. l.
Any
element a of IT (i . ) fixes the lines of (Jl . ; hence a fixes l
A
all lines ofo!J. in
J.
~
It follows that a induces a collineation
Since {l . is a hyperplane of l.
:T the collineation
a
induced by a in ✓ is a perspectivity of ✓with axis ~i . A
The collineation a fixes all lines of~; of
a must
hence the centre
lie on the improper line of ✓ common to all A
affine planes&'. of 7represented by the lines ofci) {g}. l.
In other words we have the following result: THEOREM 3.3. Let 71 be the dual plane of a derivable proA
jective plane 71 and let 71 0 be a Baer subplane of 71 belonging to the dual of a derivation set of 71 contained in 71. If IT(71 0 ) denotes the group of all Baer collineations of 71 fixing
;o
pointwise, then IT(; 0 ) is isomorphic to a sub
group of the group rx,$'('of all perspectivities in a threedimensional projective space ':f with a common axis
51
~
isomorphic to contained in
TI 0
and with centres on a line i of 9' not
£.
The investigations carried out for the duals derivable planes
TI
can be dualized step by step.
TI
of
If G
denotes an arbitrary but fixed point in a derivation set d)of
TI,
of
through the points of ;JJ  { G} which are different
TI
then it is not difficult to show that the lines
from the line containing ol)are the points of a threedimensional affine space
:r. A
By completing j t o the corresponding projective spacer it follows that the Baer subplanes of to ,:,l) are hyperplanes of r.
TI
belonging
Thus the statements of Theo
rems 3 . 2 and 3 . 3 can be formulated for derivable planes: THEOREM 3.4.
Any Baer subplane belonging to a derivation
set of a derivable projective plane is desarguesian. THEOREM 3.5.
Let IT(TI 0 ) be the group of the Baer col
lineations of a derivable projective plane
TI
fixing point
wise a Baer subplane TI 0 which belongs to a derivation set of TI. group
Then IT(TI 0 ) is isomorphic to a subgroup of the
rx,~
of the perspectivities in a threedimensional
projective space L with a common axis:J( isomorphic to
TI 0
and with centres X on a line i of L not contained in~.
52
REFERENCES
1. 2.
3. 4. 5. 6.
7. 8.
9. 10.
11. 12. 13. 14. 15. 16.
s.A. Amitsur . Rational identities and applications to algebra and geometry. J. Algebra 3 (1966): 304359. .. . J. Andre. Uber nichtdesarguessche Ebenen rnit transitiver Translationsgruppe. Math. Z. 60 (1954): 156186. R. Baer. Homogeneity of projective planes. Amer. J. Math. 64 (1942): 137152. R.H. Bruck and R.C. Bose. The construction of translation planes from projective spaces. J. Algebra 1 (1964): 85102.   Linear representations of projective planes in projective spaces. J. Algebra 4 (1966): 117172. J. Cofrnan. Baer subplanes of affine and projective planes. Math. Z. 126 (1972): 339344. P. Dembowski. Finite Geometries. New York: Springer (1968) . M. Hall. Projective planes. Trans. Arner. Math. Soc. 54 (1943): 229277. A. Herzer. Dualitaten rnit zwei Geraden aus absoluten Punkten in projektiven Ebenen. Math. Z. 129 (1972): 235257. Ableitung Zweier selbstdualer, zurn Satz von Pappas aquivalenter SchlieSungssatze aus der Konstruktion von Dualitaten. Geom. Dedicata 2 (1973): 283310. Charakterisierung verschiedener regularer Faserungen durch Schliesungssatze. To appear in Archiv der Mathematik. H. Lenz. Zur Begrilndung der analytischen Geornetrie. SitzBer. Bayer. Akad. Wiss. (1954): 1772. Vorlesungen ilber projektive Geornetrie. Akad. Verl. Ges. Geest und Portig, Leipzig (1965). T.G. Ostrom. Semitranslation planes. Trans. Amer. Math . Soc. 111 (1964): 118. G. Pickert. Projektive Ebenen. BerlinGottingenHeidelberg: Springer (1955). o. Prohaska. Endliche ableitbare affine Ebenen. Geom. Dedicata 1 (1972): 617.
53
UNITARY PLANES Martin Gotzky
In the following the intention is to develop unitary geometry of the plane, with the help of quasisymmetries. At the heart of the considerations is the theorem of the third quasireflection (see Gotzky, 1973).
Part I
deals with unitaryeuclidean geometry, and part II with unitaryminkowskian geometry (see Gotzky, 1970). I.
GENERALIZED EUCLIDEAN GEOMETRY
Let P be a projective plane and Sa set of perspective collineations
i 1.
If a is a straight line of P, let
S(a) be the set of elements of S with axis a .
Assume
the following property: (*l)
{l} u S(a) is a group for every straight line a.
Obviously {S(a); a is a line of P} is a partition of s. Let G
= be the group generated bys.
We are
interested in generated groups (G, S) with (*l), for which the following statement holds:
54
THEOREM OF THE THIRD QUASIREFLECTION.
Let a, b, c be
straight lines through a given point O with a f b. a€ S(a), S € S(b), and caS f c.
Let
Then there exists a
y € S(c) and a straight lined passing through Osuch that aSy € S(d). Fixing a straight line g of P we may consider the affine plane
A= Pg instead of P.
If S 0 is the set of
linereflections of A and A a pappian plane with Fano axiom, then Veblen (Veblen and Young, 1918, vol. 2, §52) has already stated this theorem for (, S 0 ) .
Only
(*1) does not hold. There are many examples of generated groups (G, S) with (*l) for which the theorem of the third quasireflection holds.
For instance let S 2 be the set of all axial
collineations of with S 1 the set of all shears and linereflections of an affine Moufang plane Fano axiom.
A=
P with g
Then (G, S . ) for i = 1, 2 is a generated 1
group with (*1) holding.
The theorem of the third
quasireflection holds for
(G,
arguesian and for
S) exactly
i f A is des
(G, s 1) exactly i f A is pappian
(Gotzky, 1972). For the case (G, s 2) '
let a € S (a),
s
€ S (b), and
y € S(c) with a, b, c and O, P, Q as shown in figure 1. Further let Q
= QaBy.
Then P
PaSy for every point P
on the connecting line P of O and Q if and only if aSy € S.
55
FIGURE 1
This shows that quasireflection is closely related to Desargues' theorem. Now let V = Vn{K, f, 1) be an ndimensional left vector space where K is a skew field of char. f 2, 1 is an antiautomorphism of K with 1 2
= 1, and f is an 1hermitian
form, that is a function mapping each pair a, b of vectors onto an element of Kand satisfying f{sa, b)
=
s f{a, b)
f{a' + a." , b) = f
(a
It
if
s EK,
b) + f {a", b),
f(a, b) 1 = f{b, a).
(If 1
1, then K is commutative and f a bilinear form,
hence V is a metric (orthogonal) vector space.) The set of all linear mappings = f(a,
7T
satisfying f{a
b) is a group called the unitary group U{V).
7T
I
b n)
The
subset of all linear mappings n of U(V) fixing each element of rad V := {x EV; f{x, V) = O} is also a group called the restricted unitary group U*{V).
56
Let a I rad V ands EK\ {O}, then ( 1)
xcr ( s, a)
f ( x, a) • s • a
:= x 
defines a linear map cr(s, a) fixing the hyperplane {y;
f(y, a)
= O} vectorwise.
Also, cr(s, a) E U*(V) if
and only if
(2)
S
I
l
+
S
I
Let S(Ka) S (V)
=
f (a, a). := {cr(s, a); s 1 1 + s 1 = f(a, a)} and S
u
a/rad V
(Ka) .
Obviously S(Ka) u {l} is a group. THEOREM.
Consider the generated restricted unitary group
(U*(V), S(V)).
Then, for the onedimensional subspaces
of V not contained in rad V, we have:
if Ka~ Kb and
Kc~ Ka+ Kb and also a E S(Ka), BE S(Kb), and (Kc)aS Kc, then there exist a y E S(Kc) and a subspace Kd
+ Kb with a Sy E S (Kd)
~
~
Ka
(Gotzky, 1964, 1965) .
Let n = 3 and P = PV be the desarguesian projective plane with onedimensional subspaces of Vas straight lines, twodimensional subspaces as points, and incidence defined by inclusion.
Then, for P,
(PU*(V), PS(V)) is
a generated group satisfying (*1) and by the theorem also satisfying the theorem of the third reflection. REMARK.
If S contains only reflections, the theorem of
the third quasireflection for (, S) becomes the theorem of the three reflections (see Bachmann, 1959 or 1973).
57
P = PV is called a unitary plane. f(x, y)
=
L
Ky if
0, an orthogonality relation for these planes
is defined. plane.
By Kx
If dim rad V = 1, Prad Vis a unitary affine
In this case we say
Prad Vis unitaryeuclidean if index f =
o,
Prad Vis unitaryminkowskian if index f = 1. REMARK.
If index f = 0 for (PU*(V), PS(V)), the theorem
of the third quasireflection holds even if we delete the assumption ca8
1
c; we say the stronger theorem of the
third reflection holds.
Also the theorem of the third
quasireflection keeps holding if ca8; c is replaced by
ct
c.
In this case we say that the varied theorem of
the third qusireflection holds. Next we concentrate on affine translation planes
A= P, with a point involution jong which defines an g affine orthogonality relation by a Lb if the intersecting points of a with g and b with g correspond under j (see figure 2).
FIGURE 2
58
To avoid distinction between cases we restrict ourselves to cases with j free of fixed points.
Thus there
are no selforthogonal straight lines. Assume a system S with (*1) such that: (*2)
S(a) contains an involution for all straight lines a. Obviously then there is a
unique reflection on every
straight line a of Pg and Char Pg~ 2; that is, the Fano axiom holds. We assume that the stronger theorem of the third quasireflection holds for
(G, S) with G = and assume
furthermore that (*3)
S contains all perspective collineations of G.
(*4)
G preserves the orthogonality induced by j. The question is whether A is unitary or not (if
unitary it is unitaryeuclidean).
For a better under
standing of this question, I shall prove some little statements.
(For the following discussions if
of the same group, we define
sY
:=
S, y are elements
y 1 Sy.)
LEMMA 1.
The point reflections are in G.
LEMMA 2.
Suppose a
E S(a), SE S(b), and a II b.
Then
aS
is a translation if and only if there is a point reflection C such that
aS C = 1.
59
PROOF.
a
 1
a
e = e a e is a translation.
Let aS be a transThen
lation and ea point reflection with be= a. aSe· (S 1 )e = aS where (S lation. LEMMA 3.
1
e
) Sand aS are translations, so aS
e e Moreover, as E S(a), so as =
e
is a trans
1.
Let a E S(a), s E S(b) with ola ~ s1O.
Then
there is a o E S(d) for all d such that oaS is a glidequasireflection. REMARK.
If a ES u {l} then a is called a quasireflection.
If Tis a translation and a ES
u
{l}, then Ta is called
a glidequasireflection. PROOF.
Let e be a point reflection such that de IO. C
There is a o E S(d) such that y := o aS Es.
If y E S(c)
then c IO (theorem of the third quasireflection).
Now
oaS = o(o 1 )cy is a glidequasireflection. DEFINITION 4. points XY, then
If TIE G and MXY is the midpoint of the HTI:
X ➔
MXY defines a point map called
the midpoint map of TI. If a midpoint map H preserves collinearity, TI then it is a collineation or else there is a line passing
LEMMA 5.
through all the points of image Hn. PROOF.
See Gotzky (1972).
60
LEMMA 6.
Ha preserves collinearity for all a Es.
From now on, we shall not distinguish between a point and its point reflection. PROOF OF LEMMA 6. C
f a.
Let a E S (a), 0 I a, and O I
C
I p but
We show that, regardless of how P was chosen,
there is a line m passing through 0 such that P 8 a is on m. Consider the identity O•POPa = (PPaO) 0 = PaoP.
Let
b La and b IO such that O is the product of the line reflections in a and b. and SE S(b).
Then aO
=
a'S for some a' E S(a)
By the theorem of the third quasireflection,
there is a y E S(c) such that y := ya'S ES. 3.)
Obviously Pao= P 0 , so O•POPa
= P 0 P.
(See figure
Leto E S(d).
Then dis a line through O perpendicular to the connecting line of O and POPa.
Since d does not depend on the
choice of Pon c, the connecting line also does not.
FIGURE 3
61
If
this connecting line is called m, the lemma is established. If w E G and Tis a translation, then H WT
LEMMA 7. PROOF. p
=
H H . WT
Using 1
H
MPPWT
p T
MppwH T
The uniqueness of the midpoint yields H ) T = (M M PPWT PPw 11
p
11
HwT
H H (P W) T
H H p WT
This proves the lemma. LEMMA 8.
If Tis a translation and if w ES, then HTw
and H preserve collinearity. WT PROOF .
Since S contains all the axial transformations by
(*3), it follows that TWT H
TWT
1
•T
1
ES.
Furthermore, we have
Hence we only have to prove that HwT
preserves collinearity.
But H WT
preserve collinearity. LEMMA 9. PROOF.
If a, SES then HaS preserves collinearity. Let a E S (a) and S E S (b).
If a
11
b then
aS ES u {T; T translation}, hence aS preserves collinearity.
Thus we may assume that a f band a, b IO for some point 0.
62
There is a Y
Let c be a line.
is a glidereflection (Lemma 3).
H H P aS = P yaB for all P I c.
€
S(c) such that yaS
Now Pas= PyaS and hence
Since HyaB preserves collinearity, then HaB also has to preserve collinearity. LEMMA 10.
Let O, P, Q be noncollinear points.
exists an axial collineation
o
o0
such that
=
o
There and P 0 = Q.
Let O, P, Q be noncollinear and a, b, c, R be as
PROOF.
shown in figure 4.
FIGURE 4 (1)
Assume ls(x)
Choose a'
E
I
> 1 for all lines x.
S(a), B"
E
S(b), and y
E
S(c) such that
y is an involution but a' and B' are not. y 1 Since a' := y a•y is a noninvolution, Lemmas 5 and 6 together yield that H Y is a collineation with axis a' aY. It is easy to show that Lemmas 5 and 9 together yield that H,
a Y
is a collination (use the fact that
63
at
C).
a'y Now X all XI c.
= Xa'Y
Ha•y for all XI c, so X
Therefore y•
:= H, H 1 Y fixes every point of
a ya' c; that is, y' is axial with axis c
s (c)).
=
(it might not be in
Similarly, y" := Ha" H 1 y is an axial collineation µ y B"
= HY with axis C. Because H a' a'Y we obtain
= y,
p(y'y)
Ha'
•y
P I a HS"
R = Q
y
=
Q
y II y H B"y
Q
Q I
(y"y)
Ha" µ
I
·y
b
Using RI c, we are led to
= Since (y'y)
Ha,
Q
(y"y)
Ha" µ
and (y"y)
Ha" µ
are obviously axial and
both have the same direction PQ, the product o := (y'y)Ha'[(y"y/B"rl is also axial.
This
o
satisfies o 0
=
O and p 0
Q, as
required. (2)
Assume IS (all
a~ b ~ c ~ a.
=
1 for one a I
o.
Let b, c I O,
Suppose, if possible, that IS(c) I > 1 and
choose a e S(a), Be S(b), and distinct y, y" e S(c). The theorem of the third quasireflection yields that o
=
yBa and o'
=
y'Ba are both in S.
to the sequence of inferences
o'o 1
But this would lead =
o•o 1
e S(c) u {l};
y, y', o, o' e S(c); hence Bae S(c), which is impossible.
64
So the assumption ls(c) I > 1 was incorrect . that IS(c) I= 1 for all lines c.
This proves
But in this case the
theorem of the third quasireflection yields the theorem Hence A is pappian (see Bach
of the three reflections.
mann, 1959 or 1973) and Lemma 10 holds. THEOREM. PROOF.
Pis a Moufang plane. It is sufficient to show that for every pair
a.++ b in A = P there is a line reflection with axis a in g the direction of b (Pickert, 1955).
Since there are line
reflections in A it is therefore sufficient to show that
Au~(A) is transitive on the nonparallel pairs of lines of A. Thus we want to map a
tf
b onto c }t d.
Since it is
a translation plane, we may assume that a, b, c, d IO for a point 0.
Using Lemma 10 we furthermore may assume
that a= c.
Fix e, P,
Q as in figure 5
there is an m and aµ
E
(c
II
d).
By Lemma 10,
S(m) such that Oµ = a and Pµ = Q;
hence aµ= a= c and bµ = d.
FIGURE 5
65
This proves the theorem.
Whether the plane A is unitary under the given conditions now depends on the existence of both a nondesarguesian affine Moufang plane
A=
P and a given pointg
involution jong that is free of fixed points, which together admit a generated collineation group (G, S) satisfying (*1) to (*4) and the theorem of the third quasireflection.
It is still unknown whether such planes
exist. In the next section, unitaryminkowskian planes
A=
P are investigated. g
Using an additional condition
denoted there as A.6, a full characterization of those planes is given. II. II.l.
UNITARYMINKOWSKIAN PLANES Axioms
The purpose of this section is to introduce the group plane of a given generated group (G, S) with S the set of generators of G, and further to present a set of axioms for the elements of s which ensures that the group plane has certain desired properties. BASIC ASSUMPTION. l
JS
(1)
Let (G, S) be a generated group with
such that the following properties hold: There is a partition SG = {s
(x);
x
E
G} of S which
is invariant under inner automorphisms.
(2)
S(x) u {l} is a group for all x
66
E
G.
(3)
For all
X
E G such that S(x)
u { l}
is nonabelian,
the set S(x) contains an involution.
(4)
If a 'I band B E S (b), then S(a)B
=
S(b) yields
S(b) £: CG(S(a) u { l}) • 1 (Remember that S(a) 8 : = B s (a) B.)
DEFINITION 1.1 .
P := {a$; a, 8 ES and a, B, aB involu
tory}. Suppose S(a) and S(b) are elements of the partition
P; and consider the following properties :
SG and A
E
(a)
A
CG(S(b) u {l}).
(b)
A /
(c)
S(a) u {l} s CG(S(b) u {l}).
(d)
If S(a) or S(b) contains an involution, then a 'I b.
E
S (b).
DEFINITION 1.2. a
I
A
Ib
precisely when (a) and (b) hold.
b precisely when (c) and (d) hold. The triplet (P, G, 1) with Pa point set, Ga line
set and
I
as both an incidence relation and an orthogonal
ity relation is now an incidence structure with orthogonality relation between lines .
This incidence structure is
called the group plane E(G, SG) . From now on let (G, S) always satisfy the basic assumption .
We are interested in the following set of
axioms:
67
P, there is a c E G such that c I A, B.
A.l.
I f A, B E
A.2.
I f a, b E G, either there is a c E G such that
cl a, b or there is a C E P such that C A.3.
b.
I f A, B E Panda, b E G and A, B I a, b, then A
or a A.4.
I a,
=
I a,
B
b.
Let a, b,
o
=
C
b, c.
E G and O E P such that a :/ b and Further let a E S(a) and 13 E S (b).
If
S(c) contains an involution, then there exist a y E S(c) and ad IO such that A.5.
o
:= al3y E S(d).
Let a E S(a), 13 E S(b), and a, b la where a, b, dEG. Then one and only one of the following conditions holds
A.6.
(a)
al3ES.
(b)
There is a CE P such that al3c
= 1.
Let a, b, c E G and DE P such that a, b, c ID. If neither S(a) nor S(c) contains an involution and if further a :f b :/ c, then there is a 13 E S(b) u {l} such that S(a) 13
Ex. A. A
S(c).
There exist A E P and b E G such that b I b but
I b.
(A+ b means that A I b does not hold.
a+ b means that
a jb does not hold.) THEOREM 1 . 3.
The group plane E = E(G, SG) = (P, G,
j)
is a Fano translationplane (that is, a translation plane satisfying the Fano axiom) with an orthogonality relation definable by a point involution j on the line of infinity.
68
The involution j has at least four fixed points. For all w longing tow.
G, let
E
wbe
the inner automorphism be
Then (G, S) is a generated group, withs
a set of axial collineations of E, and the following properties are satisfied: (*l)
S(a)
(*3)
Each axial collineation of Gisin s.
(*4)
G preserves the orthogonality of E.
(*5)
If a
u
E
{l} is a group for every a
G satisfies a
group of shears with axis a.
I a,
If a
E
G.
then S(a) u {l} is a E
G satisfies a .}a,
then S(a) u {l} is a group of homologies with axis a containing an involution. Theorem 1.3 holds for generated groups (G, S) satisfying the basic assumption, A.l to A.6, and Ex. A (see Gotzky, 1970).
The proof needs a large number of element
ary properties of the partition SG, which follow from the assumptions.
It includes a proof of the existence and
uniqueness of the perpendicular, and also a proof of the fact that a quadrilateral with three right angles is a rectangle. From now on, let (G, S) always satisfy the basic assumption along with A.l to A.6 and Ex. A so that, in particular, Theorem 1.3 holds. REMARKS.
Axiom A.4 yields the varied theorem of the
third quasireflection for the plane E.
69
By A.6, the group
S(a) u {l} acts transitively on the set of selforthogonal lines different from a which pass through a given point 0 on a. Reduction theorem
II.2.
Each element of G is a product of elements of S.
We are
interested in representations of a given element w
E
G
as a product of elements of S with a minimum of factors. Suppose that a, bare nonparallel lines of the group plane E and c is a further line satisfying c a
E
S(a),
.f c.
S(b) and let O be the point with O
SE
Finally let dllc with d
IO.
and a line m lo such that aB
By A.4, there are a 6
=
µ6 for someµ
E
Let a, b. E
S(d)
S(m).
Using this result, one easily proves: LEMMA 2.1. fying a
II
Let w b, c
choice of a
E
II
E
G.
There are lines a, b, c, d satis
d, and d,rd, such that for a certain
s (a), B
E
S (b), y
E
S (c), and 6
E
S (d) the
following holds:
w = aBy6. Using A.5, one deduces from Lemma 2.1 the following: LEMMA 2.2.
Let w
E
G.
Then either there are nonparallel
lines a, b such that, for certain a
E
S(a) and
BE
S(b),
we have w = aB; or there are two points A, Band a line c such that, for a certain y
70
E
S(c), we have w
= ABy.
(If
a,
Sare axial collineations such that their axes
have a point in common, then aS is called a quasirotation. The elements if
y is
ABy
EPPS are called glidequasireflections
a quasireflection, or glideshears if
y is
a
shear.) Lemma 2.2 tells us that the elements w of Gare either quasirotations or glidequasireflections, or glidequasishears.
Because w is also the inner automorphism
of G belonging tow, Lemma 2.2 yields: THEOREM 2.3.
The centre of G is trivial and therefore
G ~ G/Z(G) = G. Lemma 2.2 also yields REDUCTION THEOREM 2.4.
II.3.
If OE Pis any point of E, then
The theorem of the antiorthological iquadrilaterals
The theorem referred to in the title of this section is a special case of the theorem of the antiorthological quadrilaterals, due to Schlitte (1955), which characterizes the affine orthogonality relations induced by hermitian forms. DEFINITION 3.1.
Let i = l, 2, 3, 4 and j = 1, 2 .
Consider
4 lines ai and 2 selforthogonal lines sj such that there A.
1
71
with a 0 := a s
s
1 ,
and sj IAj, Aj+ 2 •
4 ,
s
2 ,
a
a4
3 ,
;
The a. are the sides and the s . the J
i
diagonals of the quadrilateral. 1 ,
a
,
which may be abbreviated as (ai; sj), is called
2 ),
an iquadrilateral.
A Is
Then (a 1
If there is a point
then (ai' sj) is called almost complete and A
2
is referred to as the diagonal point of the iquadrilateral (see figure 6).
A,, /
,,,
/
I I
/
/
/
/
/
I
/
/~
FIGURE 6 The following two lemmas are the core of the theory to be considered here: LEMMA 3.2
(first lemma of the iquadrilaterals).
Let
(a.; s.) be a nondegenerate, almost complete iquadrii
J
lateral with vertices Ai and diagonal point A. 2, 3 let bi be the line satisfying b:!. a line through A.
I
For i = 1,
A, a .• i
Then the following two statements are
equivalent:
72
(1)
=
(2)
A,
I
I
I
I
I
I
s 2 for i
1, 2,
I
A FIGURE 7
PROOF .
There exists exactly one line satisfying (1).
Assume if possible that there are two elements B•, BI4 such that (2) holds.
s. S l
=
s2 implies b 4 /
Then Sl
1
s~s.
€
8 (S l)
and, since
/ b~, it follows that
So there is at most one 8 4
€
€
s'
Sl
4
s·s = 1
4
4
s
1.
S satisfying (2), which means
that it is sufficient to show that (1) implies (2). Assume 8 1
,
8 2, 8 3 ,
8 4 satisfy (1) anu the assumptions
of (2). the relations: and
A.
l.
It follows that:
73
(a)
s.
A i+1' A.1
and that
1
s:1 A. 1
(b}
1
1 ai'
I
a., s 1
l
S·1
A.
Now (a) yields Ai+l
s.
for i = 1, 3
52
1
for i = 2, 4 and A 5 = A 1 . for i = 1, 3 and (b) yields
1
= A. 1
for i = 2, 4.
1
Reviewing the situation we know that Aw= Ai A 1 = A~ and s 1
I A,
A1 .
Thus Lemma 2.2 yields w
S(s 1 ).
E
This
proves the lemma. LEMMA 3.3 (second lemma of the iquadrilaterals).
Let
(ai' sj) be a nondegenerate, almost complete iquadrilateral with vertices Ai and diagonal point A. 4, 3 let bi be the line satisfying ai
be any line passing through A.
II
bi
I
A.
For i = 1, Let b 2
Then the following state
ments are equivalent: (1)
b2 11 a2.
s.1
(2)
If Si E S (b . ) satisfies 51 1 1  1 3, 4, then 8184 8382 E s (s 1 ).
PROOF.
= 52 for i
1, 2,
As in the proof of 3.2 it is sufficient to show
that (1) implies (2). Assume 8 1 , 8 2 , B3 , 13 4 satisfy (1) and the assumptions of (2).
Choose ai 1
1
v = a 1 a 4 a 3a 2
a.
E
S (ai) such that s 1 1 1
and w = 13 1 13 4 8 3 8 2 .
statements: (i)
s
1
1
II
s 2•
We prove a sequence of

is a fixed line of both v and w.
74
Define
The proof is straightforward. (ii)
There exist points X, YI s
For i = 1, 2, 3, 4, parallel to s
1
1 B.a7 l l
1
such that w = vXY.
fixes the pencil of lines
and has as its centre the pencil of perl So Biai is a translation.
pendiculars to bi and ai.
I
I
FIGURE 8 Since
vw
1
1 B.a7 l l
is a translation for i
must be a translation.
=
1, 2, 3, 4, also
So (ii) holds because of (i).
(iii)
w is a quasirotation with centre A.
(iv)
vis a quasirotation or a translation.
whas
the fixed point A and v the fixed line s
1•
Therefore the statements (iii) and (iv) are consequences of Lemma 2.2 (remember that s
75
1
Is
1
and that the elements
of Sin particular are quasirotations). (v)
vis either a shear with axis s
or a transla
1
tion. Assume that vis not a translation. quasirotation fixing s
v
fixed point Q of
Then it is a
[(i) and (iv)] and there is a
1
(the centre of the quasirotation).
Since every fixed line of a quasirotation goes through its centre, we conclude that either v
possibilities:
s
Now there are two
1•
S(s 1 ) or
E
fixed point, which is Q.
QI
v has
Therefore Q
s2.
because A1
=
C(l Sl
II
C(
1
s2,
I s 2, C( 1 Sl
v
Assume, if possible, that
C(l
has one and only one fixed point Q. the only fixed point of
exactly one
V
C( 1 I
hence
=
s2.
In this case Q
is
which also has the fixed line
oa i I
C(l Sl
I
s2 and hence,
The last equation forces
A, which is impossible for nondegenerate iquadri
laterals (ai' sj).
The conclusion is that v
E
S(s 1 ).
So (v) holds. (vi)
w
E
S(s 1 ) .
If vis a translation, then by (ii) so is wand, by (iii), it follows that w
=
If v
S (s 1 )
then, by (ii),
the collineation ~ has to be a glideshear.
But, by (iii),
w is a quasirotation.
l.
E
The conclusion is that w must be a
shear, which proves (vi) and the lemma 3.3. THEOREM 3.4 (theorem of the antiorthological iquadrilaterals).
Let (ai; sj) and (bi' tj) be two nondegenerate
76
iguadrilaterals with s i = l, 2, 3 .
\
1
Then also
It2 and a4 Ib4•
s
2
I
t
1
and a 1.
I b 1.
for
\
\
\ \ \S1 \
\
\
B, \
\
FIGURE 9
\
THEOREM 3 . 5 (theorem of the parallogical iguadrilaterals). Let (ai, sj) and (bi, tj) be nonde9:enerate ig:uadrilaterals with s. 11 t. and a. l. J J 2, 3. Then also a 4 _ 11 b 4 .
II
b. for j = l, 2 and i = l, l.

Theorems 3.4 and 3.5 are both consequences of Lemmas 3.2 and 3 . 3 of the iquadrilaterals.
This is obvious if
the iguadrilaterals in question each have a diagonal point.
The case in which the diagonals are parallel can
be reduced to the general case with diagonal point (see Gotzky, 1970) •
77
II.4.
Coordin ates 1
In this section the results of section II.3 are used to prove that the group plane E minkow skian plane.
=
(G, SG) is a unitary 
The method is to constru ct both the
skew field which coordin atizes the plane and the hermiti an form which represe nts the orthogo nality relatio n of E.
FIGURE 10 Let a E S(s) withs Is and EE S(e) with Further let t
=
sE and,
such that Ole, s, t. Ff,
crE
E
S(t).
E2
=
1.
O may be a point
Let F I s and E = F E , such that
0 f, E.
1. Compare the introdu ction of coordin ates in Wolff (1967). 78
IO
From now on, for all the lines a, b, •. • satisfy s t a t t, s t b
t
that
t, ... , let a, S, .. • represent
elements of S(a), S(b), • .. respectively, such that sa sS = t, ... B := F
(remember Axiom A.6).
t,
Further let A:= Fa,
B, ••••
Next let K be the set of all points incident with t, and define: A + B := AOB A
0
B := Fm: B
A
0
0
:= 0 =: 0
+
Then K
0
A
for all A, B
E
K,
for all A, B
E
K\ {O},
for all A
E
K.
(standing for K with respect to the operation
+) is an abelian group with unit 0, in which A has the
inverse A= OAO .
Further K0
(representing K\ {O} with
respect to the operation o) is a not necessarily abelian group with unit E = F£, in which A has the inverse A £a1£ £ (remember that £ 2 = 1 and hence t = s). = F Concerning the proof of these statements there is only one problem worth mentioning , and that is the associativity of K0 which we now prove. Ao B = A£B; therefore (Ao B) o C = (A A£B£Y.
O
B) £y =
Let D = B ° C, hence D
a'= 8£yo
1
has the fixed points O and F.
the reduction theorem yields a'
E
S(s).
A£B£y = A£cr'o = A£o =Ao D; hence (A o B)
o
C
Ao D =Ao (Bo C).
79
Since O
Thus
t
F
Since (A+ B)
O
C = (A+ B)Ey = (AOB)EY = AEYosEY
AoC + B°C, K is a skew field if and only if (i)
Ao
(B + C) = AoB + AoC
also holds.
Therefore K is a skew field if there is an
antiautomorphism
of K.
TI
CONSTRUCTION OF AN ANTIAUTOMORPHISM
OF K.
TI
be lines incident with O and let a*, b*,
Let a, b, ... be the per
pendicular lines from O onto a, b, ... respectively. mapping
j:
TI:
K with 0 F
. 1 utory.2 is 1nvo
a*O
The
0
= A*O
A
for
We shall see that
TI
E
Ko
is an antiautomorphism
of K with respect to the operations+, o. First, let D =A
a'= a£ 1 So 1
E
Considering figure 6 yields
a* o*
1
S* £*
1
E
s (s) (Lemmas 3. 2 and
Thus S*E*
2.
B.
S(s) and the second lemma of the iquadri
laterals yields a" 3.3).
O
1
a*o*
1
E
S(s
a*~* 1 u
)
= S(s).
Since E(G, SG) is a translation plane, the theorem
of the parallogical iquadrilaterals (Theorem 3.S)implies that the line reflection a with axis sin the direction of O a 1 a*O O·Oa 1 a*O t exists. Therefore (A*) = A = A = (AO)oa
1
a•aa*a holds for all A
E
K0
•
If now d
IO
such
that Ao= F 0 , then o = oao and o* = aa*a: hence (A*)o = 0 o 1 o* 0 (A) = (A )*.
So *O = O*, and therefore
80
TI 2
= * 2 0 2 = 1.
E and E* 1 = E*, hence F $*Ea* = F o*O , hence
But E*O
DTI = Fo*O = FB*Ea = (FB*O)Ea*o. If y
such that F
Es
a*Oy 1
E
S(s).
a*O
= F
y
and c Io for y
E
S(c), then
Hence
On = (FB*O)Ea*O = (FB*O)EY = Bn
O
An.
So TI is at least an antiautomorphism of K0 Second, let D =A+ B.
•
Consider then figure 11.
s
FIGURE 11
The theorem of the antiorthological iquadrilaterals (Theorem 3.4) applied to the iquadrilaterals with vertices F, B, FOB, D and F, B*, FOB*, D* yields (FOB)D Since FA 11 (FOB) D and FA* FA
I
I
I
(FOB*)D*.
FA we get
(FOB*)D* II FA*.
So F(FOB*) = A*D*, hence A*D* = OB*, hence D* = A*OB*. The last equation yields
81
=
Dn
A1TOB1T
which proves that n is an automorphism of K+ Son is an antiautomorphism of K, and the following theorem is proved: THEOREM 4.1.
K
(K, +, o) is a skew field with anti
automorphism n. Next we introduce coordinates in E
=
E(G, SG) using
K as skew field of coordinates.
Since Eis a translation plane, there is for each point Pi of E exactly on&pair Xi' Yi (ii)
e
K such that
P.l. = x.oY:. l. l.
Conversely every pair xi, Yi by (ii).
e
K determines a point Pi
We have to show that every line considered as
a set of points can be described by a linear equation. First, we describe lines parallel to s o r t . Obviously (1)
lines parallel to s are described by equations X = C,
(2)
lines parallel t o t are described by equations Y = C. Second, we describe lines incident with O.
and O :/ P 1
,
I a.
P2
Consider figure 12 .
Let a IO
By the theorem
of the parallogical iquadrilaterals (Theorem 3.5)
a perpendicular
Hence there is a line
both the lines X 1 Yi and X 2 Y; and passing through O. ~1
Y~
l.
x~ l.
for i
=
1, 2, so x .
l.
82
Y.
l.
0
A.
to Then
a
a
t
FIGURE 12
Thus (3)
lines a which are incident with O but different from sand tare described by equations X =Yo A.
Finally, let s, t ~ a IO and b I/ a. There is a point OZE zE Z EK such that b = a = a (remember that Eis a translation plane).
If now P
= XOYE is a point of b,
the above result yields: E:
x = yz O
o
A
(ozEyzEo)
E:
A=
o
E:
= (YOOZ O) o A= (Y + OZ O) o ~ zEo ~ ~ Let B := 0 o A; then X = YoA + B.
83
(YzEozo:o)
A= yoA
o
A E:
+ OZ O
o
A.
This proves (4)
lines b
parallel neither to s nor t o t are described ~ by equations X = Y0 A + B. ~
Since K is a skew field a general line equation can be derived from (1) to (4):
(5)
X o A+ Yo B + C If now V
O.
= V 3 (K) is the 3dimensional right vector
space over the skew field K, it is easy to realize that the lines of our plane can be represented by the 1dimensional subspaces of V which are different from (0, 0, E} o K. E
= PV the planes P(O,O,E)oK and
Moreover, with P
= E(G, SG) are isomorphic .
(Compare the introduction
of coordinates in Artin, 1957.) Next we describe the orthogonality of E THEOREM 4 . 2.
Let a, b be lines of E
by the vectors (A 1 , A 2
,
= E(G, SG).
= E(G, SG) represented
A 3 } and (B 1 , B 2
B 3 ).
,
Then alb
if and only if
PROOF FOR A 2
= O.
Since a, bare lines of E, it is easy
to see that the inequalities A 1 o A 2 i o i
B 1 o B 2 hold.
Now A 2 = 0 means that a II s or, since s Is, that a So a
Ib
if and only if B 2
0.
s.
But assuming A 2 = O, the
matrix equation holds if and only if B 2
84
J
= o.
PROOF FOR
A2
1 o.
Some considerations similar to those
just used show that we may assume B 2 generality.
101
Let therefore A 2
1 B2
0 without loss of •
The matrix equation holds if and only if AnoB + I 2 A:oB 1 = O, hence if and only if (A 1 oA~)n + B 1 oB 2 = 0. Using the symbols out of (4), this last equation becomes transformed into (A)n 
B
= O, or An=
aIb
The last equation is equivalent to
a0 ,
or A*= B.
(see figure 12
and remember the meaning of* used for the construction n). It remains to show that
a I b if
and only if a
Let P 1 := X 1 0Y~ 1 o 1 Q 2 := x 2 0Y! with P 1 Q2
I b.
aIb
To visualize this, refer to figure 12.
i f and only i f XIY~
I X2Y!.
I
b.
a and We have
Hence, applying the
theorem of the antiorthological iquadrilaterals to the quadrilaterals with vertices O, P 1
x2
respectively, we find that
,
aI b
E
Y1
,
X 1 and O,
Q2
if and only if a
,
E
Y2
,
I b.
This completes the proof of 4.2. MAIN THEOREM 4.3. Let f be the TIhermitian form (right linear, since V 3 (K) is a right vector space) defined by the matrix
0
E
0
E
O
0
0
0
0
Further, let V = V 3 (K, f, n).
Then E(G, SG) ~ PRad V for P
= PV and G = G = U!(K, f, n).
Hence, in particular, E(G, SG) is a unitaryminkowskian plane.
85
PROOF.
Taking Theorem 4.2 into consideration, we know
already that E(G, SG) holds.
~
PRad
v·
By Theorem 2.3, G
~
G
We now compare the generated groups (G, S) and
(PU*, PS(V)) Part I) .
(for the definition of the last group see
Since PS(V) is the set of all axial collinea
tions of PRad V which preserve the orthogonality, we infer from Theorem 1.3 that identifying E(G, SG) and PRad V the inclusion
S
~ PS(V) becomes valid.
A.6 yields PS(V) ~
S,
Because now Axiom
we get (G, S) ~ (PU*, PS(V)).
finally U* := U~(K, f, TI)
~
Since
PU* the main theorem must hold.
It remains only to remark that if U*
=
u;(K , f, TI)
is a restricted unitary group of the minkowskian type, then (PU*, PS(V)) satisfies the basic assumption and the axioms of section II . l . REFERENCES Artin, E. 1957. Geometric Algebra . Interscience Tracts, No. 3, Interscience Publishers, New York. Bachmann, F. 1959. Aufbau der Geometrie aus dem Spiegelungsbegriff. Die Grundlehren der mathematischen Wissenschaften, Band 96. Erste Auflage 1959, zweite Auflage BerlinHeidelbergNew York 1973 . Gotzky, M. 1964 . Eine Kennzeichnung der orthogonalen Gruppen unter den unitaren Gruppen . Arch . d. Math . 15: 161165 . 1965. Eine Kennzeichnung der unitaren Gruppen uber einem Schiefkorper der Charakteristik # 2. Dissertation, Kiel . 1970 . Aufbau der unitarminkowskischen Geometrie mit Hilfe von Quasispiegelungen. Habilitationsschrift, Kiel. 1972 . Mittelpunktsabbildungen in affinen Ebenen. Abh. Math. Sem. Hamburger Univ. 37 : 133146. 1973. Der Satz von der dritten Quasispiegelung. Supplement 15 in Bachmann (1973) . Pickert, G. 1955 . Projektive Ebenen. Springer, BerlinGottingenHeidelberg. 86
Schutte, K. 1955. Ein Schliessungssatz fiir Inzidenz und Orthogonalitat. Math. Ann . 129: 424430. Veblen, O. and J.W . Young. 1910, 1918. Projective Geometry, vols . 1, 2. Boston. Wolff, H. 1967. Minkowskische und absolute Geometrie. Math. Ann. 171: 144193 .
87
THE LENZBARLOTTI TYPE III
Christoph Hering
1.
INTRODUCTION
A finite projective plane admitting a collineation group G of Lenz type III is desarguesian.
The first published
proof of this theorem (see [SJ) uses the classification of finite groups with a split BNpair of rank 1.
There
fore a complete presentation of this proof necessarily must be very long.
Also, the proof is quite indirect.
It involves the characterization of a large variety of groups, namely the groups PSL(2, q), Sz(q), PSU(3, q), groups of Ree type, and sharply doubly transitive groups, although in the end, because of geometric reasons, only the groups PSL(2, q) actually occur.
Therefore it seems
worth while to look for a more direct proof which always stays within the range of the 2dimensional linear fractional groups.
For planes of odd order such a proof
actually exists, and it is the main purpose of this paper to describe this shorter, more economic way of argumentation.
88
Instead of the classification of finite groups with a split BNpair of rank 1 we use here a system of three theorems on group spaces.
At first glance, this system
looks somewhat artificial.
The main idea is to save for
Ga sufficient amount of geometric information without losing the possibility of applying induction.
This is
quite difficult and could only be achieved by partitioning the problem into three different cases.
One of the three
cases (see Theorem C) deals with groups in which all involutions are central.
Theorem 2.1 and Corollary 2 . 2
list a few simple properties of such groups.
Possibly
one can find stronger results of this kind, which might lead to further simplifications.
The proof presented
here does not use any grouptheoretical classification theorem apart from the classification of finite groups with dihedral Sylow 2subgroups by Gorenstein and Walter [2]
(which of course uses the theorem of Feit and Thomp
son) .
Even this theorem is not used in its full generality
but only in a certain quite special situation (see Proposition 3.6) . Our notation is fairly standard.
a
a and b elements of G, and Then a
b
< [a, bl
= b
Ia
1
E
ab,
[ a , b] = a
(J{, and be.$, >.
1
Let G be a group,
as well as §&subsets of G. b
1
Also,
ab, and
}G
uP, [a, oll']
=
is the centre of G,
~(lthe centralizer of t'tin G, J!Gdtthe normalizer oft( in G, G' the commutator subgroup of G, G# the set consisting
89
of all nontrivial elements in G, o(G) the largest normal subgroup of odd order of G, and ated by all involutions in G. element of order 2.
Q2
(G) the subgroup gener
An involution is always an
A dihedral group is a group with two
generators a and band the relations an= b 2 = 1, where n
:2:
bab = a 1
2.
Suppose that in addition to the group G we have a set
and a binary operation
Q
= a
x G
Q
for all a
E
Q
+
such that
Q,
and g, h
E
G.
Then
we call the pair (Q, G) together with the given binary operation a group space.
2.
a
{w
E
Q
w
{g
E
G
I a?l.
Furthermore, we denote
= w for all a
E
('.7[}
1
Q(tfl)
and G al ... an
= a. for lsisn}, if al, l.
... ,
a
n
E
n.
GEOMETRIC AND GROUPTHEORETICAL TOOLS
In this section we present the most specific grouptheoretical and geometric results which will be used in our proof of the Type III Theorem. THEOREM 2.1.
Let G be a finite group containing a sub
group Z such that Q 2 (G)
sz
sjG,
and let Ube a subgroup of G containing
z.
Choose a sub
group N of Z maximal with respect to the property that Q2
(U/N) s Z/N.
90
Then the following statements hold: (a)
Z/N is an elementary abelian 2group.
(b)
If 1 :j x E Z/N, then there exists an element
y E U/N  Z/N such that y 2 = x. (c)
The number r of conjugacy classes of involutions
in G/Z intersecting U/Z nontrivially is at least IZ/NI  1. In particular, r ~3, unless the Sylow 2subgroups of U/Z are cyclic or dihedral. (d)
PROOF.
I Z/N I :;;
I U/Z I •
~ru,
Denote G = G/N, u = U/N, and z = Z/N.
a subgroup of odd order of
z.
As
z
an involution in U/A, then l I contains an involution x and xA =
xA
A~ U.
z contains
y EU  Z.
If XA is
= 2IAI so that xA
E Z/A.
Z/A, and by maximality of N we obtain A= 1. that
Let A be
So S1 2 (U/A) ~ Suppose now
an element z such that y 2 :J z for all
For each involution x in U/ the group
is an abelian 2group containing the cyclic group as a subgroup of index 2. cyclic or =
X
Hence either is
, where Xis an involution.
In the first case there exists an element y such that y2
z and = .
that x E Z/.
As y 2 = z, we have y E Z, so
In the second case
But this implies that = So S1 (U/) 2
x E Z by assumption. ~ z and x E Z/.
:;; Z/, and by maximality of N we obtain
= 1, which proves (b).
91
Suppose now that Z contains an element a of order 4. By (b), there exists an element y E U  Z such that y 2 = a 2 .
JU,
As a E
order 8.
the group is an abelian group of It contains two cyclic groups of order 4, namely
and .
Therefore is the direct product of a
cyclic group of order 4 and a cyclic group of order 2. But n Z
= .
which do not lie in
Hence we have involutions in ,
Z,
a contradiction.
Z is
So
an ele
mentary abelian 2group. Let z 1 and z 2 be two different nontrivial elements in
z.
u
By (b), there exist elements u 1 and u 2 in
such that
u:
=
zl and u~
=
z2.
Z
Clearly ulz and u2Z are
U/Z. Suppose that these G/Z. Then there exists an
involutions in
involutions are
conjugate in
element gE G
such that g
1

u 1 gZ
=
u 2 Z.
But this implies that g
u 2 z for a suitable element
= (u2z) tion.
2
z E Z,
and hence z 2
= (g 1 ulg) 2 = g 1 u:g = g 1 zlg = zl
I
=
1
u 22
u 1g
=
=
u 22 ;
2
a contradic
So the number of conjugacy classes of involutions
in G/z intersecting
U/Z
nontrivially is at least as large
as the number of nontrivial elements in is at least three, unless
11 ,z =
2 and
z.
This number
u contains
exactly
one involution, in which case the Sylow 2subgroups of U/Z are cyclic or dihedral.
Thus the proof of (c) is
complete, and as an immediate consequence we obtain (d). Finally, we consider the special case U to the following corollary:
92
=
G.
This leads
COROLLARY 2.2.
Let G be a finite group containing a sub
group Z such that n 2 (G) s Z
s3G.
If G/Z has at most two
classes of involutions, then the Sylow 2subgroups of G/Z are cyclic or dihedral. LEMMA 2. 3 (Zassenhaus [11]). a finite group g
T
= g
1
PROOF .
G.
for all g If
X €
If oCG, €
Let , be an involution acting on
=
1, then G is abelian and
G.
G, then (x
1 TT X
)
=
T (x )
i . e . T inverts the element x 1 xT . assumption implies that the map x
1
X
= (x
1
X
T 1 )
On the other hand, our 1+
x  ix T for all x
onetoone and therefore a map of G onto itself . inverts each element of G. ((xy)1)1 = all x,y
€
,
E
G is
Thus T
This implies that xy =
((xy)T)1 = (xTyT)1 =
(x1y1)1 = yx,
for
G, so that G is abelian.
BRAUERWIELANDT THEOREM 2 . 4 (see Wielandt [10]). E = {l, e 1 , e
2 ,
e
3}
Let
be an elementary abelian group of
order 4 acting on a group G of finite odd order.
THEOREM 2.5 (Gorenstein and Walter [2]) .
Then
A finite group
with dihedral Sylow 2subgroups satisfies one of the following conditions: (i)
G/O(G) is isomorphic to a subgroup of PfL(2, q)
containing PSL(2, q) for a suitable odd prime power q. (ii) G/O(G) is isomorphic to the alternating group A 7
93
•
(iii)
G/O(G) is isomorphic to a Sylow 2subgroup of
G. LEMMA 2.6 (see (3, Hilfssatz 6]). Let G be a finite doubly transitive permutation group which contains involutions fixing 2 letters but no involutions fixing more than 2 letters.
Then the Sylow 2subgroups of Gare dihedral
groups or quasidihedral groups. LEMMA 2.7.
In PSL(2, q), q odd, the normalizer of each
elementary abelian subgroup E of order 4 is transitive on E  {l}. PROOF.
Let q be an odd prime power and G
Then G is not 2nilpotent .
~
PSL(2, q).
Hence by the Transfer Theorem
of Frobenius (7, p. 436, Satz 5.8] G contains a 2group E such that3'GE/~E is not a 2group .
As Eis a dihedral
group, it follows that Eis elementary abelian of order 4. Hence the order of 3tGE/~E is divisible by 3, so that ')'{GE
is transitive on the set of nonzero elements of E.
We finish the proof by showing that the automorphism group of G acts transitively on the set consisting of all subgroups of G which are isomorphic to E.
Let F~ G be a
second elementary abelian subgroup of order 4 .
As all
involutions are conjugate in G, we can assume without loss of generality that En F contains an involution a. be the centralizer of a in PGL(2, q).
94
Let C
Then C is a dihedral
group and E/ and F/ are involutions in (C nG)/. These involutions must be conjugate in C/, as E and F are not cyclic. LEMMA 2 . 8 (Baer [1]).
An involutory automorphism of a
projective plane of finite odd order is either a homology or a Baer involution. LEMMA 2.9 (Ostrom [8] and Liineburg). automorphisms of a projective plane
Let G be a group of
q, .i),
a
E
,:t_··, b
E .~(
a,
.!.!_ a and Bare involutions, then aB is the unique involution in G(anb, AB). PROOF.
As the commutator [a, BJ fixes each point on a
and b, it is trivial.
Hence aB is an involution.
involution fixes a and exactly two points on a.
This A Baer
involution fixes at least three points on each of its fixed lines.
Therefore the product aB is a perspecti
vi ty, which clearly must have a n b as centre and AB as axis.
Finally, if y is any involution in G(an b, AB),
then by the part of the theorem which we have already proved, a(aB) and ay lie in G(B, b). a(aB) (ay)
1
On the other hand
fixes each point on AB and of course also each
point on b, which implies that a(aB) = ay.
Hence aB is
unique . 3.
THREE THEOREMS ON GROUP SPACES
Let (Q, G) be a finite group space with the property:
95
(*) For each a En the stabilizer Ga contains a normal subgroup Na which is sharply transitive on n {a}, such that for a E n and g E G we have (Na) g = N ag In this section we derive some properties of (n, G) which will be useful for the proof of our main theorem . we introduce the following notation:
s = .
E
¢ is the representation of G on n determined by the group space (n, G).
K is the kernel of¢.
If U is a
subgroup or an element of G, then we denote u¢ =
U.
If u is a subgroup or an element of G, then n(U) is the set of fixed points of u . If U is a subgroup or an element of G and ln(U) I ~ 3, then S (U)
3, then PropoHence in this case
S = S
and Sis isomorphic to a homomorphic image of the representation group of s. Proposition 3.3.)
(Note that S
n
K s
j
S because of
By Schur [9, pp. 119120] it follows
that S ~ SL(2, q) ors~ PSL(2, q) except, possibly, if
101
q = 9, in which case the Schur multiplier of s has order 6.
However, we also know that S n K is a p  group:
Let
Y be the Sylow psubgroup of Sn K, and Pa Sylow psubgroup of S
a
containing Y.
Then P = Y
N a
x
As ls:s a
q + 1, P actually is a Sylow psubgroup of s.
I
=
Hence by a
theorem of Gaschiitz (see (7, p.121, Hauptsatz 17.4]) there A
A
exists a subgroups of S such that S = SY and Sn Y = 1. A
As Y is central, S plies Y = 1.
S, and S/S
!
~
Y.
But S' =Snow im
So S _ SL(2, q) or PSL(2, q), whenever S' =
s. There still remains the case q = 3. ~ A4
•
Let T be a Sylow 2subgroup of
subgroup of order 4 in S, so that T
g
s.
In this case S Then Tis the
Sand T(S n K)
9
S.
Now the normalizer of T contains T and the central subgroup (Sn K), hence the product T(S n K), so that Tis normal in T(S n K).
This implies that Tis characteristic
in T(S n K) and therefore normal even in S.
Hence TN CJ, is
a group and as Tis transitive, we have S = TNa. Because of Sylow's theorem, 3 Clearly T' s Sn K.
f Is
n Kl, so that Sn Ks T.
Also, by a lemma of Zassenhaus (see
(7, p. 350, Satz 13.4]) T/T'
~
(Sn K)/T'
x
Tl/T', where
T 1 /T 1 is an elementary abelian group of order 4, such that T 1 is normalized by Na and Na acts nontrivially on T 1 /T'. Here T 1 is transitive on II, so that S = Na T 1, T 1 = T, and T' =Sn K.
Thus Tis generated by two elements a and b.
We now determine the order of the commutator subgroup T'.
102
Let x, y ET and consider the commutator [x, y).
x ES n K, then [x, y) = 1 . in T 
(S n K).
If
So we can assume that x lies
As~ is transitive on T/S n K  {l}, a
there exists an element u EN such that xu = a•z, where a
z ES n K.
Furthermore,yu has the form arbsz,, where
r, s E {l, 2} and z' E Sn K. [x, y)
=
[x, y) u
Hence
= [xu, yu) = [az, arbsz' J = [a, bs).
Therefore T' = and
IT' I~ 2, as [a,b)
2
=
[a,b 2 )
= 1. THEOREM A.
Assume that there exists a subgroup U in SK
containing Kasa normal subgroup of index 2 such that IQ(U)
I
= q + 1 and S(U)/K(U) ~ PSL(2, q)
odd prime power q. PROOF.
for some
Then IQ! > q 2 + 1.
See [6, pp. 448452).
THEOREM B.
Assume that (Q, G) has the following proper
ties: (a)
If a, SEQ and a t S, then Gas contains exactly
three involutions.
One of them fixes all points in Q while
the remaining two don't fix any point different from a and
s.
All three are contained in JGaS but at least one is
not contained in
JG {a, S} .
(b)2
We consider the group space (Q(U),at'GU).
As the
three involutions in Gas are central, they are contained inoC'Gu.
Let Ebe the group generated by these involutions
and suppose that E ~J,({U){a,s}·
In the doubly transitive
group space (Q(U),ot"'GU), there exists an element x interchanging a and 6.
As x centralizes E it follows that
G{a,S} = centralizes E, which is impossible by So E ~ }(~U)aS but E / j(~U){a,s} •
(a).
Also u
so that u is an even permutation of Q and hence ln(U)I
=2
(mod 4).
~U).
for a suitable prime power  1)
2
= lnJ
Therefore we can apply induction to (Q(U),
It follows that S(U) ~ SL(2, k
S,
E
q.
q)
Because
and IQ(U) I = q + 1
q+
1 = IQ(U)
I
~
+ 1, we obtain a contradiction to Theorem A.
So all involutions in Shave at most 2 fixed points. As they induce even permutations on Q and lnl they actually all have exactly 2 fixed points.
104
=2
(mod 4),
Hence by
Lemma 2.6 the Sylow 2subgroups of Sare dihedral groups or quasidihedral groups.
Also, we can obtain information
about the involutions ins.
Clearly the kernel K contains
exactly one involution by (a).
If S contains only one in
volution, then we are finished by Proposition 3.6. So we may assume that S contains an involution t which is not contained in K.
Then by the above t has two fixed points,
w.l.o.g. a and 8.
Let Ebe the elementary abelian sub
group of order 4 generated by all involutions in Gas· Eis not central in G{a,B}' there exists an element x G{a,B} such that E
= .
E
Hence E $Sand therefore
Sn K ~En K, which is a group of order 2.
s
As
Furthermore,
S' by Proposition 3.5, and therefore Sn K $
JS
n S'
is isomorphic to a subgroup of the Schur multiplier of In particular, the Schur multiplier of
s has
s.
even order.
Hence by Schur (9, p. 108, Theorem V] the Sylow 2subgroups of Sare not quasidihedral groups.
So they are
dihedral, and we are finished by Proposition 3 . 6. THEOREM C.
Assume that each involution in S acts trivially
on n and that 4 $ lnl
=2
(mod 4).
Then S ~ SL(2, q) and
lnl = q + 1 for a suitable odd prime power q. PROOF.
Let (n, G) be a counterexample of smallest possible
degree.
By our assumption lnl is even so that bec~use of
Proposition 3.5 each element ins induces an even permutation on
n.
Hence each involution in S fixes at least 2
105
points.
Suppose that each involution in S fixes exactly 2
points.
Then all involutions in Sare conjugate by Pro
position 3.4(b), and therefore the Sylow 2subgroups of Sare cyclic or dihedral by Corollary 2.2.
But then G is
no counterexample because of Proposition 3.6.
Hence
contains involutions fixing more than two points.
S
Concern
ing such involutions we prove (1)
If u is an involution in S fixing more than 2 points
and U is the preimage of in G, then S(U) and S(U)
~
ln(U) I
SL(2, q)
PSL(2, q) for a suitable odd prime power q.
Furthermore, PROOF.
~
lrl(U)
I
= q + 1 < (lnl
 1)
!,:
2
+ 1.
As u ES, it is an even permutation, so that
=
lnl  2 (mod 4).
Hence we can apply induction to
the group space (S"l (U), LGU). SL(2, q) and ln(u)I
=
It follows that s (U) ~
q + 1 for a suitable oddprirrepowerq.
By our assumption the unique involution in S(U) must lie in K, so that S(U)
~
PSL(2, q) .
Because of Theorem A we
have the inequality lnl > q 2 + 1. Suppose now that S contains an involution fixing exactly two points .
By the above, S contains involutions
which fix more than two points.
Hence by Proposition
3.4(c) S contains an involution u such that lrl(u) ~
(lnl  1)
!,:
2
+ 1.
I
But this is a contradiction to (1).
Hence each involution in S fixes at least three points in
n. 106
(2)
If a, B, and y are pairwise different points in Q,
then SaBy does not contain any elementary abelian subgroup of order 4. PROOF.
Suppose that SaBy does contain an elementary
abelian subgroup X of order 4. such that In (a 1 l I In (a3 l I
3.
x S(T).
Let
X be a subgroup of S(T) isomorphic to the alternating group A 4
•
(A subgroup of this kind exists.
107
See (7, p.213,
Hauptsatz 8.27].)
Let F be the subgroup of order 4 and x
an element of order 3 in X, and denote E = x centralizes t and hence normalizes E.
F.
x
Now
In fact, has
three orbits in E  {l}, namely x, F  {l}, and a second orbit of length 3. We now choose an element d preimage of in G by D. ol~.
We consider the centralizer
In this centralizer we have the normal subgroup
S(D) and the product E•S(D). A
Sand the second by L. odd prime power q.
We denote the first group by
Note that S ~ PSL(2, q) for some
s,
Let a,
differ~nt points in SI (D). L
F  {l} and denote the
E
and y be three pairwise
Then L A
A
A
;); LaSyS/S s L/S = ES/S
aSy
aSy
===
n
s A
1 and therefore
s. A
E/E n
Hence LaSy is an elementary abelian 2group, and by (2) we obtain IL a..,y I s 2, so that O
=
LaSy
.
This implies that ILi s 2(q + l)q(q  1) and hence IE n §I
~ 2, so that IE n
SI =
2 or 4.
We consider at first the case IE n
sl
=
4.
In this
case the normalizer'Jt8 (E n S) contains an element y which induces an automorphism of order 3 of En
{d},
(see Lemma
Here y centralizes and hence normalizes E
2. 7) •
S
(E n
X
sJ.
Also, y has three orbits in E
(E n s) it, and a third orbit of length 3.
IF n (E n
d
_>
sJ I
=
2I
E#  { t } .
d
~
F# u

{ l}' namely
As
(E n slit and actually
Also, t i s not invariant under y.
108
This implies that is transitive on E#, which leads to a contradiction to Theorem 2.1.
sl =
Assume now that IE n
2.
A
Then L/S is an element
ary abelian group of order 4, and we have three subgroups H1 , H2 , and H3 in L containing Sas a subgroup of index 2. {HI, H2, H3} and assume that HaSy I 1. Then and H = XS. So two of the groups H al3y = L al3y =
Let H
€
A
HI , H2, and H3 have a trivial stabilizer on a, 13, and y, which because of their order implies that they are sharply triply transitive on n(D). that His of this kind.
=
x
En H.
H.
Let
Q
Choose HE {H 1 , H2 , H 3 } such
Then we have IE n HI
=
4 and L
be a Sylow 2subgroup of H containing
By Lemma 2.6, Q is a dihedral group or a quasi
dihedral group.
As IHI
=
(q
+ l)q(q  1), we have IOI ~ 8.
Hence there exists a subgroup Y such that (En H) and IY: (En H) I= 2.
Let z E Y 
(En H).
4
Y s Q
Then z does
not centralize E, since in a dihedral or quasidihedral group each elementary abelian group of order 4 is selfcentralizing.
On the other hand, z centralizes d and hence
normalizes E = x in E.
(En H).
So z induces a transvection
We consider the group acting on the project
ive plane determined by E:
If has three orbits,
then these orbits must be {t}, F#, and the remaining elements in E#.
This implies that the elation z fixes t, d,
and F so that is its axis and d its centre .
But
then z fixes a line not passing through its centre, namely
109
En H, a contradiction.
So has at most two orbits
in E#, and we can again apply Theorem 2.1. 4.
PROJECTIVE PLANES OF LENZ TYPE III
THEOREM 4. 1.
r .t
and
Let
(f, :L)
a subset of
Q
(f,Zl
Assume that
be a projective plane,
.t E ,Z,, P E
.t containing at least 3 points.
admits a finite group G of collineations
leaving invariant P and Q such that for all XE Q the group G(X, XP) is transitive on
Q
{X}.
Assume in addi
tion that each involution in G is a homology and that IQI
=2
(rr,od 4) .
Then there exists an odd prime power q such
that ~ SL(2, q),
IQI = q + 1,
and (,,
/1:>
PROOF.
We consider the group space (Q, G).
contains a desargues ian subplane of order q. Clearly
this group space has the property (*) defined in section 3.
If each involution in the subgroup S
=
lies in G(P, .t), then the first two statements of our theorem follow from Theorem C. ists an involution t
ES  G(P, .t).
homology whose centre z lies on
Assume that there exThen t must be a
.t and whose axis a passes
through P, because obviously Gleaves invariant the line
.t.
By Proposition 3.5 each element in S induces an even
permutation of Q.
This implies that t must have fixed
points in Q, so that Zand an
llO
.t lie in
Q.
Also,
(Q, G)
is doubly transitive, and hence G contains an element x interchanging Zand an .t.
But then tx E G(a n .t, ZP).
By Lemma 2.9 the product ttx is an involutory homology in G(P, l).
Also, this lemma implies that t, tx, and ttx are
the only involutions in the stabilizer G2
, an"0
have all properties required in Theorem B. cases ~ SL(2, q) and
suitable prime power q. that
(f, c\C)
l~I
•
Hence we
So in both = q + 1 for a
By [4, Theorem 2.8] we now obtain
contains a subplane of order q.
THEOREM 4.2.
Let
be a projective plane of finite
odd order which contains a line .t and a point P not incident with .t such that (y,:;/;1 is (X, XP)transitive for each point XE .t . PROOF.
Then
(r
~ is desarguesian.
Let ({,~) be a minimal counterexample and denote
G
=
.
We consider the group space (.t, G), which again has the property (*) of section 3.
By Baer's Involution Lemma
2.8, each involution in G is a homology or a Baer involution.
Suppose that G contains a Baer involution t.
Then
the set/J/ of fixed points o f t generates a subplane of order q and
(,,£)
has order q 2 for some odd q.
This sub
plane contains P and .t, and because of 3.1 it is again (X, XP) transitive for each X tion,
Of is
desarguesian.
E
.t n
tf .
Hence, by induc
This implies that q is a prime
power and S(t) induces a group of collineations of fwhich
111
is isomorphic to SL(2, q). 3. )
(We define S(t) as in section
Hence the group of permutations of
/j,
n
l induced by
S(t) is isomorphic to PSL(2, q), but this is impossible by Theorem A. Thus each involution in G is a homology. (mod 4), then Theorem 4 . 1 implies that guesian .
Assume that Ill
=0
(mod 4).
((,£:>
If Ill
=2
is desar
Then each involu
tion in G lies in G(P, l), as otherwise it induces an odd permutation of l .
But IG(P, l)
I
Ill  2
=2
(mod 4).
This implies that the Sylow 2subgroups of G contain only one involution.
So they are cyclic or quaternion groups,
and Proposition 3 . 6 implies that G ~ SL(2, q) for a suitable prime power q .
As in 4 . 1 this finishes our proof.
REFERENCES 1. 2. 3. 4. 5.
6. 7. 8. 9.
R. Baer. Projectivities with fixed points on every line of the plane . Bull . Amer. Math. Soc. 52 (1946): 273286. D. Gorenstein and J . H. Walter. The characterization of finite groups with dihedral Sylow 2subgroups. I . J. Algebra 2 (1965) : 85151 . c. Hering. Zweifach transitive Permutationsgruppen, in denen 2 die maximale Anzahl von Fixpunkten von Involutionen ist . Math . z. 104 (1968): 150174 . On projective planes of type VI. To appear . C. Hering and W.M. Kantor. On the LenzBarlotti classification of projective planes. Arch . Math . 22 (1971): 221224. C . Hering, W.M. Kantor, and G.M. Seitz. Finite groups with a split BNpair of rank 1. J. Algebra 20 (1972) : 435475. B. Huppert. Endliche Gruppen I . BerlinHeidelbergNew York: Springer Verlag (1967). T.G. Ostrom. Double transitivity in finite projective planes. Canad. J. Math. 8 (1956): 563567. I. Schur. Untersuchungen iiber die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen. J. Reine Angew. Math. 132 (1907): 85137. 112
10. 11.
H. Wielandt. Beziehungen zwischen den Fixpunktzahlen von Autornorphisrnengruppen einer endlichen Gruppe. Ma th • Z• 7 3 ( 19 6 0 ) : 14 6 15 8 • H. Zassenhaus. Kennzeichnung endlicher linearer Gruppen als Perrnutationsgruppen. Abh . Math . Sern. Univ. Hamburg 11 (1936): 1740.
113
SOME RECENT RESULTS ON INCIDENCE GROUPS
H. Karzel
1.
INTRODUCTION
For the study and foundation of absolute planes, it is useful to consider the associated kinematic space which is also called a group space.
To each line X of an abso
lute plane there corresponds exactly one line reflection X fixing all points of X and preserving the structure of the absolute plane (compare [12], section 17).
Let be
the set of all line reflections, r := the group generated by all line reflections, 0 2 := {XY : X, YE D}
(we de
note the line and the line reflection by the same letter), and J := {yE r: y 2 = 1, y # l } .
If a Er, let~ := {X ED:
a.XE J}; when aE 0 2 \ {l}, the set~ is called a pencil. L be the set of all pencils.
Let
Two pencils~, SE Lare call
ed joinable i f ~ n ~#~,and a pencil which is joinable to every other pencil is called a proper pencil . L 0 denote the set of all proper pencils.
Let
The pair r, Dis
then a reflection group; that is, r, D fulfills the following four axioms ([12], section 19):
114
(S1)
Dis a system of generators of r.
(S2)
If~ E Land X, Y, Z E
(S3)
L0
(S4)
For each XE D, the cardinality
'cl,
then XYZ ED.
°I ¢.
!{a
E Lo: XE
a}!
°I 1, 2. If r is a reflection group, one can prove that D 2 is a subgroup of rand that, for each a E D 2 \ {l}, the set ~ 2
:= {XY: X, YE
a}
forms a commutative subgroup of D 2
•
The kinematic space associated with a reflection group r, Dis defined as follows: the point set o~ the kinematic space is the subgroup G := D 2 and the set of lines is G := {a•i 2 : a, BE D 2 ,
B I l}; this means that each line
is a subgroup or a coset of D 2
•
One proves that the pair
(G, G) is an incidence space (see section 2).
Therefore
the set G is provided at the same time with a group structure and an incidence structure.
Both structures fulfill
the following compatibility conditions: V9.,
G For each y E G, the mapping Y9.,: {G+ E;, ➔
yf;,
is an automor
phism of the geometric structure (G, G). V r
For each y E G, the mapping y
r
:
{G+ G E;,
➔
E;,y
is an automor
phism of the geometric structure (G, G). Vk
Each line X
REMARK .
E
G with 1 E X is a subgroup of G.
The kinematic space of an absolute plane is a
dimensional) projective space or a
(3
(31)slit space (see
section 2) if and only if the incidence structure of the
115
absolute plane is a projective plane or an affine plane (see [3),
[15) respectively); such an absolute plane is
called a (generalized) elliptic or euclidean plane, respectively. This situation led us to introduce the concept of an incidence group [1] : A triple (G, G, • ) is called an incidence group if (G, G) is an incidence space,
(G,
•) is a group, and the
compatibility condition vi is valid . This paper will be a continuation of the reports [4] and [11) on incidence groups.
In section 2, we give the
geometric notations used and, in section 3, the algebraic ones .
For the algebraic description of the class of punc
tured affine incidence groups, G. Kist has introduced the concept of a nearfield extension, which is a generalization of the normal nearfields used for the algebraic representation of projective incidence groups (4.1).
Many
of the known infinite proper nearfields F do not contain subnearfields K such that the pair (F, K) is a normal nearfield, but they do contain subnearfields such that they are nearfield extensions.
For instance, the Kal
scheuer nearfields (section 3) are nearfield extensions over lR and over~, but not normal. Examples of incidence groups are obtained from algebraic structures (section 4), for instance , from associative unitary algebras . An important class consists of the 116
kinematic spaces which are derivable from kinematic algebras.
In section 5, we state the representation theorem
for punctured affine incidence groups by nearfield extensions of G. Kist. A type of generalized incidence groups are the socalled 2incidence groups (section 6), which can be described by pairs of neardomains.
The case in which the
underlying geometric structure is an affine space or plane will be investigated. Finally a geometric characterization of projective kinematic spaces is briefly discussed in section 7. 2.
GEOMETRIC CONCEPTS
Let P be a set and Ga subset of the power set of P; the elements of P will be named points and the elements of G lines. (Il)
The pair (P, G) is called an incidence space if: For each pair of distinct points p, q E P, there exists exactly one line A E G such that p, q EA; this line will be designated by p, q.
(I2)
For each XE G, the cardinality
!xi
is at least 2.
For the following, let (P, G) be an incidence space. A subset Tc Pis called a subspace if t implies t
1 ,
t
2
c T.
1 ,
t
2
ET, t
1
# t
Let T be the set of all subspaces.
Then Tis closed with regard to intersections and therefore the closure of a subset Mc P defined by
117
M
:= n{T:
2
is a subspace.
Mc TE T}
inf{
IMI: M
For TE T we define by dim T :=
T}  1 the dimension of a subspace.
of dimension 2 are called planes. G(p)
Subspaces
For a point p E P let
:= {X EG: p EX} and for a subspace TE T let G(T)
:={XE G: X
c
T}.
called an mline if:
A line LEG is
Let m E lN u {O}. (1)
ILi
~
3 if m = 0, and (2) for
each XE P\L, there exist exactly m distinct lines H1 , •••I
H
m
E G(x) n G({x} u L) such that H. n L =¢for i
i E {l, 2, •.. , m} and each XE G({x} 1 for all i
u
E {l, ... , m} satisfies X
Ix
L) with n
L
~
¢.
n
Hil
Let G
m
be the set of all mlines. REMARKS.
An incidence space (P, G) with G = G0 is a pro
jective space.
If G = G1 we call (P, G) a pseudoaffine
space; if in addition the relation on G defined by "A
II B A = B or A n B = ¢ and dim AuB = 2" is tran
sitive, then (P, G) is an affine space.
This last condi
tion can be proved if there is a line A E G with IAI
~
4.
(Compare [ll), pp. 7981). For every subset Uc P we define the trace space (U, G ) by G u
u
: = { X n U: X E G,
IX n U I
space is again an incidence space.
~
2}.
The trace
If (P, G) is a pro
jective space and La subspace of (P, G), then the trace space (P\L, G(P\L)) is called a slit space and if dim P = n and dim L = r it is called an (n  r)slit space. An internal characterization of slit spaces is given in
ll8
[11].
If (P, G) is a pseudoaffine (affine) space and q
a fixed point of P, then the trace space (P\{q}, G(P\{q})) is called a punctured pseudoaffine (punctured affine) space.
Each line of (P\{q}, G(P\{q})) is either a 1line
or a 2line.
G. Kist [13] has given the following internal
characterization
an incidence space (P, G) is a punctured
pseudoaffine space if G = G1 u G2 and if the conditions (Pl) and (P2) hold, and a punctured affine space if (P3) also holds: (Pl)
Let Ebe a plane with G 2 (E) for each point x E E,
(P2)
:= G(E) n G2 f ¢; then
IG(x) n G2 (E) I = 1.
To every line A E G1 there exists a line BE G2 such that A II B (that is, A n B = ¢ and dim
= (P3)
2).
The relation 11
1
:= II
n
(G 1 x G1 ) is transitive
or there is a line A E G with IAI 3.
AuB
~ 4.
ALGEBRAIC CONCEPTS (NEARDOMAINS, NEARFIELDS, NEAR
FIELD EXTENSIONS) A set (F, +, ·) provided with two operations+: F x F and•: F x F
➔ Fis
➔
F
called a neardomain [5], p. 123) if
the following conditions hold: (Fl)
(F, +) is a loop (0 denotes the neutral element) such that a, b E F and a+ b = 0 implies b +a= 0.
(F2)
(F*, •) with F* := F\{0} is a group.
119
(Fl)
For all a, b, c
(F4)
For every a
(FS)
For all a, b for all
X
E
F, a• (b + c)
E
a•b + a•c.
F, 0•a = 0.
E E
F* there exists d
F, a + (b + x)
A neardomain
=
(FI
(F, +) is a group.
+,
.)
=
a,b
E
F such that
(a + b) + d
a, b·x.
is called a nearfield if
A set (F, +, o, •) provided with three
binary operations is called a dicksonian nearfield (or regular nearfield) (2)
[2] if (1)
(F, +, o) is a nearfield,
(F, +, ·) is a field, and (3) for all a
ping ~a: F
+
F; x
+
a 1
•
(a ox)
F* the map
E
(a 1 denotes the inverse
with respect to·) is an automorphism of the field (F'
+' . ) .
REMARKS.
(1) We know that every finite neardomain is a
nearfield but we do not know at present whether there are neardomains which are not nearfields.
(2) With the ex
ception of seven finite nearfields, all known nearfields are dicksonian nearfields. A pair (F, K) is called a normal nearfield if Fis a nearfield, Ka subnearfield~ F, and K*
~
F* [1].
As H. Wahling [17] proved, K is even a field and (F, Kl a left vector space . Normal nearfields are used for the algebraic representation of desarguesian projective incidence groups [1], [4] .
For the description of desarguesian punctured affine
incidence groups, G. Kist [13] introduces the concept of a nearfield extension: 120
A quadruple ((F, +, o), of a nearfield (F, +, (F, +, o), a mapping• a+~
a
0 )
(K, +,
0 ),
·,
~)
consisting
a subnearfield (K, +,
o)
of
: F x K + F, and a mapping~
in the symmetric group of the set K is
called a nearfield extension if the following conditions are valid: (El)
• (K x K) =: K•K
c
K and (K, +, •) is a field such
that the identity 1 0 of (K,
0 )
and 1. of (K, •)
coincide. (E2)
( (F, +),
(E3)
For all a, b a
o
(b•>..)
(K, +, ·), ·) is a right vector space. €
= (a
F*, and for all O
A EK
holds
b)·~a(>..).
In [13] Kist proves the properties (3.1) of nearfield extensions and shows how one can get examples of nearfield extensions. (3.1)
Let (F, K, ·, ~) be a nearfield extension, then: (a) For every a E F* the mapping~
a
is an automor
phism of (K, +, ·). (bl (3.2)
(K, +, o, •) is a dicksonian nearfield.
Let (F, +,
every y, z E F,
o)
be a nearfield, KF := {x E F: for
(y + z)x = yx + zx} the kernel (the kernel
of a nearfield is always a field!),
•:=
0
IF x KF, and~
the function mapping each a E F* on the identity of SK. Then (F, KF, ·, ~) is a nearfield extension.
121
F
The same
is true if His a subfield of KF. REMARK.
With (3.2) we can get examples of nearfield
extensions from the Kalscheuer nearfields : be the quaternions over the reals IR, c
f ]H l X where
E
Let (Ill,+,
·)
IR*,
+ IlI +
e ic log(a • a)
•X•e
ic log(a • a)
denotes the involutorial antiautornorphism
of (Ill,+,·)
(a·a is an element of m::= {r
E
1R:0 < r})
and IlI
1H
X
]H
➔
~ { a ob
(a, b)
~
0
if
a/ 0,
if
a = 0.
Then (Ill,+, o) is a Kalscheuer nearfield. (IH, +,
0 )
The kernel of
is the field of complex numbers(«:,+,
·)
(in
a: := 1R + iIR the multiplications o and • coincide). Therefore ((Ili, +, o), (1R,+,
o),
o, 1/J") with
(«:, +, o), o, 1/1') and ((Ii,+, o),
1/1~ := 1/Jal«:(=ida:) and 1/1; := 1/JallR
(=idlR) are nearfield extensions. (3.3)
Let ( (F, +, 0) ,
.:
F
{ (a,
X
K
;q
(K, +, 0) ) be a normal nearfield,
➔
F
➔
a•\ := \oa
and
122
1/!:
l
F*
....
SK
a
....
1/!a :
Then ( (F, +, 0) ,
:
{
....
K
....
>,_a = ao>.. oa 1
(K, +, 0) ,
.
is a nearfield extension
ljl)
and ((K, +, • ) is an antiisomorphic field to (K, +, REMARK.
0 )).
(3 . 3) shows that every normal nearfield can be
considered as a nearfield extension.
However there are
nearfield extensions which are not normal nearfields. For instance, the Kalscheuer nearfields do not contain any subnearfield K such that ((Ill,+, o),
(K, +,
0 ))
is
a normal nearfield. Let (F, +, o, •) be a dicksonian nearfield and
(3.4) (K,
+, o) be a subnearfield such that a
all a E F* and all A EK . with
®
= . IF
X
K and ljJ :  a
1
Then ((F, +, o),
{
K
....
K
>,_
....
a
1
• (ao>..)
EK for
(K, +, o),
@,
· (ao>..)
is a nearfield extension. 4.
INCIDENCE GROUPS
An incidence group (G, G, ·) is called 2sided if both
v1
and Vr are valid, and linearly fibred or a kinematic space if v 1 , Vr, and Vk are valid.
If the incidence structure
(G, G) is for instance a projective, affine, or slit space we call (P, G, •) a projective, affine, or slit incidence group.
123
ljl)
Examples of incidence groups 1.
Let (V, K) be a left vector space and G :={a+ Kb:
a, b EV, b i O}; then the pair (V, G) is an affine space and the pair (V, +)where+ is the vector addition is a commutative group. at: V
➔
V; x
➔ a+
Since
for each a EV the mapping
xis bijective and maps lines on lines,
(V, G, +) is a commutative affine incidence group. 2.
Let (A, K) be an associative algebra over the field
K, such that K is in the centre of A and let Ube the set of all units of A.
Now we can apply an affine and a pro
jective derivation: (a) By the affine derivation, let G :={a+ bK: a, b EA, b i O} and (U, G) be the trace space (section u 2) belonging to the set of units.
Then (U, Gu' ·) is a
2sided incidence group, since U is a group with respect to the multiplication given in the algebra A.
In the
special case that (A, K) is a division algebra, we have U = A* := A\{O} and (A*, GA, ·) is a punctured affine incidence group (as they will be discussed in section 5). (b) For the projective derivation, if A*/K* := {K*·x: x EA*} where A* := A\{O} and K* := K\{O}, let l
2
be the
set of all 2dimensional vector subspaces of (A, K) and X :
{
A*
X ➔
➔
A*/K*
the canonical map.
K*x
Then (A*/K*, G) with G := {x(L*): LE l
124
2 }
is the projective
space corresponding to the vector space (A, K) and since K* ~Uthe set x(U)
= U/K* is a group.
If we again take
the trace space as incidence space, then (U/K*,G U/K*' ·) is a 2sided incidence group. a division algebra, group.
In the case that (A, K) is
(A*/K*, G, •) is a projective incidence
We obtain examples of kinematic spaces from kine
matic algebras [7]:
(A, K) is a kinematic algebra if for
each x EA the element x 2 belongs to the set K + Kx.
The
quaternions Ill over the reals :rn. constitute a kinematic division algebra and the corresponding kinematic space is the same, which is associated with the classical elliptic plane.
The kinematic space of the classical euclidean
plane can be derived from the fourdimensional algebra (A, m) with basis {l, i,
£,
id and i
2
=
1, e: 2
=
O,
ie: = e:i. The projective incidence groups are the most intensely studied ones [4].
The main theorems on projective inci
dence groups are the following: (a)
(4 .1)
Let (F, K)
be a normal nearfield (section 3),
G := F*/K* and G := {x(L*): LE L 2 } , then IT(F, K) (G, G,
:=
·) is a desarguesian projective incidence group
[ 1, 4 l •
(b)
Let (G, G, ·) be a desarguesian projective in
cidence group, then there is exactly one normal nearfield (F, K) such that (G, G, •) and IT(F, K) are isomorphic [ 1, 4 l •
125
(4.2)
Let (G, G, •) be a desarguesian projective incidence
group and (F, K) the corresponding normal nearfield; then (a)
(G, G, ·) is 2sided if and only if F is a field
(b)
(G, G, •) is a commutative incidence group if
(16].
and only if Fis a commutative field [4]. (c)
(G, G, •) is a kinematic space if and only if
(F, K) is a kinematic division algebra.
If G is not com
mutative, then (F, K) is a quaternion division algebra (this means that K is the centre of F and [F: Kl= 4).
If
G is commutative, then K is a commutative field of characteristic 2 and Fa pure inseparable extension of K (6]. Similar results on slit incidence groups can be found
in [11]. 5.
PUNCTURED AFFINE INCIDENCE GROUPS
The punctured affine incidence groups (G, G, •)  here (G, G) is a punctured affine space (section 2)  were recently studied by G. Kist in his dissertation [13].
He
got the following main results: (5.1)
(a) Let ( (F, +, o),
(K, +, o), o,
field extension (section 3) with K {a+ b•K: a, b
E
~
~2
F*, a J b·K} u {bK*: b
be a near
ljl)
F
,
E
~
K, and G :=
F*}; then
(F*, G, o) is a desarguesian punctured affine incidence group with G1 ={a+ b•K: a, b
126
E
F*, a J b·K} and G2 =
{b•K*: b (b)
E
F*}. Let (G, G,
o)
be a desarguesian punctured affine
incidence group; then there exists exactly one nearfield extension (F, K,
~) such that (G, G,
o)
and the inci
dence group derived from the nearfield extension are isomorphic and the groups (F*, REMARK.
o)
and (G,
o)
coincide.
In section 3 we have seen that every normal near
field can be considered as a nearfield extension and that there are nearfield extensions which do not come from normal nearfields.
Those punctured affine incidence
groups which can be represented by normal nearfields or even by field extensions are characterized by G. Kist (13). 6.
2INCIDENCE GROUPS
In this section we shall consider the following generalization of incidence groups, which are closely related to the punctured affine incidence groups: A tripel (P, G, f)
is called a 2incidence group if
(1)
(P, G) is an incidence space.
(2)
(P, f)
is a sharply 2transitive group (this
means that, for each two pairs (a 1 , a with a y(a
1
1)
(3)
'I a 2 , b 1
:/
b2
,
bl' y(a 2
)
=
b
Each
y
E
r
2 ),
(b 1 , b 2 )
there is exactly one
y
E
E
P x P
r such that
2 •
is an automorphism of the incidence
space (P, G).
127
In this section, we say that two lines X, Y €Gare parallel, denoted by X II Y, if X = Y or if dim and X n Y =
¢.
XuY = 2
An automorphism a of (P, G) will be called
a dilatation if, for all
X
e G, a
(X)
II
X
and a translation
if a= 1 or if a is a dilatation without fixed points. To every 2incidence group, there corresponds a usual incidence group: (6.1)
Let (P, G, r) be a 2incidence group, a, e € P with
(:= the trace space {X a 'I e, p a := P\{a}, G : = GP a a r := {y e r: y(a) Xe G, Ix n P I ~ 2 ), and a}. a
P:
n
a
Then
a
ra operates regularly on Pa: hence, for each x € Pa there is exactly one tion
x era
such that x(e) = x.
The multiplica
defined by {Pax Pa (x, y)
➔
p
a
➔ x(y>
makes P
a group, such that (P, G, •) is an incidence a      a a
(6.2)
Each desarguesian punctured affine incidence group
~
(G, G, ·) can be extended to a 2incidence group (P, G, r) such that for any a, e e P, a ,f e, the incidence group (Pa' Ga,
·) and (G, G, •) are isomorphic.
Let ((F, +, o},
~K, +, o), ·, ~) be the nearfield extension corresponding to (G, G, ·), G :={a+ boK: a, b € F, b 'IO} and r
:=
{[a, b]: a, be F, b 'IO} the sharply 2transitive group consisting of all mappings
128
[a, bl + box,
G,
then (F,
= (F*,
G,
f) is a 2incidence group extending (G, G, ·)
o).
For the proof, it is sufficient to remark that condi
(E3)
tion
of a nearfield extension implies that boK =
b•K. Let (P, G, f) be a 2incidence group, KEG and
(6.3)
rK := {y Er: y(K) = K}.
Then (K, rK) is a sharply 2tran
sitive group. PROOF.
Let (a 1
,
a 2 ),
(b 1 , b 2 ) E K
x
K with a
b 2 , and y Er uniquely determined by y(a 1 ) b2 .
Then K = a
1,
I a 2, b 1 I
1
= b1,
y(a 2 )
=
a2 = bl, b2 and y(K) = y (al), Y (a2) =
bl, b2 = K, soy E r K
.
For the further discussion of 2incidence groups, we need the following theorems

from the theory of sharply
2transitive groups stated for instance in section 11 of
[ 51 : (6.4)
Let (F, +, •) be a neardomain; for each pair (a, b)
E F x F*, let [a, bl be the mapping [a, bl: F a+ b·x and r := {[a, bl: a, b E F, b IO}.
+
F; x
+
Then (F, f)
is a sharply 2transitive group. (6.5)
Let (P, f) be a sharply 2transitive group, J :=
129
{y
f:
E
y
1, y ;
2
l} and 0, 1
E
P two distinct points,
then: (a) a
E
For all p, q
=
J such that a(p) Let a, 8
(b)
E
E
P with p ; q, there is a unique
q and a(q)
J, a
i 8;
=
p.
then aS is free of fixed
points. (c)
If one a
J has a fixed point, then each
E
has a fixed point and to each point x ponds exactly one x
+
J
E
P, there corres
E
J fixing x.
There exist two operations+: P
(d)
•: P x P
E
1
x
P
+
P and
P such that (P, +, •) is a neardomain (section
3); 0 (resp. 1) is the neutral element with regard to+ (resp.
•);
r
consists of all mappings [a, bl: P
a+ b•x with a, b each x
E
E
P and b
t 0
and
r0
~
+
P; x
(P*, ·).
For
+
P* := P \ {0}, let x+ be the involution interchang
ing O and x; then+ is given by +A a
+ b :=
C
O (b)
if a
t
0
if a = O
or
i
if
a
if
a= 0
0
according as the involutions of J have fixed points or not. (e)
The neardomain (P, +, .) has the characteristic
2 (that means 1 + 1
=
0) if and only if the elements of J
have no fixed points.
130
By Theorem (6.5) the sharply 2transitive groups (P, r) decompose into two classes depending on whether each involution
1
E J
has a fixed point or not.
A 2in
cidence group (P, G, r) is said to be of type 1 if each 1
E
J has a fixed point; otherwise it is of type 2.
From (6.3) and (6.5) we obtain: (6.6)
Let (P, G, r) be a 2incidence group, 0, 1
distinct points, K : = ~ . a n d (P, +,
P two
E
•) the neardomain
associated with (P, r) whose neutral elements are O and 1. Then
r
consists of all mappings [a, bl
with a, b
E
P, b
:P
➔
P; x
➔
a+
b•X
I 0, K is a subneardomain, and each
line XE G can be represented in the form X =a+ b·K with a, b
E
P and b
I 0.
The last statement follows from the fact that
r
is
2transitive on P and, hence, it is transitive on the set
G of all lines. Conversely we have: Let (F, +, ·) be a neardomain,
(6.7)
subneardomain of (F, +,
•),
(K, +,
•) a proper
(F, r) the corresponding
sharply 2transitive group (see (6.4)) and G :={a+ b•K: a, b
E
F, b i O}.
PROOF.
= u + •v•b·K
Then (F, G, r) is a 2incidence group.
Let X := a + b·K v(a + b·K) E
E
G
and [u, v)
E
r,
then [u, v) (X)
u + (v•a + v•b·K) = (u + v•a) + d
G (see section 3); this shows that
131
r
u,v•a
preserves
the structure (F, G).
Let a, b E F be two distinct points.
By section 3 we obtain a+ (a+ b)•l =a+ (a+ b) = 0 + d
a,a
·b = b.
joining a and b.
Therefore X :=a+ (a+ b) • K is a line By the sharp 2transitivity of r, it is
enough to show that K is the unique line joining the points 0 and 1.
Let 0, 1 Ea+ b•K; then there is A EK such
that O =a+ bA.
Thus a= b•A = b(A) and a+ b•K = b(A)
+ b·K = b• (A+ K) = b•K.
Since 1 Ea+ b•K = b · K implies
the existence ofµ EK with 1 = b•µ, we see that b = µEK and therefore a+ b•K = b·K = µ 1 •K = K.
1
Each line
a+ b•K E G contains (at least) the points a and a+ b. Therefore (F, G) is an incidence space and (F, G, r) a 2incidence group. By the Theorems (6.6) and (6.7) we see that there is bijective correspondence between all 2incidence groups (P, G, f) and all pairs ( (F, +, • ) ,
(K, +, ·) ) of near
domains, where K is a subneardomain of F.
We shall now
study this correspondence for the class of 2incidence groups, where (P, G) is an affine space. First we need the following lemma: (6. 8)
Let (P, G, f) be a 2incidence 9:roue of t:r:ee 1.
Then: (a) For all x E p and all
X
(b) For all x E p and all y E G with =
cp.
132
A
E G with x E x, x(X) A
X
J Y, x(Y)
x. n y
(c) For all x
E
P, the involution xis a dilatation.
PROOF.
(a) From (6.3),
rx> fore Jx
is a sharply 2transitive group of type 1.
(X,
= {y: y
tions of (b)
rx
€
X}
(6.5), and (6.6) it follows that There
(see (6.Sc)) is the set of involu
and x(X) = X as required.
Suppose, if possible, that z
ix, ~(x, z) = x;"z by (a), and x;"z A
{z} so x(z) = z.
€
n
y n ~ (Y) ; then z
Y
= x, z n ~(Y)
=
Since x thus has the two distinct fixed
points x and z, the sharp 2transitivity implies that~=
x
1, contradicting
E
J.
(c) The statement (c) is a consequence of (a) and (b), since by (a) any plane E containing xis fixed by (6. 9)
x.
Let (P' G, r) be an affine 2incidence groue with 3 for x
IXI
~
(K'
+, 0)) are an associated eair of neardomains and
€
T := J.J =
G and dim P
{xy:
x, y
€
~
2.
sueeose ( (P = F, +, 0) '
p}
or T := J u { l}
according as (P, G, f) is of tyee 1 or of tyee 2.
Then:
(a) Tis a groue consisting of translations which oeerates regularly on the set of eoints and Tis isomorehic to (F, +), and
if (t', G, f) is either of type 1 or of type 2
dim P = 2. (b)
(F, +,
nearfield, if
0 )
T
is a nearfield and (K, +, o) a subis a group.
133
(c)
is a translation plane
If dim P = 2, then (P, G)
and [F: K] = 2 (that is, F = K + aK for a E F\K). (d)
If (P, G) is desarguesian and Ta group, then
there exist two mappings •: F x K that ((F, +,
0
),(K, +,
0
),
➔
F and~: F*
SK such
• , ~ ) i s a nearfield extension.
We prove first the statements (a), incidence groups
➔
(c) for 2
(b),
(P, G, f) of type 1.
Results (6.8c) and (6.Sc) imply that Tis a set conLet a, b E Pandy E J be the
sisting of translations.
involution interchanging a and b, so ya ET and ya(a) = b. Since by assumption (P, G) is an affine space of dim P
~
2,
this implies that Tis a commutative group and (P, G) a From (6.Sd), we obtain T
translation space. and therefore (F, +,
0
)
is a nearfield.
Now let (P, G, f) be of type 2 and l
and X E G with
EJ
would imply
dim P = 2
because Hence
l
l (X)
{1(a)} = 1(X)
assumption, that
l
(P = F, +)
~
" X .
Then {a}= X
X = {a}
n
l
(X) II X
n
Let
l(X)
contradicting the
has no fixed points. we have
dim P =2.
Therefore and
for all
X E G .
is a translation.
Since any two distinct points can be interchanged by an involution,
(P, G) is a translation plane and T =Ju {l}
is the group of all translations. T
~
From (6.Sd) we derive
(F, +) and (F, +, o) is a nearfield. It remains to prove statement (d) for both types.
Theorems (6.1) and (6.5), the incidence group (F*, G 0
134
,
By 0 ),
with F* = F 0 = F \ {O} and G0 : = {X \ {O}: XE G} corresponding to the points 0, 1
F, is a desarguesian punctured
E
affine incidence group and (F*,
o)
=r
0 •
Applying the
theorem (5.1) of Kist there is a nearfield extension
((F,
$,
that I!>= Problem:
), o.
(K, +, ),
•,
1/J)
representing (F*, G 0
Statement (a) finally implies that$=+ .
(P, G, f)
Let
of type 2 with
dim P
be an affine 2incidence group and
3
be a projective double space
Then there is a quaternion divisionr• algebra (F, K) such that (P, G, 11 i, 11 r) is derivable from 11
the kinematic space associated with (F, K)
(in the sense
of section 4, Example 2(b) or Theorem (4.1)).
140
(7.6)
Let (P, G, Iii, llr) be a projective double space
with
Iii=
llr and dim (P, G) = 3.
Then there is a com
mutative field K of characteristic 2 and a pure inseparable extension field F of rank [F : Kl = 4 such that (P, G, Iii, llr) is derivable from the kinematic space associated with (F, K) •
REMARK.
Every pure inseparable extension field F of a
commutative field K of characteristic 2 produces examples of projective double spaces with
Iii=
llr•
In the meantime H.J . Kroll has proved in [14a] that there are no other examples of projective double spaces II R. = II r
with
Are there any (projective) incidence groups
Problem: (G,
G,
•
.)
with
(that is, where
vi , Vk
which are not kinematic spaces Or in the algebraic
is not valid)?
V r
language for the projective case:
besides the kinematic
division algebras, are there normal nearfields such that element
[F, K]
x2
~
3
and, for each
is contained in the set
x
E
F\K,
(F, K) the
K + Kx ?
H. wahling has recently solved this problem for projective incidencegroups: implies
He proved that
vr •
141
Vi
and Vk
REFERENCES 1.
E. Ellers and H. Karzel. Kennzeichnung elliptischer Gruppenraume. Abh. Math. Sem. Univ. Hamburg 26 (1963): 5577.
2.
Endliche Inzidenzgruppen. Hamburg 27 (1964): 250264.
3.
H. Karzel. Verallgemeinerte elliptische Geometrien und ihre Gruppenraume. Abh. Math. Sem. Univ. Hamburg 24 (1960): 167188.
4.
  Bericht uber projektive Inzidenzgruppen. Deutsch. Math. Verein. 67 (1964): 5892.
5.
Inzidenzgruppen I. Vorlesungsausarbeitung von I. Pieper und K. Sorensen, Hamburg (1965).
ti.
Zweiseitige Inzidenzgruppen . Univ. Hamburg 29 (1965): 118136 .
7.
Kinematic spaces. Istituto Nazionale di Alta Matematica, Symposia Matematica 7 (1973): 413439.
8.
H. Karzel and H.J. Kroll. Eine inzidenzgeometrische Kennzeichnung projektiver kinematischer Raume. Arch. Math. 26 (1975): 107112 .
9.
H. Karzel, H.J. Kroll, and K. Sorensen. Invariante Gruppenpartitionen und Doppelraume. J. Reine Angew . Math. 262/263 (1973): 153157.
Abh. Math. Sem. Univ.
Jber.
Abh . Math. Sem .
10.
Projektive Doppelraume. 206209.
11.
H. Karzel and I. Pieper. Bericht iiber geschlitzte Inzidenzgruppen. Jber. Deutsch. Math.Verein. 72 (1970): 70114. H. Karzel, K. Sorensen, and D. Windelberg. Einfilhrung in die Geometrie. UTB 184, Gottingen (1973).
12. 13.
Arch. Math. 25 (1974):
G. Kist. Punktiertaffine Inzidenzgruppen und Fastkorpererweiterungen. Abh. Math. Sem. Univ. Hamburg 44 (1975).
Zur Struktur geschlitzter Doppelraume. H.J. Kroll. J. Geometry 5 (1974): 2738. 14a. H.J. Kroll. Bestimmung aller projektiven Doppelraume. Appearing in Abh. Math. Sem. Univ. Hamburg. 15. H. Meissner. Geschlitzte Gruppenraume. Abh. Math. Sem. Univ. Hamburg 32 (1968): 160185. 14.
142
16.
H. Wahling. Darstellung zweiseitiger Inzidenzgruppen durch Divisionsalgebren. Abh. Math . Sern . Univ. Hamburg 30 (1967): 220240.
17.
Invariante und vertauschbare Teilfastkorper. Abh. Math. Sern. Univ. Hamburg 33 (1969): 197202.
143
TOPOLOGICAL HJELMSLEV PLANES
J.W. Lorimer
Hjelmslev planes are generalizations of customary projective and affine planes where two distinct points may be joined by more than one line and two distinct lines may meet in more than one point.
Two points P and Qare neighbours
(P ~lP Q) when they possess more than one joining line. Two lines land mare neighbours (projectively) if they intersect in more than one point and are neighbours (affinely)
if each point incident with either l or m possesses
a neighbour point incident with the other of l or m.
In
these cases we write l ~L m. An incidence structure H
=
is said to be
a topological Hjelmslev plane if His a Hjelmslev plane (cf.
[2],
[4]) with the following properties:
(i)
lP and Lare topological spaces.
(ii)
lP and
~L are closed sets of IP x lP and
L x L
respectively. (iii) The joining of two nonneighbouring points is a continuous function; and the meeting of two lines with
144
a unique intersection point is a continuous function. Moreover if the plane is affine we also assume the additional property: (iv)
L(P, g), the unique line through P parallel to
g, is a continuous function of P and g. Examples of such planes can be constructed over certain topological rings.
A local ring with its unique
maximal ideal equal to its set of twosided zero divisors is an Hring if for any two elements a and bat least one is a multiple (left and right) of the other (cf.
(2)).
Such a ring is a topological Hring if it is a topological ring whose units form a multiplicative topological group. The subring Dn of the (n x n) matrices over a topological division ring F consisting of matrices of the form (dij) where dij
=
O if i
Hring (cf.
>
j and di,i+kl
=
dl,k is a topological
[ 1)) •
In fact from the constructions in (2),
(3), and (4)
we have the following result. THEOREM 1.
Every desarguesian Hjelmslev plane can be co
ordinatized by a topological Hring and conversely. If His a Hjelmslev plane, then the "quotient geometry" H/~
=
1
=
l.
f: (X, Ri)
+
(Y, Si}
(i EI) is a mor
Y is a function and for all i (f(x), f(x'))
EI
E R .• l.
we shall refer to R(2i)iE{l} as R(2) and call it the category of graphs.
The elements of R . are called i1.
(coloured)arrows.
We shall say "f preserves R." if f: l.
x + y is a morphism from (X, Ri) to (Y, Si) in R(2).
A
loop of R. is a point x such that (x, x) l.
E Ri' an isolated
point of R . is a point such that y(x, y) l.
I Rl.. and (y, x}
I Rl.. ; a twocycle in Rl.. is a pair (x, y) such that (x, y) and (y, x) E R. Aut(X, R.) . I = {fl f and f1 E Hom (X, Ri)iEI' l. l.E (X, R.). I)}; l. l. E
(X, Ri)iEI is rigid if Hom (X, R . ) . 1 (X, R.). I l. 1.E l. l.E
176
Stab (
'.: : 1 • X
=
X
X,y
{f I f Aut(X, R. ) . If(x)
) ( X, R. ) . I i
iE
i
iE
f(y) = y}.
THEOREM 5 (CayleyFrucht).
Let G be a group and (X, R) g gE: G
be a graph with IGI colours defined by X = IGI and (g 1
,
g2)
LEMMA 1.
Let G be a group.
Then there exists a multi
coloured graph (X, R.) . I such that i
(i)
iE:
Aut(X, R.). I " G. i iE:
(ii) For N < G (including N = 1 and N = G) E:
(x, y)
Ri(N) such that Stab(x,y)((X, Ri)iEI) "N .
CONSTRUCTION.
Let I= {a, b, c}U{G x N} where N
(N = 1 and N = G included)}.
{N I NLIG
Let
G G X ={NI N < G} u• {Hg I Hg€ H' H < G},
Ra= {a rigid graph on{~ IN< G}}, G Ng) Rb= {(N,
Rc
I
G Ng E: N},
{(Ng, Kh) INg ~ Kh and N ~ K},
R(N,g) = {(Nh, Nh') LEMMA 2 .
I Nh
=
Nh'g}.
If G is a multicoloured graph, then there exists
a graph G1 with the same automorphism group, and the effect of stabilizing an arrow of G can be achieved by stabilizing an arrow of G 1 [4]. THEOREM 6 .
If G is a group, then there exists a graph
(X, R) such that A(X, R) "G, and if a~ ~o is a cardinal IXI = a!GI
(31,
(41.
177
THEOREM 7.
If G., i

l.
€
I is a wellordered sequence of
groups, then there exist graphs (Xi, Ri) with X. l. R.
C
l.
+
,
X., J
and A(X . , Ri) " G. [ 8] • Rj, i < j , l. l.
THEOREM 8. (X 1
C
If G 1 , G 2 are groups then there exist graphs
R 1 ) and (X 2
,
R 2 ), and an arrowonto map~:
(X 1 , R 1 )
(X2, R2) such that A(XI, RI) "Gl and A(X2, R2) "G2 (5). Furthermore by a technique found in [4] and [8] one
can assume that all graphs which appear in the above theorems have no loops, isolated points, or twocycles, and that each point is related to at least two others.
These
assumptions are for technical reasons which will reveal themselves in the particular construction and proofs we shall use. RESULTS NEEDED FROM PROJECTIVE GEOMETRY We shall need the following wellknown results from projective geometry: THEOREM 9.
The collineation group of Pi=oo is isomorphic
to the subgroup of the collineation group stabilizing i THEOREM 10.
[7].
Given a partial plane p there exists a free
completion of this plane to a projective plane [2]. THEOREM lL
If a partial plane is confined (i . e. contains
at least three points on each line and at least three lines
178
through every point) then the free completion has the same collineation group as the partial plane and furthermore the isomorphism is given by restriction [2]. The configuration V The configuration Vis given
by the following table where
the columns are lines and the entries points: A
C
E
A
B
D
D
C
E
A
B
B
N
0
F
K
N
0
K
M
K
N
C
L
L
G
L
K
M
G
G
0
0
D
H
F
H
M
F
H
E H
FIGURE 1 LEMMA 3 .
V has no collineations other than the identity
and is confined. The pasting of Von a graph v,
The process we shall use is analogous to the sip process [8].
Let G
=
(X, R) be a graph without loops, isolated
points, or twocycles and with each point connected to at least two others.
We define the configuration (G) in the
179
following way: The points of V(G): P(G) = {[{PD  {K, O}} The lines of
X
R] u
x}.
V(G): L(G)
The incidence of V(G): IG by: (i)
(p, r) IG (.Q,, s) r
=
s and p ID
s) s
=
(x, y), K ID
(iii) y IG (.Q,, s) s
=
(y I
(ii)
X
IG
( _Q,
I
z) ,
.Q,;
.Q,;
0 ID .Q,.
Intuitively, we place a copy of Vin place of every arrow of R pasting the points Kand/or O together whenever they correspond.
If we represent the configuration Vas in
figure 2 and paste it on the graph of figure 3 we get the configuration of figure 4.
D
0
K
FIGURE 3
FIGURE 2
THEOREM 12. If G = (X, R) is a graph such that G has no loops or isolated points, and every point is connected to at least two others, then: 1.
V(G) is a confined configuration.
2.
There is an isomorphism between the collineation 180
D
FIGURE 4 group of V(G) and the automorphism group of the graph G. 3.
There is a oneone correspondence between onto
maps from G to G* (where G* also has the properties given in the hypothesis) and onto homomorphism from V(G) to V(G*). 4.
Stabilizing an arrow of G corresponds to stabiliz
ing a line of V(G). Theorems 14 now follow from the analogous theorems for graphs and considering the plane F(V(G)). REFERENCES l. 2. 3. 4.
R. Frucht. Herstellung von Graphen mit vorgegeben abstrakter Gruppe. Composite Math. 6 (1938): 239250. M. Hall. Projective planes. Trans. Amer. Math. Soc. 54 (1943): 229277. z. Hedrlin, A. Pultr, and P. Vopenka. A rigid relation exists on any set. Comment. Math. V. Carolinae 6 (1965): 149155. Z. Hedrlin and J. Lambek. How comprehensive is the category of semigroups? J. Algebra 11, No. 2 (1969): 195212.
181
5. 6. 7. 8.
z. Hedrlin. On endomorphisms of graphs and their homomorphic images . In: Proof Techniques in Graph Theory, edited by F . Harary. Academic Press (1969). D.R. Hughes. On homomorphism of projective planes. Proc. Symp. Appl. Math. 10 (1960): 4552. R. Lingenberg. Grundlagen der Geometry 1. Bibliographische Institute, Zurich (1969). E. Mendelsohn. On a technique for representing semigroups as endomorphism semigroups of graphs with given properties. Semigroup Forum 4 (1972): 283294.
182
RECENT ADVANCES IN FINITE TRANSLATION PLANES
T.G. Ostrom
I.
GROUPS GENERATED BY GIVEN TYPES OF COLLINEATIONS
The work on rankthree affine planes, reported on by Luneburg at this same conference, is certainly a phase of the "recent advances" but I shall make no further mention of it here.
Most of what I have to say was reported on at a
Conference on Projective Planes held at Washington State University.
Copies of the Proceedings may be obtained
from the Washington State University Press.
My report in
those Proceedings contains what I hope is a complete listing of known finite translation planes and some of the properties of the collineation groups.
Since then I be
lieve that Rao has some more examples of flag transitive planes.
Prohaska and Walker have an example, not yet pub
lished, of case (e) in the theorem I shall soon be stating. I believe that the list is complete with these additions. Some of what I shall talk about is in my "Finite Translation Planes."
183
If
TI
is a translation plane of dimension rover its
kernel K, then the (affine) points of
TI
can be identified
with the elements of a 2rdimensional vector space over the field K; the lines through the zero vector Oare rdimensional vector subspaces and the scalar transformations are the homologies with centre O and axis t 00 •
The system
of lines through the origin is called a spread. Although some of what we have to say does not require finiteness, we shall assume that everything is finite . "Point" always means "affine point" unless we say "point at infinity." K
=
GF(q) .
The letter q will denote the order of K:
The stabilizer of O in the collineation group
is a group of semilinear transformations  a subgroup of fL(2r, q).
We call it the translation complement and its
intersection with GL(2r, q) the linear translation complement.
The characteristic of K will be denoted by p. The first result, I am told, is implied by some un
published work of John Thompson.
In the following form
it is due in part to Hering and in part to myself. THEOREM.
Suppose that the translation complement contains
(affine) elations.
Let S be the subgroup of the transla
tion complement which is generated by all of the elations. Then one of the following holds: (a)
Sis an elementary abelian pgroup and (if Sis
nontrivial) only one line through O is the axis of a nontrivial elation .
184
(b)
S
~
s
SL(2, p) for some s.
The net whose lines
through the origin are the axes of elations can be embedded
r
in a desarguesian affine plane of order q .
= 2 and S
s
(c)
p
(d)
S contains a normal subgroup N of odd order and
~
Sz(2) for some s.
index two in S. (e)
p = 3 ands~ SL(2, 5).
We have a desarguesian
net as in case (b) . Details of the proof are given in [7],
[13], and [18] .
The first part is concerned with the group generated by two elations with different axes.
It depends upon the
fact that the group generated by two matrices
lo a)1
r1
over a field of characteristic p
~
[i ~)
and
s
3 is SL(2, p ),
where GF(ps) is the smallest subfield containing a .
A re
presentation can be used so that two elations with different axis can be represented by p 1
=
lrrr 1o)
and P 2
__ (ro
A 1 ),
where I is an r by r identity matrix and A is an r by r submatrix with an eigenvalue a. Let K' be an extension of the field K which contains the eigenvalue a of A and let We dimension 2r over K'.
w be
a vector space of
If we think of the elements of Was
ordered rtuples of elements of K' then the vector space Ve V obtained by restricting the rtuples to elements of K can be identified with the one in which the plane represented.
Furthermore
acts as a reducible
group of linear transformations on We 185
TI
w.
There is an
is
invariant subspace of dimension two on which it turns out to act faithfully.
The restriction to this subspace is
conjugate to the group generated by ( 11 From this, one concludes that
~ SL(2, ps).
p
= 3, there is the additional possibility that
~
SL(2, 5).
For p2 >
3. Some modifications are required for characteristic 2 or 3.
Note the remark we have appended to the listing for
[ 18].
A remark or two about this theorem.
The second part
of conclusion (b) suggests that one might be able to con
186
struct examples by modifying a desarguesian plane .
I have
been able to carry this out (15]. Let G be the (linear) translation complement (all elations in the translation complement will be in the linear translation complementj and let G 1 be the subgroup generated by all the elations. theory, if G
~
G1 ,
As a matter of pure group
then G modulo C(G 1 )
(the centralizer in
G of G 1 ) is isomorphic to the group of automorphisms of G 1 induced by taking conjugates.
In the present case, C(G 1 )
is very restricted since it must leave each elation axis invariant.
In fact it is precisely the group fixing all
elation axes . We would like to find other theorems of this type for planes in which the translation complement contains no elations . For large dimension it may be helpful to try to characterize what I call minimal non f.p . f. groups. DEFINITION.
A group of linear transformations is fixed
pointfree if no nontrivial element of the group fixes any nonzero vector. DEFINITION.
A normal subgroup G 1 of a group G of linear
transformations is said to be minimal nonf . p.f. with respect to G if: (a)
G 1 is not f.p.f.
(some nontrivial element of
G 1 fixes some nonzero vector).
187
(b) tained in
If His any normal subgroup of G properly conG1,
then His f.p.f.
The minimal property en
sures that if a is any nonf.p.f. element of G 1 , then G1 is generated by all of the conjugates of a with respect to G.
See (17). Foulser (3) has shown that for p > 3 the arguments
used on groups generated by elations work as well for groups generated by Baer pelements.
A Baer pelement is
an element of order p that fixes pointwise a subplane whose order is the square root of the order of the full plane. Thus we have the following problem: determine the groups generated by Baer pelements for p = 2 or 3 subject to the restriction that the group contains no elations. Most of the really interesting known finite translation planes are of dimension two over the kernel, i.e., the vector space has dimension four.
I think it would be
a big advance if we could list all of the possibilities for dimension two.
This can now be done, in principle,
for planes of even order since Wagner, in a paper not yet published, has determined all subgroups of GL(2, q) for q even.
It remains, of course, to be determined how these
groups can act on translation planes of dimension two. The task will probably be easier if we only look at minimal nonf.p.f. groups.
188
When both the dimension and order are odd, we have a good knowledge of the abstract groups but a shortage of examples of planes and of the representations of these groups that can act as collineation groups. There may be some point here in a thorough investigation of the small dimensions, say 3 and 5, so that the dimension of the vector space is 6 or 10 respectively. Let G be the linear translation complement and let G be the induced group on the line at infinity. GL(2r, q) and PGL(2r, q).
G is
Thus G
c
the image of Gin the homomorphism onto
If both rand pare odd, a result of Hering
[8] says that the Sylow 2groups in Gare cyclic or dihedral.
By use of the GorensteinWalter theorem, I have been
able to show that the nonsolvable minimal nonf.p.f. groups with respect to Gare isomorphic to SL(2, u) for some odd u except possibly when A 7 is a factor group .
In my origi
nal version I also included PSL(2, u) as a possibility, but Liineburg has pointed out that this case can be eliminated because PSL(2, u) contains too many involutions. Nothing here tells us that u must be a power of the characteristic .
If u is not a power of p, a result of Hering 2d + l where dis the di
and Harris [4] implies that u
~
mension of the vector space .
For planes of dimension 3,
dis 6 and the minimal u is 13. 27 do admit SL(2, 13)
[SJ.
Bering's planes of order
I know of no other example of
a translation plane of odd order and dimension which admits
189
a nonsolvable collineation group.
Hering has another re
sult which shows that the stabilizer of a line is always solvable in this case. In another direction, we might ask: "What about groups generated by homologies with affine axis?"
More specifi
cally, suppose that P and Qare points on i 00 and there is a nontrivial (P, OQ) homology of order u.
It can happen
that P and Qare in a single orbit of length two or that both are fixed by the translation complement.
It also
might happen that the subgroup generated by conjugates of the given homology contains elations.
These are all of
the possibilities in a desarguesian plane unless u
= 2, 3,
or 5 in which case we can get a dihedral group, SL(2, 3) or SL(2, 5).
I have been able to show that all of this is
still true for translation planes of odd dimension (16). Let me remind you that two homologies with the same centre and different axes (or the same axis and different centres) generate a group which contains elations. The proof depends on looking at groups generated by a pair of homologies and which are, in a certain sense, minimal but such that the orbit of P contains points besides P and Q.
Neglecting a few complications which arise
if P and Qare in the same orbit we get Frobenius groups with complement of order u if the group is solvable.
This
turns out to be possible only for u = 2 or 3. For characteristic two, if the group is nonsolvable
190
it must contain
involutions .
At odd dimension the invo
lutions are elations; if there are no elations the group is solvable by FeitTheompson. For both the solvable and nonsolvable case, we do what amounts to reducing the problem to the desarguesian case: we represent the two generating matrices in the form and [
A2
B 2 j)
C2
D2
and show that the submatrices A 1 , B1
. . • , C 2 , D2 generate a ring which is a field.
,
The Goren
steinWalter theorem gives us possibilities for the permutation group induced on the orbit of the centre P.
With
a little casebycase argument, we can show that we always can choose a representation so that the submatrices commute and do generate a field . Our conjecture for translation planes of dimension two is :
If Pis in an orbit of length greater than two
(where there is a
(P, OQ) homology of order u) and if the
group generated by homologies of order u contains neither elations nor Baer pelements, then u = 2, 3, or 5 . II.
REDUCIBILITY CONSIDERATIONS
In the general theory of linear groups, the groups acting on vector spaces of relatively small dimension are less complicated than when the dimension is higher.
When the
group is reducible one is enabled to restrict the consideration to groups of smaller dimension.
191
We shall also call
attention to certain other notions that amount to a kind of reducibility.
We shall illustrate this approach by
looking at the case where the dimension is already small . For larger dimensions, there are more cases to consider but the same methods can be applied in principle.
In gen
eral, if cr is an element of the linear translation complement, the set of points fixed by cr is a vector subspace, which we shall denote by V(cr).
Furthermore either V(cr) is
a subspace of a line through the origin or is the set of points of an affine subplane.
In the latter case, the
dimension of V(cr) must be even, say 2t, and the intersection of V(cr) with each line through the origin is a vector subspace and the dimension is either zero or t. Now, specifically, suppose that G is the linear translation complement or a subgroup of the linear translation complement of a finite translation plane of dimension two over K, i.e., the vector space has dimension 4; the subspaces of dimension two are lines through the origin or are Baer subplanes.
We can make a sort of classification
in terms of reducibility as follows: 1.
G has an invariant subspace on which it does not act
faithfully.
The subgroup fixing this subspace pointwise
is normal in G. (a)
G has precisely one invariant line t through 0
and G does not act faithfully on t.
192
Note that t i s the
axis of a nontrivial group of elations.
All of the semi
field planes are of this kind. (b)
G has precisely two invariant lines through O
and G does not act faithfully on one or both. tains affine homologies.
Here G con
There are many examples of this
kind. (c)
G either has no invariant lines through the or
igin or at least acts faithfully on the lines it does fix but G has a unique invariant Baer subplane and does not act faithfully on this subplane.
In this case we have a
pgroup fixing the Baer subplane pointwise.
Some planes
of this type are derived from those of type l(a) . (d)
Faithful on all invariant lines, a unique pair
of invariant Baer subplanes. of these. (e)
Not faithful on at least one
Analogous to case (b). Faithful on all invariant twospaces but there
are invariant !spaces on which the action is not faithful . I know of no examples of planes of this kind. 2.
G is reducible but acts faithfully on all of its in
variant subspaces.
There must be at least one invariant
line or at least one invariant Baer subplane.
The collin
eation group is isomorphic to some subgroup of GL(2, q). The groups generated by elations (with more than one elation axis) are of this type (for odd characteristic) but the full translation complement is irreducible in the known cases. 193
DEFINITION.
A group G of nonsingular linear transforma
tions operating irreducibly on a vector space Vis imprimitive if V = V 1 each
o
$
•••
$
Vk and for each i = 1, ..• , k and
in G there is a j such that
v.o i
=
v ]..
If no such
decomposition is possible G is said to be primitive . Thus an imprimitive group is irreducible but is very much like a reducible one and, in fact, contains a normal reducible group of relatively small index.
For instance:
if P and Qare points on £ 00 such that every element of G fixes or interchanges P and Q and if G is irreducible then G is imprimitive. 3.
G is irreducible but imprimitive and does not act
faithfully on its subspaces of imprimitivity.
All of the
nearfield planes and most of the generalized Andre planes are of this type.
That is, we have (P, OQ) homologies and
P, Qare in an orbit of length 2.
There will be an analo
gous situation in some of the planes derived from these in which the subspaces of imprimitivity are Baer subplanes. 4.
As in 3 but the action is faithful on the subspaces of
imprimitivity. It may happen that G is irreducible and even primitive but is reducible in some other sense.
Suppose that our basic
vector space V be regarded as the space of ordered 4tuples over the kernel Kand that K' is an extension of K.
194
Let W
be the vector space of 4tuples over K'.
If G is repre
sented by matrices with elements in K, G acts on W.
It
can happen that G is reducible or imprimitive on W.
It
turns out that if G is primitive on V it must act faithfully on its irreducible constituents in W.
It also turns out
that no 3dimensional subspace of W can be an irreducible constituent of G.
Hence G must be isomorphic to a subgroup
of GL(2, qi) for some i; in fact G could be isomorphic to a 1dimensional linear group  but see below . DEFINITION.
If G is irreducible over every extension K'
of K, then G is said to be absolutely irreducible. There is another way of effectively reducing the dimension.
Let H be an abelian group acting irreducibly on
a vector space V of dimension dover K. need d
= 4.)
(Here we do not
Then His cyclic and the elements of V can
be identified with the elements of GF(qd) in such a way that His represented by multiplication maps x+xa in the larger field . This generalizes as follows.
Let G be a primitive
subgroup of GL(n, q) and let A be a maximal abelian normal subgroup of G.
Suppose that (IAI, q) = 1 and A is f . p.f.
Then A is cyclic and the ring of endomorphisms generated by A is a field
K.
If
IKI =
qt then t
I
n, Vis a vector
space of dimension n/t over a field isomorphic to K, and G is isomorphic to a subgroup of fL(n/t, qt).
195
If Gas a
subgroup of GL(n, q) is primitive, then A corresponds to a group of scalar matrices in fL(n/t, qt). Although it does not directly apply here, we might mention the following as stated in Dixon [l], Corollary 4.2A.
If G is an irreducible primitive linear group over
an algebraically closed field, then every normal abelian subgroup consists of scalars and is in the centre. This kind of reduction of dimension is, in certain cases, closely related to certain methods by which planes are constructed .
This may, in some sense, be considered
to be a version of Cliffords theorem (see Dixon [1]) . Hering uses a slightly different version in [9] .
A proof
of the present version is given in [19]. In [11] and [14], I attempted to take a broad look at constructions .
Whether their originators looked at
them that way or not, most of the successful constructions of translation planes can be looked upon as examples of one or the other of the two folllowing general methods or combinations of the two. 1.
To construct a plane admitting a translation com
plement acting in a certain way with a small number of orbits, get an explicit representation of what the group must be like .
By usually ad hoc or trialanderror methods
choose certain initial lines through the origin; the others are defined as their images. 2.
Modify some known plane.
196
Usually the original
plane can be taken to be desarguesian and the modification involves replaceable nets.
The replacements very often
amount to the following: In a desarguesian affine plane r r A A of order q coordinatized by GF(q) = K, think of Kasa vector space of dimension rover K
=
GF(q) so that the
plane is a vector space of dimension 2r over GF(q).
Let
W be a vector subspace of dimension r such that for each ~
a~ 0 in K the image of Wunder the mapping (x, y) +
(xa,
ya) is either equal to W or intersects W only in the zero vector.
We now replace the lines through O which inter
sect W with Wand its images under this cyclic group of order q
r
 1.
This group continues to act as a collinea
tion group of the new plane and is, in fact, normal.
This
procedure can be carried out simultaneously in several different parts of the plane. earlier considerations .
Now let us return to our
Planes constructed in this way
will admit the kind of reduction we have just been discussing. Suppose that we have a translation plane of dimension rover GF(q) and that the translation complement has a normal abelian subgroup H such that V
=
V1
$
V 2 where V 1
and V 2 are rdimensional subspaces left invariant by Hand irreducible under H.
Suppose also that the linear trans
lation complement is irreducible and primitive. has an image different from V 1 or V 2 •
Then V
In fact Vis a union
of Hirreducible subspaces of dimension r which can be
197
thought of as lines through the origin of a desarguesian affine plane and H can be identified with an irreducible group of scalar maps (x, y)
+
(xa, ya).
The given plane
can be constructed from the desarguesian plane by a modification of the procedure we have been describing. REMARK.
These ideas are very close to the ones which led
to my characterization of generalized Andre planes [12) and the improvement by Liineburg. at this same conference.)
(See Liineburg's lecture
An equivalent characterization
was made independently by Rao, Rodabaugh, Wilke, and Zemmer (22).
The known planes of odd order which contain elations
appear to have the property that the translation complement is irreducible and primitive but that a minimal nonf.p.f. group is reducible  i.e. there is an invariant subplane. The full linear translation complement is subject, however, to the kind of reduction mentioned above.
This is almost
equivalent to saying that these planes were constructed from desarguesian planes. If a group is absolutely irreducible and if its order is relatively prime to the characteristic, it is isomorphic to a group of linear transformations on a vector space of the same dimension over the field of complex numbers.
The
primitive complex groups are known (2) for small dimensions. I have examined those of dimension 4 [19); some of them cannot act as collineation groups of translation planes
198
of dimension two. All of this survey does not quite amount to a complete explicit listing of possibilities.
For instance, we have
not tried to list all planes of dimension two with a pair of points P, Q on £ 00 in a single orbit of length two such that there is a nontrivial (P, OQ) group of homologies. Nevertheless, all possibilities but one are included in some sense .
Let me try to explain this last case.
For
a plane of dimension two an element of order pin the translation complement must be of one of three types: tion,
(b) Baer pelement,
is !dimensional.
(a) ela
(c) the pointwise fixed subspace
The missing case is the one in which we
have an absolutely irreducible group in which the pelements are of this third type.
We do have examples due to
Hering [6] and the guess is that others which might exist all have essentially the same nature. A similar analysis can be carried out, say, for planes of dimension three over the kernel  the vector space has dimension 6.
The number of ways a pelement can act, and
the possible geometric nature, are somewhat greater. The number of examples of known planes is much smaller. Here is the place where we can use more examples of translation planes.
We do, however, have much more knowledge
as to the possible abstract groups which can appear.
199
III.
ORDERS OF ELEMENTS
In this section we are interested in the possible orders of elements of the translation complement and how they act. Again, we shall illustrate what can actually be a more general approach by reference to the case of planes of dimension two. Now IGL(4, q) I = q 6 (q + 1)
2
(q  1)
4
(q 2 + 1) (q 2 + q +1)
so that if o is an element of prime order, then lol divides p(q  1) (q + 1) (q 2 + 1) (q 2 + q + 1). q4

1 if it is fixed point free.
divides q 2

Now lol must divide
If o is a homology,
lol
1: if for some line i through the origin, the
dimension of V(o) i is one, then lcrl divides q 2

q =
q(q  1). Thus the order of every nonf.p.f. element (if prime) must divide q 2 vide q 2 origin.
1.


1.
Conversely, suppose that lcrl does di
Then o fixes at least two lines through the
If iol divides q + 1 but lol
some !space pointwise invariant.
i 2 and o leaves
The number of !spaces
in a two space is q + 1 and if o fixes two !spaces pointwise in the same twospace it must fix this twospace pointwise.
This all adds up to the fact that if 2
i lcrl
divides q + 1 then either o is a homology or o is fixed point free. Suppose that 2
i lol divides q  1.
Then o leaves
at least two lines through the origin invariant and on
200
each of these there are at least two invariant !spaces. In this case if cr has an invariant 1space which it does not fix pointwise, there is a scalar\ such that cr\ does fix the onespace Q pointwise, the 1space in question. If cr leaves all !spaces on some line invariant, either a is a homology or there is a scalar A such that cr\ is a homology. Now let us go back to looking at things more generally. Suppose that cr is an element acting on a vector space of dimension t over GF(q) and suppose that cr has an irreducible constituent V 1 of dimension s.
s must divide q 1. Now suppose t is a prime factor of q 1 but does not divide
point free on Vl and that
la I
Then must be fixed

Ia I

qs  1 for any s < t.
Then must act irreducibly on
the vector space of dimension t. qprimitive divisor of q
t
 1.
We say that lcrl is a (This notion can also be
defined for numbers which are not prime.) Now an odd prime factor of q 2 + 1 is a qprimitive divisor of q 4

1.
Hering [9] has investigated subgroups of GL(t, q) t which contain qprimitive factors of q  1. (His results are stated for the case where q is a prime.
If I am not
mistaken, they still hold if q is a prime power.) In Hering's notation, let r be a prime qprimitive factor of q
n
 1 and let G be some subgroup of GL(n, q)
whose order is divisible by r.
201
Lets be the normal sub
group generated by all relements and let F be the Fitting subgroup of G.
Then SF/Fis simple.
If this factor group
is trivial then the Sylow rsubgroup R must be in F.
Since
Fis nilpotent, this implies that G has a normal abelian irreducible subgroup and is isomorphic to a subgroup of fL(l, qn). Another result in the same paper states that Fis in the centralizer in G of S except possibly when r = 2a + 1 for some a.
= n + 1
We want to use this to examine the
possibility that SF/Fis a cyclic group of prime order. Suppose that F n Sis in the centre of Sand is not fixed point free.
Then the subspace pointwise fixed by any ele
ment of F n Sis invariant under s contrary to the fact that S contains elements of order r which must be irreducible.
Hence we may assume that F n Sis f.p.f.
Further
more, if r divides the order of F n S we may again conclude n that G is isomorphic to a subgroup of fL(l, q ). Thus if SF/F = S/S n F has prime order and fL(l, qn) either r
=
n + 1
=
2a + 1 or F n Sis f.p.f.
and its order is not divisible by r. have IS/Sn Fl = r. free.
GI
In this case we must
We claim thats must be fixed point
If not, the rth power of an element of S which is
not f.p.f. must be in F n Sand hence must be f.p.f. the nonf.p.f. elements must have order r. impossible if r is a qprimitive divisor.
202
Hence
But this is We conclude
that if SF/Fis a solvable simple group at least one of the following must hold:
=
n + 1
=
2a + 1,
(a)
r
(b)
n G 2' fL(l,q),
(c)
Sis fixed point free.
Fixedpointfree linear groups are Frobenius complements and their structure is well known. When r is not merely a qprimitive divisor of q
n
 1
but is a pprimitive divisor, Hering has shown that SF/F cannot be one of the known simple groups of Chevalley type except possibly when Sis essentially all of SL(n, q) or r divides (n + 1) (2n + 1)
(9, 10].
For n = 4, corresponding to planes of dimension two, the exceptional values of rare 3 and 5.
A pprimitive
divisor must divide q 2 + 1 and 3 does not do so for any q. For completely reducible solvable groups or for groups whose order is prime to the characteristic there are theorems due respectively to Ito and FeitThompson which require Sylow subgroups to be normal for primes too large compared to the dimension.
See Dixon (1).
REFERENCES Note that further references are given in (11), (20) 1.
(14), and
J.D. Dixon. The structure of linear groups. Nostrand Reinhardt (1971).
203
Van
2. 3. 4. 5. 6. 7. 8. 9.
11 . 12. 13. 14. 15. 16. 17. 18.
19. 20. 21. 22.
W. Feit. The current situation in the theory of finite simple groups. Actes de Congres International des Math. 1 (1970): 5593. D.A . Foulser. Baer pelements in translation planes. J. Alg . (to appear). M.E. Harris and C. Hering. On the smallest degree of projective representations of the groups PSL(n, q) . Canad. J. Math. XXIII (1971) : 90102. C. Hering. Eine nichtdesarguesche Zweifach transitive affine Ebene der Ordnung 27. Abh. Math. Sero. Hamb. 34 (1969): 203208. A new class of quasifields . Math z. 118 (1970): 5657. On shears of translation planes. Abh. Math. Sero. Hamb. 37 (1972): 25868. On 2groups operating on projective planes. Ill. J. Math. 16 (1972): 581595. Transitive linear groups and linear groups which contain an irreducible subgroup of prime order . Proc. Conf. on Proj . Planes . Wash. State Univ. Press, Pullman (1973) . T.G. Ostrom . Vector spaces and construction of finite projective planes. Arch. Math. 19 (1968) : 125. A characterization of generalized Andre planes . Math. Z. 110 (1969): 19. Linear transformations and collineations of translation. J. Alg. 14 (1970): 405416. Finite translation planes, Lecture Notes in Math. No. 158 . SpringerVerlag. A class of planes admitting elations which are not translations . Arch . Math. XXI (1970): 214217. Homologies in translation planes. Proc. London Math . Soc. XXVI (1972): 605629. Normal subgroups of collineation groups of finite translation planes. Geom . Ded. 2 (1974): 467483. Elations in finite translation planes of characteristic 3. Abh. Math. Sero. Hamb . (to appear shortly). Note: The condition that the group is generated by elations was left out in the statement of the main theorem at the beginning of the paper. This condition is definitely a part of the hypothesis. Nonmodular collineation groups of translation planes. Submitted to Geom . Ded. Classification of finite translation planes. Proc. Conf. on Proj. Planes. Wash. State Univ . Press, Pullman, WA (1973) . D.S . Passman. Finite Permutation Groups. W.A. Benjamin Pub. Co. (1965). M. L.N. Rao, J . W. Rodabaugh, F.W . Wilke, and J.Z. Zemmer. A new class of finite translation planes ob
204
tained from exceptional nearfields. 11 (1971): 7292.
J. Comb. Thy.
Note: This research was supported in part by the National Science Foundation of the U.S.A.
205
RECENT RESULTS ON PREHJELMSLEV GROUPS
Edzard Salow
Let G be a group generated by an invariant set S of involutions.
Let P be a nonempty invariant subset of {x ;
= ~
x ESS A x involutory} with P n S
The elements of
Sare called lines, those of P points.
A relation
I
is
defined in the following manner: if A E P, b E s, then A I b: Ab is involutory, if c, d E s then c I d : cd E P. In the first case one says "A is incident with b" and in the second case "c is orthogonal to d ."
The triple (G, S,
P) is called a PreHjelmslev group if Al. V A E P, b E s 3 c E s : c A2 . Y A E P, b, c, c ' E S :
I
A, b.
(A , b
C,
C
1
> C
CI) •
A3.
VA, B, C
E
P, d
E
S
(A, B, C
d
>
ABC
E
P) .
A4.
V a, b, c
E
S, D
E
P
(a, b, c
D
+
abc
E
S) .
(G, S, P) is called a
(nonelliptic) Hjelmslev group if
206
= {x; x ESSA x involutory}.
in addition P
In this paper we only consider PreHjelmslev groups (G, S, P) which have the following property, called Axiom W: 3a, b, c, d ES: a
I
b Ac
I
d A a, b, c, d pairwise
intersect uniquely. This property is equivalent to the existence of a E G'
THEOREM 1.
s,
Let (G,
P) be a finite PreHjelmslev group
with Axiom Wand CG(P) = {l}. £, k,
with the following properties:
~
(1) ring
Mis a finite unitary module over the commutative
R with 1 and (2)
(P, S,
Then there exist R, M, ~,
~
I>
1
2.
xM
=
{O} implies x
u, v ER Aw EM A (3c, d ER: uc + vd
w)
Es~
x
M, {R(u, v, w);
= l)}, I).
Here
(a, b) is defined by ua +vb+ w = 0.
is equivalent to R(u, v, 0)
(3)
0 for x E R.
is an embedding of the incidence structure
into the incidence structure (M
R(u, v, w) I
=
Es~.
k and k + 1 are units in R.
P~
M
R(u, v, X
M.
£ is a symmetrical
bilinear form on M such that (x, y)tz = (x, z)£y for all x, y, z EM. of R.
The values of 1 are elements of the radical
A symmetrical bilinear form f on Rx Rx Mis de
fined by ((u, v, w),
(u', v', w'))f := uu'k + vv' + (w, w') 1. g 2 is equivalent to
(q 1 , q 2 )f
=
0.
_If g E Sand g~
unit in R.
207
=
Rq then (q, q)f is a
(4)
~
is a monomorphism from G into the orthogonal
group of the metrical module (Rx Rx M, f).
g~ is a sym
metry along g~ for all g ES. By a symmetry along Re (c ER x Rx Mand (c, c)f unit in R) we mean the mapping oRc: x
~ x 
(3) in Theorem 1.
(,
s,
for x ER x Rx M.
Given R, M, k, 1, f with the properties (1),
THEOREM 2.
P
unit} and
2(c, x)f((c, c)f) 1 c
Let S := {oRq; q ER x Rx MA (q, q)f
:= {oRqoRr;
oRq' oRr ES A (q, r)f = O}.
Then
P) is a finite PreHjelmslev 9:roup with Axiom W
and c (P)
=
{l}.
(,
s,
P> is a Hjelmslev 9:roup i f
and only i f 0 and 1 are the only idempotent elements in R. It is possible to formulate Theorem 1 in a more grouptheoretical manner.
R. Stelting has given a characteriza
tion of the groups of finite Hjelmslev groups by grouptheoretical properties [6).
A similar statement is the
following: G is the group of a finite PreHjelmslev group if and only if
(*) G has a normal 2complement K.
A Sylow 2group
of G has a generating set T of involutions with TTT = T. There exists an involution A
E
TT
such that CK({A, b})
{l} for all b ET. Now we can reformulate Theorem 1 in the following manner. 208
=
THEOREM 3.
Let G be a finite group with the properties (*)
such that there. exists a" G' with CK (A) = CK (a).
Then
IG/CG(G') I ~ 2 or G/CG(G') is isomorphic to a subgroup of the little orthogonal group Hof a metrical module, which contains H". By the little orthogonal group we mean the group generated by the set of all symmetries. The difficulties in the proof of Theorem 1, which is yet unpublished, arise from the fact that there are some strange phenomena which can occur in the finite PreHjelmslev
groups with Axiom W:
There may exist points A, B with
no or with more than one joining line.
There may exist a
line g with the property that there are no points A, Bon g, for which all joining lines carry the same points. may exist lines
a, b, c with a
I
There
b such that c intersects
a and b both in more than one point.
This last phenomenon
cannot occur in finite Hjelmslev groups. Among the PreHjelmslev groups which are treated in Theorem 1, those which are singular, i.e. for which PPP= P, play a distinguished role.
The bilinear form£ in Theo
rem 1 belonging to a singular PreHjelmslev group is zero. If in proving Theorem 1 one would follow the usual way, one would try to coordinatize the PreHjelmslev group by first coordinatizing a suitable incidence structure.
But
there is a better way, namely to replace the incidence structure by a singular PreHjelmslev group.
209
This makes
good sense, because a singular PreHjelmslev group can be coordinatized directly by halfrotations [2].
The reduc
tion to the singular case is explained in the following proposition. Let (G, s, P) be a finite PreHjelmslev group
PROPOSITION.
with Axiom W and Z(G) = {l}. eing from s
X
Let 0 E p and m be the map
s to the set of rotations around 0 with
(g, h)m = (O, g) (0, h) for g, h E s. perpendicular on g through O).
({O, g) denotes the
Then for all f Es there
exists an automorphism Kf of (P, s,
I)
the points on f and for which (g, h)m for all g, h Es. e,
f
E
s
A
(
e,
f) m =
o)
~ KeKf
=
((gKf' hKf)m)
1
Define T := {Kf; f Es} and Q := {KeKf;
o} •
(, T, Q) is a finite singular
PreHjelmslev group with Axiom Kh' A E Q
which fixes f and
w.
The mapping T: h ET
(e, f lines withe, f
is an isomorphism from (P, S,
is equivalent to (g, h)m
=
I)
I
~
=
A and (e, f)m
2!!. (Q, T,
I).
Kgl Kh
Q.
For infinite PreHjelmslev groups there doesn't exist such a general coordinatization theorem as Theorem 1.
But
in [3] we show that an infinite PreHjelmslev group (G, S, P) with Axiom W can be embedded into an orthogonal group over a commutative ring R with land½ if (G, the following two additional properties: B have
a
joining line.
s,
P) has
(1) All points A,
(2) Every line g is incident with
points A, B, which have gas unique joining line.
210
Here
(P,
s, I l
is embedded into the incidence structure (M, N, I)
with M := {Rx;
X
element of a basis of R
{Rf; f element of a basis of (R Rf); xf
= O}.
X
R
X
X
R
X
R}, N :=
R)*} and I
:= { (Rx,
The embedding is locally surjective, i.e.
if a point C is an image, then every line through C is an image too.
For an arbitrary commutative ring R with 1 and
1
2 (M, N, I) generally doesn't satisfy property (1).
There
fore not every commutative ring with 1 and½ can occur. That is the reason why it would be more satisfactory to take a weaker axiom about joining lines instead of (1) and (2), for example the following one: AXIOM Y.
There exist A E P, b ES such that for every point
Con b there is a line through A which intersects b uniquely in C. But it is not known how to coordinatize an infinite PreHjelmslev group with Axiom Wand Axiom Y.
Nevertheless
in [4] and [5] the assumption of Axiom Wand Axiom Y has brought some deeper results. At first we only assume Axiom Wand !Pl > 1 for the PreHjelmslev group (G, S, P).
Let~ be a set of rotations
(By a rotation we mean a product of two intersecting lines.)
CP(~) is the point set of a Subpre
Hjelmslev group [1].
If A E P we denote {CP(~);
rotations with A
E
CP(~)} by~.
lattice by inclusion.
Let KE~·
211
~
a set of
~ becomes a complete Then there is exactly
one partition [Kl of P which has K as an element and which is induced by a homomorphism from (G, S, P) on a PreHjelmslev group.
Now we consider CG([Kl).
subgroup of G.
CG([Kl) is a normal
The factorization by CG([Kl) gives a homo
morphism on a PreHjelmslev group, which identifies exactly the elements of K with A.
Let TIA be the mapping from~
into itself with LTIA := Cp(CG([Ll)
n
CG(A)) for L
€
~
Then TIA is an antihomomorphism from the lattice '7A_ into itself with Tit= TIA.
If in addition (G, S, P) satisfies
Axiom Y, then TI~ is the identity of 7A. ment of Hjelmslev's Reciprocity Theorem. every KE
7,,.
If BE P then for
there exists an LE '.F8 with [Kl
sequently the set point A.
This is the state
[Ll.
Con
T:= {[Kl; KE ~A} is independent of the
The order relation o f ~ induces an order relation
on ~such that (~, .=) is isomorphic to (7, !>) for all A E P.
The mapping TI, which is induced on 7by TIA, is in
dependent of the point A too. The mapping
TI is interesting in several respects.
Again let A E P and KE
9i·
Then there exists a locally
surjective embedding of the incidence structure of the PreHjelmslev group (G, S, P)/CG([Kl) into the incidence structure of KTIA.
(The point set of the incidence structure of
KTIA is KTIA itself, and mis a line if there exists b ES with m ={XE KTIA; X
b}.)
After having got this embed
ding, one can easily prove for a PreHjelmslev group (G, P) with Axiom Wand Axiom Y:
212
If U is an antiatom of the
s,
lattice '1', then {G,
s, P)/CG{U) is a Hjelmslev group with
Axiom Wand Axiom Y, in which no double incidences occur,
i.e. no lines a, b intersect in more than one point. Finally we want to consider more general partitions A homomorphism which is induced by an element of 7"
of P.
we call an elementary contraction.
A product of such ele
mentary contractions we call a contraction.
There are con
tractions which are not elementary contractions.
The set
of all partitions of P which are induced by contractions we denote by
6.
On & there exists exactly one associative
operation o with the following property: E §:",
~
If U E G, [CP (ti)]
is the homomorphism from {G, S, P) induced by U,
and t i s the homomorphism from {G,
s, P)~ induced by
[CP{ti~)], then U o [CP(ti)] is the partition of P which is induced by
~t
mutative.
7generates t.
It is an astonishing fact that
O
Now we assume that (G,
is com
s, P) is
a PreHjelmslev group with Axiom Wand Axiom Y, such that the ascending chain condition is valid for
r.
Then if
{U.}. lN is a sequence of elements of 7, {U 1 ° ... oU} llE nnE IN
is finite.
It follows that '.7°'has only a finite number of
atoms and antiatoms.
The structure of (G, S, P) can be
clarified even more.
There is a finite chain U0 < U1
is given by
P1  P2  P3 + P4 = n  2·d(~, y).
We may use these notions to establish THE PLOTKIN BOUND.
If the parameters of a binary (M, n, d)
code satisfy n < 2d, then M PROOF.
2d/(2d  n).
~
The M code words may be used to form the rows of
an M x n matrix P.
According to (2.A), the Hamming distance
between distinct code words is at least d precisely when
~
n  2d
for all distinct~•
i
E
Sn.
But the inner products are
given as entries in the symmetric M x M matrix PPt.
222
There
fore, for an (M, n, d)code, (2 . B)
PPt
~
elementwise
=
nI + (n  2d) (J  I)
2dI + (n  2d)J.
If jM is a column vector of M l's, then (j~ P) (Pt jM) is the square of the ordinary euclidean norm of j~P, so (2.B) implies that 0 < J.t PPtJ. M ~ 2dM M If n
+ Q(~) + Q(X) is a bilinear form B(~,
=
on V; it follows that Q(9} linear form is alternating.
x>
0 and that the associated bi
If this bilinear form is also
nonsingular, it is known that there is a basis ~ 1 , ..• , ~ 2m of V with respect to which the quadratic form Q is given by
or and that the number of zeros of Q is 2 2mi + 2mi or 2 2mi  2
m1
respectively.
Let Q represent the set of all zeros
of a nonsingular Q. We may now define the orthogonal graph: the vertices are the elements of Q\{9}
(alternatively, one may use the
elements of V\Q), and two distinct vertices~• y are ad
234
jacent precisely when Q(~ + y)
=
0.
As before, we find
that the orthogonal graph is a rank 3 graph under the orthogonal group {a" GL(V)
I
Q(~a)
= Q(~)
for all x" V}.
We mention that the symplectic and orthogonal graphs may be characterized in a more geometric manner (cf.
(10),
[ 13)) :
SHULT'S THEOREM.
For a regular graph G which is not a com
plete graph, the following are equivalent: (i)
G is isomorphic to a symplectic or an orthogonal
graph; (ii)
For each pair of adjacent vertices a, b of G,
there is a vertex c adjacent to both a and b such that each further vertex is adjacent to an odd number of vertices of the triangle {a, b, c}. 4.2.
Kerdock sets
We now investigate some related concepts following Kerdock, Patterson, and Goethals.
Suppose Bis the set of all al
ternating bilinear forms on V(2m, 2), allowing for singular forms and even the zeroform.
Each form B(~, y) may be
regarded as a matrix product
~tax where Bis a symmetric 2m
x
2m matrix with entries in GF(2),
such that the main diagonal consists entirely of zeros.
235
Since there are m(2m  1) positions in such a matrix above . m(2m1) the main diagonal, we find there are exactly 2 elements in B. Consider a subset S of B such that whenever B. and l. If the matrices B . ES the form B. + B . is nonsingular. J l. J representing B. and B. have the same first row, then the l. J matrix representing Bl.. + B. has first row 0, and B. + B . J l. J is singular. Therefore, the first rows for the matrices of the B. in S must all be distinct, so S contains at most l. 2 2ml forms. For this maximal case, we define: a subset K of Bis called a Kerdock set if IKI
=
2
2m1
and, whenever
Bi and Bj are distinct elements of K, the form Bi+ Bj is nonsingular.
We give the following construction of a If V(2m, 2) is considered to be GF(2 2ml) e
Kerdock set .
GF(2), each element~ in V may be written uniquely as~ a+ s where a element a
E
GF(2
2m1
) ands
E
GF(2).
The trace of an
GF(2 2ml) is defined by
E
Tr(a) =
2m2
l
k=O
and it is known that
=
Tr(a 2 )
[Tr(a)]
2
=
Tr(a)
E
GF(2),
Tr(a + B) = Tr(a) + Tr(B), Tr(l) = 1. For each y
E
GF(2 2m 1 ), we define an alternating bilinear
form BY on V(2m, 2) by By(a, B) = Tr(aBy 2 ) + Tr(ay)•Tr(By),
236
B (a, I;) y
Tr (ay) ,
By (I;, I;)
o,
for all a, B to V(2m, 2) . K =
{B
y
GF(22m1) and I;
€
GF (2), extending linearly
€
We shall show that
I
y E GF(2 2m 1 )}
is a Kerdock set .
It obviously has the right number of
elements, so it only needs to be shown that BY+ B0 is nonsingular when y
i
o.
Suppose then that BY+ B 0 is singular.
Since the kernel of By+ B 0 must have even dimension, there exists a E GF(2 2ml)\{O} which is also in the kernel; it follows that Tr(ay) + Tr(ao) and
=
0
Tr(a8y 2 ) + Tr(a8o 2 ) + Tr(ay) ·Tr(By) + Tr(ao) •Tr(Bo)
for all BE GF(2 2m 1 ) . Tr(aB(y + 6) that y
=
o.
2)
=
= Tr(ao) =
If Tr(ay)
=
0, then
0 for all BE GF(2 2ml) and we conclude
On the other hand, if Tr(ay)
then Tr{B(y + o) (a[y + o] + 1)} and we again deduce that y
=
o.
=
=
Tr(ao)
=
1,
O for all B E GF(2 2ml)
This establishes that K
is a Kerdock set. 4.3.
Kerdock codes
Suppose that Bis an alternating bilinear form on V(2m, 2) and that Q and Q' are two quadratic forms associated with Bas in §4.1; it is then an easy exercise to show that Q + Q' is one of the 2 2m linear forms on V(2m, 2).
Thus,
there are 2 2m quadratic forms associated with each alter
237
0
nating bilinear form
on V(2m, 2).
The count of alternat
ing bilinear forms in §4.2 shows that there are exactly m ( 2m+ 1 ) 2m m ( 2m1 ) . = 2 •2 elements in the set Q of all quad2 ratic forms on V(2m, 2). We may represent each Q
E
Q as a vector in V(2 2m, 2)
whose 2 2m coordinates O and 1 are the values Q(~) over all
~
V(2m, 2).
E
From these binary "words" of length 2 2m, we
form a Kerdock code by fixing a Kerdock set Kand taking as code words {Q, 1 +QI the bilinear form B(Q) EK}. ' . f orms in . . S ince t h ere are 2 2 m l b i' 1 inear K, 2 2 m qua d ratic
forms for each bilinear form, and 2 code words for each quadratic form, we have a total of 2 4 m code words.
Since
the smallest number of l's in a combination is 2 2ml  2ml, the Kerdock code is found to be a binary (2 4 m, 2 2m, 2 2ml  2
m1
5.
)code.
EQUIANGULAR LINES IN Rd
5.1.
Background
If a collection of n lines through a point O in Rd has the property that each pair of distinct lines determines the same angle~, the collection is called a set of equiangular lines.
A unit vector p. may be chosen along each line of i
such a collection in either of two directions; it follows that the inner product between distinct unit vectors is given by 238
p 2 , and (C  p 1 I)· (C  p 2 I) = 0. We shall set forth several consequences of this latter definition.
The multiplicities of the eigenvalues p 1 and
p 2 will be denoted by µ
(6 . A)
1
and µ 2 respectively, so
µlpl + µ2p2 = O.
It follows that (6.B)
If p 1 + p 2
(6.C)
If p 1 + p 2 = 0, then p 1 = p 2 = ~ a n d C 2 =
~
0, then p 1 and p 2 are odd integers.
(n  l)I; furthermore, it may be shown that n = 2 (mod 4) and that n  1 is the sum of two squares.
Examples have
been constructed when n  1 =pk= 1 (mod 4), and also for n = 226; the first unknown possibility occurs when n
= 46. Solving (6.A) for µ2, we find µ2 = p2l

p3 l P1

pl P2
and we deduce (6.D)
If p 1 = 3, then n = 10, 16, or 28.
If p 1 = 5, then
n = 16, 26, 36, 76, 96, 126, 176, or 276.
Moreover, exam
245
ples of each possibility are known, except for P 1 = 5 and n
=
76 or 96.
Furthermore
This is a consequence of the following THEOREM 6.1. most
½,a,a
PROOF.
(cf . also [21])
The number of equiangular lines in Rd is at
+ 1).
... ,
Suppose p 1,
pn are unit vectors along n equi
angular lines in Rd I so 1 fixed and t E
l
:m •
If D is the conjugate class of
some element a Es in B 1 , then D forms a twosided distributive quasi group homeomorphic to 1R2 if we set
269
x
y = y
o
I
xy for x, y ED.
Two guasigroups Da and 0 8 defined in this manner are isomorphic if and only if a= 8± 1 • Also, E(D) ~ 1R 2 • and the connected component of Eis isomorphic to the vector group rn. 2 • (4)
Let~= PSL 2 (JR).
In~, let L 2 be the class of
rotations which are conjugate to a given rotation a. fining x
O
y = y
1
xy for x, y E L 2
,
we obtain a rightdis
tributive, but not leftdistributive guasigroup. groups L
a1
and L
a2
exists some element 0
1
a 1o
De
Quasi
are isomorphic if and only if there
o in the group PGL 2 (1R) such that
= a2•
Hence we see that the group PSL 2 (JR) is the only locally compact group which leads to minimal, locally compact, connected, and locally connected quasigroups which are right, but not left, distributive. In the remaining cases in which we have a classification of rightdistributive connected quasigroups we must assume that the group E generated by right translations is a Lie group.
Only in this case does our machinery work.
For this purpose we define:
A topological rightdistri
butive quasigroup D will be called a guasigroup of Lie type if the group I generated by all the right translations of Dis a Lie group. We hope that this definition is no great restriction,
270
for it is rather natural and in fact we can prove that E
is a Lie group in many cases.
Some instances of this will
be mentioned below: (1)
Dis an almostcomplex compact manifold, and the
right translations of D preserve the almostcomplex structure. (2)
Dis a Riemannian manifold and the right trans
lations are at the same time isometries of D. (3)
Dis a manifold admitting an affine connection
for which every right translation is an affine mapping. (4)
Dis a C 1 differentiable compact manifold, and
the automorphism group E which is (topologically} generated by the
right translations possesses a neighbourhood of the
identity consisting of equicontinuous automorphisms. The main case in which I have solved the classification of rightdistributive connected quasigroups of Lie type is the case of compact quasigroups.
For these quasigroups,
we have the following: THEOREM.
Let D be a compact connected rightdistributive
quasigroup of Lie type.
Then Dis isomorphic to one of
the following quasigroups:
(1) n
~
2.
Let T
n
be the ndimensional torus group with
Let a be an automorphism of Tn such that the nor

malizer of the monothetic group in the group E
is .
=
T n
(Since the continuous automorphism group of T
271
n
is

discrete, we have = .)
Let T (a) be the class of

conjugate elements of a in E.
n
Then we get a twosided
distributive quasigroup on Tn(a) which is homeomorphic to T, where we define x n

o
y = y
1
xy for all x, y ET (a). 
n
We shall denote these quasigroups by Tn(a) .
Two such quasi
groups Tn(a 1 ) and Tn(a 2 ) are isomorphic exactly if there exists some isomorphism S from T (a 1 ) onto T (a) for which 
(2)

n

n
2
Let the Lie group 6 be the direct product of two
Lie groups A and B with the following structure: If At 1, then A is the direct product of groups Ai with i = 1, ••• , n, n > 1, such that A 1 is either the identity or a torus group of dimension at least two, and A. 
for i If B
l.
> 1 is isomorphic to either PS0 8 (lR) or Spin 8 (lR).
t
1, then Bis the direct product of Lie groups Bk,
k where every B itself can be written as a direct product of . h amalgamated central subgroups isomorphic groups B.k wit J
to the Klein four group.
Here B~ either does not occur or,
if it does, it is a twodimensional torus group S0 2 x S0 2 ; B~ for j > 1 appears to be isomorphic to Spin 8 (E.). Let Ebe the semidirect product of 6 with a discrete cyclic group~ generated by an automorphism a of 6 which acts on 6 in the following way: k
a normalizes all the groups A. and B.; for i, j > 1, 

l.  
J

it acts on A. and B~ as a triality automorphism (of order l.

J
3); finally, a induces on B~ (2.!!, A 1 ) an automorphism such
272
that the normalizer of the cyclic group¢ in Bk¢ (in A.¢) 1

1
is identical with¢. Now let 0(6, a) be the conjugate class of a in[. Defining x•y = y
1
xy for x, y
E
0(6, a) yields a right
distributive guasigroup which is not leftdistributive. This guasigroup is homeomorphic to the topological product A1 x
n TI
i=2
(A . /C . ) x (Bk/ 1
1
m TI
j=l
kc,) J
where all the C. and kc, are homeomorphic to the compact 1

J
Lie groups of exceptional type G2
•
Two guasigroups 0(6 1 , a 1 ) and 0(6 2 , a 2 ) are isomorphic if and only if there exists an isomorphism B of 6 1 to 6 2 for which a 1 8
= a2 •
In our final classification theorem, we determine all those rightdistributive quasigroups of Lie type which are homeomorphic to the plane IR2 , and which we therefore call the planar guasigroups.
The classification of planar quasi
groups appears to me to be justified because the only surfaces which admit a regular idempotent multiplication are the plane and the torus. THEOREM.
We state
Let D be a planar rightdistributive guasigroup
of Lie type.
Then Dis isomorphic to one of the following
guasigroups: (a)
Let [ be the group of proper euclidean motions,
and let a be a rotation through an irrational multiple of
273
TI.
Let S(a) be the class of conjugate elements in L. we define x
O
y = y
1
xy for x, y
E
If
S(a), then S(a) is a
twosided distributive guasigroup. The right translations of S(a) always generate L.
Two
quasigroups obtained in this fashion are isomorphic exactly if there exists a continuous automorphism y of L with a1Y
=
a2. (b)
Let a be an element of GL 2 (lR) Let L
discrete subgroup in GL 2 (IR).
a
which generates a
be the semidirect
product of the euclidean translation group 1R 2 with . Let J(a) be the class of conjugate elements of a in L. Then J(a) forms a planar twosided distributive quasigroup, if we define a product x
o
y = y
1
xy, for x, y E J(a).
The group (topologically) generated by the right translations of J(a) is isomorphic to L. Two quasi groups J (al) and J(a2) are isomorphic exactly if there exists some isomorphism 8 of the group L L
a2
al
onto
which maps a 1 to a2. (c)
The group generated by the right translations of
D is PSL 2 (lR), which has been discussed earlier. (d)
{git,
Let ti be the group of matrices u
u, v) =
[:
Let a be the involution
a
V
t
a
0
(g
u, v,
0
0 0
l
274
t
~]
t , a fixed}· JR,
> 1
and let L be the semi
direct product of~ with .
Then the class R of conjugate
elements of a in E forms a planar rightdistributive quasigroup if we take conjugation as multiplication. In R, the leftdistributive law is violated.
All the
quasigroups obtained in this manner are isomorphic because the isomorphy type of~ is independent of the real number a in the definition of the matrices g(t, u, v). (e)
{git,
Let
~
be the nilpotent group of matrices
u, v) =
IJ;
u 1
(~
0
u, r, t
which is homeomorphic to JR 3 , and let a
(g 0~
~)
E
p
]Rt j be the matrix
p
with a real number O
~
p
~
1.
We denote by Ethe semidirect
product of~ with the cyclic group . p Then the class S a
p
p
of elements which are conjugate to
in E forms a planar twosided distributive quasigroup
if we set x.y
=
y 1 xy for all x, y ES. p
Two twosided
quasigroups S
and S defined in this manner are isomorP1 P2 phic exactly if P 1 = P 2 •
REFERENCES I.
2. 3.
J. Aczel. Vorlesungen uber Funktionalgleichungen und ihre Anwendungen. Birkhauser Verlag, Basel (1961). A. Borel. Sur la cohomologie des espaces fibres principaux et des espaces homogenes des groupe~ de Lie compacts. Annals of Math. 57 (1953): 115207. B. Fischer. Distributive Quasigruppen ~ridlicher Ordnung. Math. z. 83 (1964): 267303.
275
4. 5. 6. 7. 8. 9. 10.
H. Freudenthal. Oktaven, Ausnahmegruppen und Oktavengeometrien . {Vorlesungsausarbeitung) Utrecht 1960. H. Hopf. Ober die Topologie der Gruppenmannigfaltigkeiten und ihre Verallgemeinerungen . Annals of Math. 42 (1941) : 2252. Fundamentalgruppe und die zweite Bettische Gruppe. Comm. Math . Helvet. 14 {1942) . H. Kneser. Surles varietes connexes de dimension 1. Bull. Soc. Math. Belgique 10 {1958): 1925 . H. Kneser and M. Kneser. Real analytische Strukturen der AlexandroffHalbgeraden und der AlexandroffGeraden. Archiv der Math. 11 {1960) : 104106. T.A. Springer and F.D . Veldkamp . On HjelmslevMoufang planes. Math . Z. 107 {1968): 249263. K. Strambach. Rechtsdistributive Quasigruppen auf Mannigfaltigkeiten . Math. Z. {to appear).
276
PLANES WITH GIVEN GROUPS
Jill Yaqub
Let TI be a projective plane, and let 6 be a collineation group of TI.
Then 6 may be "given" in one or more of the
following senses:
(1) 6 is isomorphic to a specific ab
stract group 6 1 (e.g. "6::::. PSL(3, q) "),
(2) 6, or some sub
set of 6, acts on TI in a prescribed way (e.g. "6 is transitive on the points of TI," or "6 is of Lenz class~ IV," or "6 contains involutory perspectivities"),
(3) 6 has a
given abstract property (e.g. "6 is abelian," or "6 is solvable"). We shall be concerned mainly with finite planes, though
in some cases it is sufficient to assume only that 6 is finite (see sections 3 and 4).
If TI is a plane "with given
group 6," it is assumed that 6 is faithful on TI (i.e. that
o
if
E
6 fixes all points of TI, then
o=
cuss examples of two types of problem: 6
~
1).
We shall dis
I, given that
6 1 (as in (1)), and possibly that there is some relation
between the orders of
TI
and 6 (e.g.
6 : PSL(3, q)"), determine
"TI
is of order q 2 and
(i) the structure of TI,
277
(ii) the
possible actions of 6 on TI; II, given that TI admits 6, where 6 has a partially prescribed action on TI (as in (2)), determine (i) the structure of TI, of 6 on TI,
(ii) the possible actions
(iii) the abstract structure of 6.
A problem of type I is usually solved as follows. (a)
The known abstract properties of 6 are used to show
that TI~ TI(6), where the points and lines of TI(6) are specified objects within 6 (such as subgroups or cosets), and incidence is described entirely in terms of 6.
(b)
An
isomorphism is established between TI(6) and TI' (6'), where
TI' is a known plane which admits 6' way.
~
6 acting in a known
It then follows that TI~ TI' and that 6 acts on TI as
6' acts on TI'.
This procedure is illustrated in some detail
in section 1, where we describe the work of Dembowski [6], [7],
[8] and Unkelbach [26] on planes of order q 2 which
admit a collineation group 6
~
PSL(3, q).
A problem of type II can be reduced to a problem of type I if it can be shown that 6 (or some sufficiently large subgroup of 6) is isomorphic to a specific abstract group. (This approach was first used by Liineburg, 1964a, c. 1 ) To achieve this, it is usually necessary to apply an appropriate deep grouptheoretic characterization theorem to a homomorphic image
r of some subgroup of 6.
Thus the fea
sibility of solving a type II problem by this method depends on the existence of the "right" characterization theorem. 1. References cited only by author and date can be found in the bibliography of Dembowski [SJ. 278
Even then, to show that
r
satisfies the hypotheses of the
characterization theorem and to determine 6 from
r
may re
quire a complicated grouptheoretic analysis involving other deep results.
In section 2 we outline the proof of
a type I result due to Hoffer (17), and then show (for the easiest case) how Kantor (21) reduces a related type II problem to the problem solved by Hoffer. It is usually very helpful to know that 6 contains perspectivities other than the identity.
.Sis a "(P, the line
,Q,
,Q,)
perspectivity" if it fixes all points on
and all lines on the point P; if P
"elation," while if then P and
The collineation
,Q,
PI
.Sis a "homology . "
,Q,,
€
,Q,,
.Sis an
If o "'I 1,
are uniquely determined by .S, and are called
its "centre" and "axis" respectively.
For the basic pro
perties of perspectivities, see Dembowski (5), §3.1.
The
existence of involutory perspectivities in a collineation group of even order can often be deduced from a fundamental theorem of Baer, 1946b. DEFINITION.
The subplane TI'
TI each point x € TI 
TI'
~
TI
is a "Baer subplane" of
is on exactly one line of
TI',
and dually. Baer subplanes are "maximal," in the sense that if TI'
c
TI" 5 TI, where TI"
is a subplane and
of TI, then either TI"= TI' or TI"= TI.
TI'
a Baer subplane
It is easily shown
that if TI' is a Baer subplane of a plane TI of finite order
279
n, then n = m2 and n' has order m. THEOREM (Baer's Theorem on Involutions). volutory collineation of n (i.e. o 2
=
Let n be an in
1 but o ~ 1).
Then
either (i) o is a (P, t) perspectivity for some (P, t), £E_ (ii) the points and lines of n which are fixed by o form a Baer subplane n' of n.
If n is of finite order n, then
in case (i), PE i n is even, and in case (ii), n
=
m2
and n' has order m. An involution of type (ii) is called a "Baer involution."
In studying problems of type I or II, with ltil
even, it is usually the first objective to show that ti contains involutory perspectivities, and that ti, or at least some "large" subgroup of ti, contains no Baer involution. An interesting feature of both proofs in section 2 is the way in which the authors use a theorem of Seib (25) to handle possible Baer involutions in ti.
(For more about
the role of Baer involutions, see Kantor (22).) In sections 3 and 4, we describe some recent results on finite collineation groups which contain nontrivial elations and nontrivial homologies respectively.
Section
3 is centred on a paper by Hering (16) whose results are in a sense complementary to those of Piper, 1965, 1966a. In section 4 we state and briefly discuss results of Piper, 1967, Brown (3),
(4), Hering [14), and Kantor [23).
Our startingpoint is Chapter IV of Dernbowski's
280
"Finite Geometries" [5], and we have systematically excluded any detailed discussion of work described there.
We have
tried to illustrate techniques by giving some parts of the proofs in detail, but there are some obvious omissions, particularly as regards arguments used in applying characterization theorems.
For these, see, for example, Llineburg
1964a, c, Hering and Kantor [20], Kantor [21],
[23].
We
have made no attempt to give a comprehensive survey of recent results; a few additional references on closely related topics are included in the bibliography. lowing are suggested as basic references:
The fol
Dembowski [5],
Dickson, 1901, Feit [9], Gorenstein [12], Hall, 1959, Hughes and Piper [18), Huppert [19], Pickert, 1955, and Wielandt, 1964.
I am grateful to Professor W.M. Kantor for sending me a preprint of [23] and allowing me to describe its results before publication, to Professors
c.
Hering, H. Llineburg,
and S.K. Wong for helpful discussion, and to Dr. J. Sunday for assistance in preparing the manuscript. NOTATION.
Points and lines will be denoted by capital and
lower case letters respectively.
Collineation groups and
their elements will usually be denoted by capital and lower case Greek letters respectively.
c 6 (E) = {o E 6 = {o E 6
I
I
If Eis a subgroup of 6,
co= oo for all o EE} and N6 (E)
o 1 Eo 5 E} are the "centralizer" and "normalizer"
281
of E in 6 respectively.
The centre of the group 6 is de
noted by Z6 and the centre of the skewfield K by ZK. (Note that we regard the class of commutative fields as a subclass of the class of skewfields  i.e. a "skewfield" may have commutative multiplication.) then, for x ES,
group on the set S,
= x} is the "stabilizer" of x in 6. K>
If 6 is a permutation 6
X
=
{o
divide n.
II
6
As usual, , X1Y2 + Y1v(x2, Y2ll = u(x1, Y1)Y2 + v(x1' Y1)v(x2, Y2). (To prove (3) and (4), expand the identity [e(x 1 + y 1e) J (x 2 + y 2e) = e[ (x 1 + y 1e) (x 2 + y 2 e) J .
Note that (3) and (4) correspond to the fact that the set of matrices of the form [ (x ) u x,y multiplication.) THEOREM 1.5.
(y ,] is closed under v x,y
Conversely, Dembowski proves the following.
Given a pair of maps u, v: K
x
K
+
K which
satisfy conditions (1) to (4), there exists a right nearfield F of dimension 2 over K such that, for some basis {l, e}, u, v are nearfield functions of F over K.
288
PROOF OUTLINE.
Define multiplication on F = K
K by
$
(x1, Y1> !FE L(f, A).
(We
write! as a rowvector for typographical convenience, but it should in fact be regarded as a columnvector.) x E W 
L = {L (f, A
{Q}},
J
Let
f E F and A E GL (3, K) }.
Let H = H(F, K) be the incidencestructure (P, L, G). THEOREM 1.6 .
(i)
His a projective plane.
Then:
It contains
a Baer subplane H 0 whose points are the xF with x E V  {Q} and whose lines are the L(f, A) with f EK . a collineation group in general
r
r
~
r
H admits
GL(3, K) which fixes H 0 as a set;
is not faithful on H.
the representation of
(ii)
(III)
on His N = {AI
289
I
The kernel of A EK* n ZF}.
(iv)
The kernel of the reeresentation of r ~ Ho is No =
{AI
>. E ZK*}.
H  Ho
r
(v)
is fla9:transitive on H  Ho, and
(GL(3, K), A, B) ,* where A, B are the sub9:roues
:::::
of GL(3,
which contain all matrices of the form
K)
: ]. I :,. v(x,y)
lu(x,y)
0
0
reseectively; here u(x, y), v(x, y) are nearfield functions of F over K with respect to some basis {l, e} . For the proof of Theorem 1.6, which involves a fair amount of calculation, see [7].
The plane H = H(F, K) is
a "generalized Hughes plane" provided that Fis not a skewfield; if Fis a skewfield, H(F, K) is desarguesian . group r
GL(3, K) acts as follows.
~
Then the point and line maps ~F
+
The
Let CE GF(3, K).
C(~F) and L(f, A)
+
L(f, CA) are clearly bijective and preserve incidence, hence induce a collineation of H. The representation of H  H0 given in (v) is a "HigmanMcLaughlin representation" If a group
r
(Higman and McLaughlin, 1961).
acts on an incidencestructure I= (P, L, E)
(not necessarily faithfully), and is transitive on the flags of I, let A= rx, B = rt, where (X, t) is a flag of I XE P, t
E
L, and XE t) .
within conjugacy in r. as follows.
(i . e.
Then A and Bare determined to
Define the incidence structure J
"Points" are the leftcosets sA of A in r,
*This notation is explained below.
290
r,
"lines" are the leftcosets nB of Bin nB ~An nB is not empty. and
r
and ~A is on
Then J is isomorphic to I,
acts on J by left multiplication.
We write J
=
(r, A, B). Let K
=
ZF f/N
=
GF(q) and consider Theorem 1.6 (iii), (iv) .
K, then N
= N0
and
r
induces the collineation group
PGL(3, q) on H, where f/N is faithful on H0 •
~
If
In (8),
(2.9), Dembowski states that H admits a collineation group ~
PGL(3, q) acting faithfully on H0 only if ZF
r
true that such a group is induced by However,
r
contains a subgroup E
~
=
5, 11, 29), E
ful on Hand on H 0 • (mod 3), SL(3, q)
~
n N
=
E n No
only if ZF = K.
=
t
1
{ 1}.
(mod 3)
=
cial Hughes planes of orders 5 2
PSL(3, q). ,
11 2
,
=1
qt
1
Hence the spe
29 2 satisfy the hy
potheses of Theorem 1.2 and must be accounted for. case q
(i.e.
Thus E is faith
Moreover, as noted earlier, for PGL(3, q)
It is
SL(3, q), and for the
special nearfields of order q 2 with q for q
= K.
In the
(mod 3), E induces PSL(3, q) on H, acting faith
fully on H 0 , provided that ZF
= K.
For the special Hughes
plane of order 7 2 , E induces SL(3, 7) on H but PSL(3, 7) on H0 •
Thus it is fairly clear that in this case H does
not admit a collineation group~ PSL(3, 7), and, as we shall see, Unkelbach's proof excludes the possibility. In view of this difficulty, we shall modify the final stages of the proof of the main theorem in (8), by working in E instead of
r.
We shall prove Dembowski's result only
291
in the case q % 1 (mod 3), since the case q
=1
(mod 3)
For q J 1 (mod 3),
is handled in Unkelbach's paper (26).
we have just seen that E induces on Ha collineation group
~ PSL(3, q) which is faithful on H0 • Eis flagtransitive on H  H0 •
Thus, by Theorem 1.3,
It follows that, for our
purposes, Theorem l.6(v) can be replaced by Theorem l.6(v) '. THEOREM l.6(v) '. H  H0 ~ (E
1,
If K
= GF(q) with q % 1 (mod 3), then
E 1 n A, E 1 n 8), where E 1
=
SL(3, q) and A,
Bare the groups given in Theorem l.6(v). THEOREM 1.7 (adapted from [8]).
Let TT be of order q 2 ,
where q is a primepower% 1 (mod 3).
Let n 0 be a Baer
subplane of TT, and suppose that TT admits the collineation group~~ PSL(3, q), which fixes n 0 as a set.
Then TT is
either desarguesian or a generalized Hughes plane. PROOF.
(1) n 0 is desarguesian, and~ acts as its little
projective group. If~ fixes n 0 pointwise, it must be semiregular on the points of TT  TT , since TT 0
must divide q 4

0
is maximal.
q, which is not the case.
PSL(3, q) is simple,~ is faithful on n 0 •
But then
l~I
Hence, since The result now
follows from Dembowski, 1965c. (2)
~
is sharply transitive on the set of ordered
pairs of lines in TT  n 0 which intersect in TT  TT 0 (and dually).
292
The number of such ordered pairs of lines is (q 4
q)q 2 (q 2


1), and this is the order of 6.
suffices to show that if
o
Suppose that
= 1.
E 1r 
o
o
fixes such a line pair, then
E 6
fixes g, h
1T O ,
1r 
E
Let G, H be the unique points of
1r 0 •
Hence it
where X = g n h 1r 0
which are
on g, h respectively and let x be the unique line of
X.
Let Y
=
(Gu H) n x.
1r 0
on
Then G, H, Y are 3 distinct col
o.
linear points which must be fixed by
Since
o belongs
to the little projective group of the desarguesian plane 1r 0 ,
it follows that
axis R,
=
G u
H.
But
o
induces a perspectivity of
o
also fixes X /
1r
O,
/
R.
1r 0 ,
with
Hence
o=
1
by Theorem 1.4.
(3) 6 is flagtransitive on a flag of from (2)
1r 
1r 0 ,
then
1T

1r 0
z
1T

1r 0 ,
and if (P, R,) is
(6, 6P, 6R,).
This follows
(or Theorem 1.3), and the theorem of Higman and
McLaughlin, 1961, already mentioned. (4) Let (P, R,) be a flag in
unique line of
1r 0
1T

1r 0
,
let g be the
on P and Q the unique point of
1r 0
on R.
Then 6P, 6R, are Frobenius groups, whose kernels are the group of all elations with axis g and the group of all elations with centre Q respectively.
The group 6P n 6R,
is a common Frobenius complement of 6P, 6R, and is sharply transitive on points t Q, I gin in
1r 0
and on lines t g, / Q
1T O •
It follows from (2) that 6P is sharply 2transitive on the set S of points of
1r 0
293
which are not on g, so that
in particular 6P acts as a Frobenius group on S (i.e. 6P is transitive on Sand only the identity fixes more than one The group of all elations in 6 with axis g is a
point).
normal subgroup of 6P which is fixedpointfree on S, and hence is the Frobenius kernel of 6P.
Dually for 6£.
6P n 6£ is the subgroup of 6P which fixes
Q
Since
€ S, it is a
Frobenius complement of 6P, and, dually, of 6£. n is uniquely determined by n 
(5)
n0
•
See [SJ,
p.317. (6)
1
Let K = GF(g), g
 {(O, O)}.
1 (mod 3), and let K = K x K
Let n 0 be the desarguesian plane over K, re
presented by homogeneous point and line coordinates x =
! =
(x, y, z),
[t, m, nJ respectively.
a group of collineations of n 0 tions x Q
=
+
,
Suppose that 8 is
represented by transforma
Ax with A€ SL(3, g), and that 8 fixes the point
(0, 1, 0) and the line g
=
[O, 1, OJ and is sharply
transitive on the set L of lines~ g, / Q.
lLx~ fr
• =
Then
0
X
€ SL(3, g) y)
0
(x, y)
€
v(x,
where u, v are nearfield functions for some F over K, and a 22 is determined by the fact that the matrix has determinant 1. First consider 8 as a group of linetransformations
t
+
!M, where M € SL(3, g).
294
Since Q and g are fixed,
!] ·
0
M=
1
k [~
0
with k 3 ( ad  be) = 1.
Then [1, 1, 0)M = k[a, 1, bl and [0, 1, l)M = k[c, 1, d) .
e
Since
is sharply transitive on L = {[2, 1, n)
(2
EK}, we see that {(a, b)} =Kand that the map¢: with ¢(a, b) = (c, d) is bijective .
t
n)
K+ K
It is easily verified
that c, d, as functions of (a, b), satisfy (1) to (4) of Theorem 1.5 (where we define c(0, 0) = d(0, 0) = 0).
Hence
there exists a nearfield F of dimension 2 over K, and an element e ME
E
F  K such that e(a + be) = c + de for all
e. The linetransformation 2 + 2TM is represented in
pointcoordinates by~+ A~, where
cl
0
k2
[
b
aj
0
Then {l, e'} is also a basis for F over K,
Let ee' = 1. and since 1
0 •
:
E
ZF, e'e = 1.
Since e[(a + be)e'J
(c +
de)e' and since Fis right distributive, e(ae'  b) = ce'  d, whence multiplying on the left by e', b + ae' = e' (d  ce').
Thus b, a are nearfield functions of d, c
with respect to the basis {l, e'} . field of order 5 2
,
11 2
nearfield functions
,
or 29 2
,
If Fis a special near
then ad  be
E
ZF for any
c, d of a, b: this can be verified
by inspection for the representations given in [SJ, p.231,
295
and it can be shown that, for a given f E F, ad  be is independent of the choice of the basis element e. here k
3
Since
(ad  be)= 1, since K has cyclic multiplication,
and since q j
l
(mod 3), it follows that k E ZF.
not special, then certainly k E ZF.
Hence
y
0
X
0
A
But then e' (k 2 d 
ce') = k 2 (b) + k 2 ae'.
k 2 ce') = k 2 e' (d 
If Fis
0
u(x, y)
v(x, y)
0
where u, v are nearfield functions of (x, y) and det A= 1. This implies that A is uniquely determined by its first row (given the basis {l, e'}), where a 22 is determined by the fact that det A= l.
Since lel= li q
"" s4
satisfies ( *) '
=
rR
=
=> q
"" A4
7, 13, or 19,
=
7, 25, or 31, and rR "" As => q For (i) to (iv) ,
49, 61, 79, or 109.
=
I Rr I
q(q
q(q  1)/2, q(q + 1)/2, q(q  1) respectively. JgJ
q(q  1),
(vi),

1) '
Thus since
(iii) can only occur if the remaining orbits
on g are of types (v) to (vii). (v),
19, 25, 31,
Unkelbach now shows that
(vii) cannot occur, for the most part by a
straightforward analysis of the orbit lengths. ample, if rR ""As with q
=
=
25, then JgJ
600,
(For exJRrJ
=
130,
and possible lengths for the remaining orbits on g are 300 (from (ii)), 325 (from (iii) and (vi)), and 130 (from (vii)). However, the equation 130x + 300y + 325z tion in nonnegative integers with x
~
=
1.)
600 has no soluThe case rR""
As' q = 19, requires a special argument, involving the Sylow 5groups of As. (iv).
We are finally left with (i},
(ii), and
Since LR contains the involution o, LR is the com
plete inverse image of rR in L, and we obtain the groups listed in (2). (3)
Z(~Q
,g
) contains a subgroup 0 of index 2 which
consists of (Q, g) homologies. Let RE g.
Then NL(LR) is transitive on the fixed
points of LR in the orbit RL. Lis transitive on
g and
In cases (a) and (c) of (2),
it is known that JNL(LR) J
thus LR has exactly 2 fixed points.
=
2JLRJ;
In case (b), L has 2
orbits and JNL(LR) I = JLRI, so that LR has exactly one fixed point in each orbit.
Let R, R' be the fixed points of LR.
303
Now LR~ ~Q
LR .
,g
, so that Z(~Q
,g
) commutes elementwise with
Thus, at most, Z(~Q ,g ) exchanges Rand R'.
But Z(~Q , g )
induces (faithfully) a group of (Q, g) homologies of the desarguesian plane n 0
,
and hence is cyclic .
Thus Z(~Q
,g )
contains a cyclic subgroup e of index 2 which fixes R, R' . · · R, r R'r. an d h ence f ixes a 11 points in
This means that e
consists of (Q, g) homologies. (4)
~P, l contains a cyclic subgroup
Consider
Q1
of index 4.
Q
1)/6, as in
= ez, withe, of order (q 
(3), and Z the cyclic subgroup of LR, with order (q + 1) or (q + 1)/2, whose existence is assured by (2).
Since
e £ Z(~Q,g), Q 1 is abelian, and since le n zi
(because
(q 
q + 1)
1,
=
2),
IQ 1 1 =
l ~p, £ 1/d, where d
Since ~P,l is a Frobenius complement and Q 1 is cyclic by Q
(2).
Q1
~ 2
=
2 or 4 .
is abelian,
Hence Q 1 contains a cyclic subgroup
of order l~P,ll/4. At this point we have a good deal of information about
~p,t·
In order to pin i t down completely, Unkelbach con
siders "Singer cycles" of ~Q
,g
.
The group G = GL(2, q)
contains a conjugacy class of cyclic subgroups T 1 of order q2

1 which are sharply transitive on the nonzero vectors
of the 2dimensional vectorspace over GF(q). group ~Q,g order (q 2
~ 
Hence the
GL(2, q)/Z 0 contains cyclic subgroups T of 1)/3, the "Singer cycles" of ~Q
,g .
These have
the following properties, which can be deduced from the corresponding properties of the groups T 1 (see (19], pp .
304
187188).
Denote the normalizer in 6Q
Then (i) N(T) not in l:
=
=
=
T
of T by N(T).
(semidirect), where i is an involution
SL(2, q);
2(q + l);
,g
(ii)
(iv) i f ,
E 6Q
l
=
jT n i::j ,g
q + l;
(iii)
jN(T) n l:I
and has odd order dividing
q + 1, it determines a Singer cycle T such that ,
ET;
1
(v) if Tis a Singer cycle, if TE T, and if the order of T does not divide q  1, then and T have the same normalizer in 6Q (5)
,g
6p,£ 5 N(T), where Tis a Singer cycle of 6Q,g·
Consider 6P,£ as a permutation group on the 4 cosets of the cyclic subgroup~ defined in (4).
Let K be the
kernel of the representation, so that K is the intersection of all conjugates of~ in 6p,i·
=
24.
Then j6P,£/KI divides 4!
divides 3 x 24/q + 1, since l6p,£1 only happen if q
=1
=
If IKI divides q  1, so that jKjx
(mod 3)).
=
(q 2

q  1, then x
1)/3.
This can
7 (since we assume q is a primepower
=
For q
7,
l6p,£1
=
16 and 6P,£ is cyclic
or a generalized quaternion by (1).
The Sylow 2groups of
N(T), where Tis a Singer cycle, are in this case Sylow 2groups of 6Q
,g
(both have order 32).
theorems, 6P,£ 5 N(T) for some T, if q
Hence, by the Sylow
=
7.
If q
~
7, then
by properties (iv) and (v) above, there exists TE Kand a Singer cycle T of 6Q
,g
normalizer N(T) in 6Q
,g
such that and T have the same .
But K is normal in 6
is characteristic in K since K is cyclic. N (T).
305
P,£
and
Hence 6P,£
.s
If q
=3
(mod 4), the Sylow 2groups S of N(T) are n quasidihedral (i.e. are of the form with A 2 = i 2 1+2n1 = 1 and iAi = A , where here is a Sylow 2group of T).
They contain exactly one cyclic subgroup of index
2, namely , and one generalized quaternion group U of index 2, namely .
It follows that N(T) contains,
to within conjugacy, just 2 subgroups of index 2 which have a cyclic or generalized quaternion Sylow 2group, namely T and O(T)·U, where O(T) denotes the maximal normal oddorder subgroup
of T.
Thus llP,t = T
or
O(T) •U.
If q  1 (mod 4) , the Sylow 2groups s of N(T) are of 1+2n1 2n the form with A , where = i2 = 1 and iAi = A again is a Sylow 2group of T.
Besides , they con
tain exactly one cyclic subgroup W = of index 2, and no generalized quaternion subgroup of index 2.
Hence here
llP,t =Tor O(T)•W (a nonabelian group). Thus finally, llP,t is determined to within conjugacy in both cases, and, by (1), there are only 2 possibilities for
TI.
Since the desarguesian planes and the Hughes planes
satisfy the hypotheses of Theorem 1.9, they are the only planes which do so. Theorem 1.2 follows from Theorems 1.8, 1.7, and 1.9. REMARKS 1.
Unkelbach states that cases (a),
(b) of (2) cor
respond to the desarguesian and Hughes planes respectively,
306
and that (c) cannot occur. For if 6P,i
~
This is clear if q
If q
(mod 4).
T, the Sylow 2groups of 6P,i are the Sylow
2groups of In N(T), whence l6P,i n II q + 1.
=3
=l
=
2(q + 1), not
(mod 4), then case (c) does not occur.
However, the fact that 2
II IT
n
II
implies that the elements
of order 4 in a Sylow 2group of In N(T) are not in T, n2 and hence must be iA 2 and its inverse: these are not in Hence l~p,i n II = q + 1, not 2(q + 1), so that the
W.
Hughes planes must also arise from (a). 2.
Theorem 1.2 could probably be proved without co
ordinates, using Unkelbach's approach. ption q
=l
However, the assum
(mod 3) is used quite strongly in the proof of
Theorem 1.9,
(2), and without it one would certainly have
additional difficulties with those values of q for which there exists an irregular nearfield of order q 2 •
(In par
ticular, there are 4 nonisomorphic nearfields of order 11 2
,
including GF(ll 2 ) . )
Apart from q = 7, all the irreg
ular nearfields have order q 2 with
qt
1 (mod 3), and here
these are handled by Dembowski's proof, in which it is unnecessary to distinguish between nonisomorphic planes of the same order. ADDENDUM.
The special Hughes plane of order 7 2 has the
interesting property that the group generated by all its elations is isomorphic to SL(3, 7), not PSL(3, 7) Theorem 3.2(d)).
(cf.
The following result, which characterizes
307
the plane in question, was proved by Llineburg during the course of this conference. Let TI be a plane of order q 2 , where
THEOREM (Llineburg). q
=1
(mod 3), and let TI 0 be a Baer subplane of TI.
Suppose
that TI admits a collineation group~~ SL(3, q) which fixes as a set.
TI 0
Then TI is the special Hughes plane of order
72, 2.
UNITALS AND PSU(3, q)
DEFINITION.
U is a "unital of orders" U is a confi
guration with exactly (s 3 + 1) points, s (s + 1) points on each line, and s
2
2
(s 2
s + 1) lines,

lines on each point.
If U is embedded in a plane TI of order s
2 ,
then for
each point XE U there is a unique line T(X) on X, the "tangent line at X," such that T(X) /_ TI
is either a tangent or a line of
+ (s 3 + 1) = s
4
+ s
2
u,
u.
Every line in
since s
2
(s 2

s + 1)
+ 1.
The following are examples of unitals.
(1)
Let
e
a unitary polarity of the desarguesian plane of order q 2 Then the absolute points and nonabsolute lines of a unital of order q. 0(P), and dually . ) 3 2 r+ 1 ,
e
be •
form
(The point Pis "absolute" PE (2)
For each Ree group G(q), q =
· a unita · 1 o f or d er q wh ic . h a dm.its G ( q ) as t h ere is
a 2transitive automorphism group.
However, such a unital
cannot be embedded in a plane TI of order q 2 in such a way
308
that G(g) extends to a collineation group of 1966b) .
(3)
Ganley [10) has shown that unitals occur in
the planes over Dickson semifields. DEFINITION.
(Liineburg,
TI
See also Seib [25).
Let e be a unitary polarity of the desarguesian
plane of order q 2
Then PSU(3, q) is the group of all col
•
lineations in PSL(3, q 2 ) which commute withe. THEOREM 2.1.
Let e be a unitary polarity of the desargue
sian plane of order q 2 commutes withe.
,
and writer= PSU(3, q), where r
Then:
(i) The absolute points and non
absolute lines of e form a unital, U, of order q.
(ii)
/rl = (q 3 + l)q 3 (q 2
(iii)

1)/d, where d = (q + 1, 3).
r is 2transitive on the points of U.
x
(iv) If
Eu,
rx
contains a normal subgroup Q of order q 3 which is sharply transitive on U 
{X}; the centre of Q is an elementary
abelian group T of order q, and the elements of Tare (X, T(X)) elations.
(v) If Jl Eu,
/rJll = (q + 1) 2 q(q  1)/d
and rJ/, contains a normal cyclic subgroup Hof order (q + 1)/d whose elements are (6(Jl),
JI,)
homologies.
(vi)
If q is
~ , each involution in r has centre X and axis T(X) for some XE U.
If q is odd and if o is an involution in r,
then a has axis
JI,
and centre 6(Jl) for some
JI,
EU;
Cr(o)
has 2 orbits on u, of lengths (q + 1) and (q 3

tively (namely Un
r is simple
JI,
and U 
if q > 2.
309
(Un
JI,)).
(vii)
q) respec
For proofs see Huppert [19], pp.242244, or Hughes and Piper [18], pp.5763.
Most of the results can be de
rived quite easily from the fact that, in homogeneous coordinates, 0 ( x 1 , x 2
,
x 3 ) 
[xq , xq 3 2
,
. a canonica · 1 x ql is 1
form for 0. A polarity of a desarguesian plane is "unitary" it corresponds to a hermitian form in the associated 3dimensional vectorspace.
For a general plane of order s
2
(not necessarily desarguesian), a polarity is said to be "unitary" its absolute points and nonabsolute lines form a unital of orders.
The two definitions are equi
valent for the desarguesian planes of order q 2
By a theo
•
rem of Baer, 1946a, a plane of order n can admit a unitary polarity only if n is a square. The aim of this section is to describe briefly the proofs of two theorems concerning unitals and PSU(3, q). The first is of type I and the second of type II, in the sense of the Introduction. THEOREM 2.2 (Hoffer [17]).
Suppose that TI is of order q 2
and admits a collineation group~~ PSU(3, q). TI
is desarguesian,
~
acts on TI as
r
Then (i)
(ii) for some unitary polarity 0 of TI,
acts in Theorem 2.1.
THEOREM 2.3 (Kantor [21]).
Let TI be of order q 2
,
0 a uni
tary polarity of TI, and Uthe unital formed by the absolute points and nonabsolute lines of 0.
310
Suppose that TI admits
a collineation group 6 which commutes with 0 and is flagtransitive on U.
Then
TI
is desarguesian and 6 2 PSU(3, q).
We first state a theorem due to Seib, which plays an important role in the proofs of Theorems 2.2 and 2.3.
It
is proved by a simple but ingenious counting argument. THEOREM 2.4
(Seib (25]) .
embedded in a plane tion of
TI
TI
Let Ube a unital of orders,
of order s 2 •
which fixes U as a set.
exactly (s + 1) fixed points of a,
Leto be a Baer involuThen (i) U contains (ii) these points are
collinear i f s is even but form an oval i f s is odd. PROOF OUTLINE FOR THEOREM 2.2. wise
Assume q > 2 (since other
is certainly desarguesian).
TI
(1)
6 fixes no point or line and has a pointorbit 0 6 is 2transitive on 0, and acts on O
of length q 3 + 1.
as r = PSU(3, q) acts on the subgroups r , XE U, in the X
representation given in Theorem 2.1. Since q > 2, 6 is simple, and hence is faithful on each point or line orbit of length> 1.
If 6 fixes a line
£, then it fixes all points on£, since it is known that 6 cannot be represented nontrivially on less than q 3 + l points if q i 5, and on less than 50 points if q = 5.
Sim
ilarly 6 must fix each line on a point XE£, so that 6 fixes
TI
pointwise.
no fixed point.
Thus 6 has no fixed line, and dually
The maximal subgroups of PSU(3, q) were
311
determined by Mitchell, 1911, and Hartley, 1926, so that all possible orbit lengths x with l be listed.
p
= 3.
(q + 1, q 2
It follows easily that pr= 3 2
,
+ 1
3
q + 1)

whence q = 2.
Henceforth we assume q > 2, since otherwise
TI
is des
arguesian .
(3)
~
contains an involutory (X, T(X)) elation, for
each x Eu. Let I be a Sylow 2group of~. odd, I fixes some point XE u.
Then, since lul is
If all involutions in~
are Baer involutions, then there exists a Baer involution o E ZI.
By Theorem 2.4, the q + 1 fixed points of o in U
are on a line i EU, where XE i. ~
X
Clearly I fixes i.
I~ X I
is transitive on the q 2 lines of U on X, so that
= q2
1~ x,t I
This implies that I is not of maximal 2power
order in ~x' since q is even. involutions,
But
~
Hence, by Baer's theorem on
contains involutory elations.
It is easily
verified that an involutory elation which fixes XE U must have centre X and axis T(X).
Since~ is transitive on the
points of U, it contains an involutory (X, T(X)) elation for each XE U. (4)
If~ contains a Baer involution fixing XE U,
then all involutory (X, T(X)) elations commute. Leto be a Baer involution which fixes XE u, and hence fixes the points oft n U for some ion X, t by Theorem 2.4.
~
T(X),
Let a, $ be involutory (X, T(X)) elations.
315
Then (oa)
is an (X, T(x)) elation which fixes all points
2
of i n U, and hence is the identity. clearly oa is a Baer involution.
=
(oa)S
S(oa).
whence aS (5)
Thus (aS)o
=
o(aS)
Thus oa = ao, and
Similarly, aS = So and
=
(oa)S
=
S(oa)
=
(Sa)o,
=
Sa.
~
contains a normal subgroup N, transitive on the
points of U, all of whose involutions are elations. By (3), for each XE
~
u.
contains an involutory (X, T(X)) elation If 6 also contains Baer involutions, then
by (4), for given XE Uthe involutory (X, T(X)) elations form, together with the identity, an elementary abelian 2group which is clearly normal in each Sylow 2group of 6 . X
The existence of N now follows from a theorem of Shult (24]
(6)
(deep grouptheoretic)
(see also Goldschmidt (11]).
~ contains a subgroup K ~ PSL(2, q
3 ),
Sz(q
1/2 ) , or
PSU(3, q), acting on the points of U in the usual 2transitive representation; q is a power of 2.
(Here Sz(s) de
notes the Suzuki group of order (s 2 + l)s 2 (s  1), where
s = Each involution in N is in fact an (X, T(X)) elation for some XE
u,
as is easily verified.
by (2), since otherwise O(~) q > 2.
t
Also O(N) = {l},
{l}, and we are assuming
Since each involution in N fixes exactly one point
of U, it now follows from a theorem of Bender [2] that N contains a normal subgroup K isomorphic to one of the groups listed in (6), and that K acts on the points of U in the
316
usual 2transitive representation. If K ~ PSL(2, q 3 ) or 3/ Sz(q 2 ) , then, for XE u, 6 contains a group of order q 3 or q
X
3/,
respectively which is an elementary abelian 2group
2
and hence consists of (X, T(X)) elations .
This is impossi
ble, since 6 contains at most q (X, T(X)) elations. K
~
Hence
PSU(3, q), and we can apply Theorem 2.2.
REMARK.
The proof of Theorem 2.2 relies on the comprehen
sive analysis of the maximal subgroups of PSU(3, q) made by Mitchell, 1911, and Hartley, 1926, but requires no modern group theory; the proof of Theorem 2.3 involves three recent and very deep grouptheoretic results.
This contrast
is typical of the difference between the grouptheoretic methods used in solving type I and type II problems. 3.
COLLINEATION GROUPS WHICH CONTAIN ELATIONS
Finite collineation groups which contain nontrivial elations have been studied extensively, in particular by Wagner, 1959, and Piper, 1963, 1965, 1966a. this fundamental work is given in [SJ, §4.3.
An account of Here we shall
describe the main results of a recent paper by Hering [16]. THEOREM 3.1. group of TT. in 6. in M, in 6}.
Let TT be a finite plane and 6 a collineation Let Ebe the group generated by all elations
Let A= {a
z
= {z
I
a is the axis of a nontrivial elation
Z is the centre of a nontrivial elation
Then, up to duality, one of the following holds:
317
I
Z =
~
II
z
{Z},
III
For some P, t with P /
IV
For some a, A= {a},
V
For some Z, a with z E a,
= (j).
lines on
A= {a}, for some Z, a with Z Ea.
For some
VII
The points of
VIII
~=a and
z
£
l~I
z
X
E
~}.
> 1. £
[Z]
any
z
a, A
z) , I~ I > 1 and l~I > 1. a, z = a but A i [Z] for
VI
of
t, Zs t and A= {XP
(the set of
E
z.
and the lines of A form a subplane
11.
There exists a subplane hyperoval Hof
11'
of order 4 in
11'
11
and a
(i.e. a set of 6 points, no 3 col
linear), such that~=
11'

Hand A is the set of
all lines which meet Hin 2 distinct points. IX
If x E ~, then the stabilizer ~xis of type II . ~
X
does not fix any pointline pair .
If XE~, then ~xis of type II, IV or IVd (the dual of IV). IV.
There exists x EA such that ~xis of type
Also~ fixes no centre or axis.
REMARKS 1.
The theorem is a generalization of the "Lenz clas
sification"
([15], p.126) .
Note that types I to VII are
entirely analogous to the Lenz classes with the corresponding number. 2.
The theorem is stated in terms of the sets~,~
However, it can also be regarded as a classification according to the figure formed by those incident pointline 318
pairs {Z, a) such that 6 contains a nontrivial {Z, a) elation .
For if Z E Zand a E ~, there exist a', Z' such
that 6 contains a nontrivial (Z, a') elation a and a nontrivial (Z', a) elations.
If Z Ea, and if z
i
Z',
at
a',
then it is easily verified that aSa 1 s 1 is a nontrivial (Z, a) elation (Wagner, 1959).
Thus 6 contains a non
trivial (Z, a) elation z E ~, a E ~, and Z Ea. PROOF OF THEOREM 3.1 (1)
If Z =~and if E fixes a EA, then 6 is of
type II, IV, IVd, V, or VI. If E fixes a, then~ Ea. a, then E fixes 6(a), since E
(If 6 E 6  E does not fix ~
6.
Thus E fixes an 6(a),
which is thus the only possible centre.) or IV if A= {a}, of type II or IVd if
l~I
6 is of type II = 1, and is
otherwise of type V or VI. We can now assume that E fixes no axis, and dually no centre.
VI .
(2)
For each a EA, 6
(3)
If E fixes no centre or axis, and if there exists
is of type II, IV, IVd, or a This follows immediately from (1).
a E A such that 6
a
is of type V or VI, then 6 is of type
VII or VIII.
If E fixes a point F, then F I that F E a for each a by hypothesis
l~I
~
E
2.
A.
~
by hypothesis, so
But this implies F
E
~, since
Hence 6 fixes no point or line.
319
The conclusion now follows from results of Piper, 1965, 1966a.
(In 1966a, Piper does not exclude the possibility
that the centres and axis of elations in 6 should form a set of disjoint subplanes of order 2.
Hering states that
this can be ruled out by applying a theorem of Bender [2).) We can now assume:
(*)
~ , ~
If a EA then 6
a
is of type II, IV, or IVd.
Note that (*) => (*) '.
(*)' If Z E Zand if a E ~, then 6Z, 6a are of types II, IV, or IVd.
(4)
If I fixes no centre or axis and if 6 satisfies
(*), then 6 is of type III, IX, X, or Xd. Assume 6 is not of type X or Xd.
Then for each a E ~,
is of type II. If I fixes (P, t), then PI t, for othera wise it is easily shown that PE Z or t EA. If I fixes 6
some (P, t) with
PI
t, then 6 is of type III, and other
wise 6 is of type IX. REMARK.
Examples of groups of each of the types I to X
(and, of course, their duals) can be found in desarguesian planes.
In particular, the plane of order 4 admits a group
of type VIII (Piper 1966a), in the desarguesian plane of order q 2
,
PSU(3, q) is of type IX (see section 2), and in
the desarguesian plane of order q = 2r, the group generated by the elations which fix a given conic (~ PSL(2, q)) is of type X (see Dembowski (5), p.185).
320
The remainder of (16) is concerned with the structure of the group E. THEOREM 3.2. group of
1T
tions in
1::,.
I
Let
be a finite plane,
1T
I::,
a collineation
and let E be the group generated by all elaLet p be the smallest prime such that
tains an elation of order p. pointwise}.
Then:
(a) If
t:,
Let K
I::,
E
I
con

0 fixes Z

is of type IV, Eis an ele
mentary abelian pgroup.
(b) If
group of class 2.
I::,
(c) If
= {o
I::,
t:,
is of type v, Eis a p
is of type VI and .if T is the
group of all elations with axis a, then either (i) E/T "' r SL(2, q) , where q = p I (ii) p = 3 and E/T " SL(2, 5) ,
(iii) p
=
2 and E/T
not divide jE/Tj.
(d) If
=
"PSL(3, q), where q in
t:,
has order p.
2.
,
t:,
2 and 4 does
is of type VII, then E/E n K
r
p , and every nontrivial elation
Also E n K ..; E' n Z (E)
commutator group of E). E/E n K "A 6
=
Sz(22r+1), or (iv) p
"
(e) If
t:,
(where E' is the
is of type VIII, then
and every nontrivial elation int:, has order
Again En K £ E' n Z(E). For proofs, see (a) Baer, 1942,
(c) Hering [ 151,
(d) Piper, 1965 and 1966a,
The fact that E n K proved in [ 161 •
(b) Hering, 1963,
£
(e) Piper 1966a.
EI n Z(E) for types VII and VIII is
It means that E is a "Schur extension" of
E* = E/E n K, so that the theory of the Schur multiplier can be used in determining E from E*. Huppert,
(19), p.628.)
321
(See, for example,
The corresponding problem for classes III, IX, Xis not completely solved (and appears to be difficult).
How
ever, if~ contains involutory elations, Hering is able to determine the group generated by all involutory elations in~
The proof depends on the following deep grouptheo
retic result, which is also due to Hering (13). THEOREM 3.3.
Let
r
be a finite group which is transitive
lwl
on the set W, where
> 1.
Suppose that for XE
w, rx
contains a normal subgroup N of even order such that NY= {l} for all YEW  {X}. in
If t i s the normal closure of N
r (i.e. the group ), then either t = N•O(t)
and N is a Frobenius complement, or t SU(3, q), or PSU(3, q), where q = 2 THEOREM 3.4.
r
~ ~
SL(2, q), Sz(q), 4.
Suppose~ satisfies the condition (*), fixes
no centre or axis, and contains involutory elations.
Let
Ebe the group generated by the involutory elations in~If 4 divides IE!, then
I~
SL(2, q), Sz(q), SU(3, q), or
PSU(3, q), where q = 2r ~ 4. PROOF.
Let a be an involutory (X, a) elation in~
By
(*), we can assume (**)
for each axis a' on X, Xis the only centre on a'.
(Otherwise the dual holds, and the proof is dualized throughout.) Let W = {o(x)
I
o E ~}.
322
Then if YE
w,
Y also
satisfies (**).
Let N be the group of all elations in 6
which have centre X. Y
€
w
Then N
6X,
4
{X}, NY= {l} by (**).
since Xis not fixed by 6. in 6, S must fix some Z
€
Thus
w.
is odd, and
lwl
> 1
It follows from (**), applied
s.
Then Es i.
3.3 to 6, W, N, and i.
lwl
If Sis any involutory elation
to Z, that Z is the centre of closure of Nin 6.
INI is even, and if
Let i be the normal We can now apply Theorem
If i = N•O(i) with Na Frobenius
complement, then N contains a unique subgroup N 1 of order 2, and all involutions in i belong to N 1 •o(i), whence 4 Otherwise, i ~ SL(2, q), Sz(q), r SU(3, q), or PSU(3, q), where q = 2 ~ 4, and finally E
does not divide IEI.
~
since these groups contain no proper normal subgroup of even order. REMARKS
1.
If 6 contains involutory elations and is not of
type II, then Eis determined, by Theorems 3.2 and 3.4. This is a very important and impressive result, which obviously has many possible applications to problems concerning collineation groups of finite planes of even order. 2.
One cannot reasonably expect to obtain an analog
ous result for type II by these methods.
For the known
finite planes, groups of type II are elementary abelian. 3.
Another application of Theorem 3.3 will be given
in section 4, and it seems that it may well be useful in other geometrical contexts. 323
4.
COLLINEATION GROUPS WHICH CONTAIN HOMOLOGIES
Homologies are in general more difficult to deal with than elations, and the results to be discussed here are not yet as complete as those described in section 3 .
We shall
mainly be concerned with groups which contain involutory homologies, but first mention two results which also apply to homology groups of odd order. THEOREM 4.1 (Piper, 1967).
Let n be a finite plane, and
let~ be a collineation group which contains nontrivial homologies and fixes no point or line.
If there exists a
line£ such that£ contains the centre of a nontrivial homology in~ and£ is the axis of 2 nontrivial homologies in~ which have different centres, then the centres and axes of homologies in~ form a desarguesian subplane n 0
,
and~ restricted to n 0 contains the little projective group of n 0 • THEOREM 4.2 (Brown (3),
(4)).
Let n be a finite plane,
and let~ be a collineation group of n which fixes no point or line and contains homologies of orders.
Then the num
ber of orbits of centres (axes, centreaxis pairs)
of or
ders homologies in~ is 1 i f s > 2, and is 1 or 2 i f s = 2. Theorem 4.1 is deduced from Piper's results on elations (1965).
The proof of Theorem 4.2 depends mainly on count
ing arguments; additional numerical information is given
324
about the 2orbit case for s = 2.
Theorem 4 . 1 was used in
the proof of Theorem 1.8. REMARK.
It is natural to ask whether Bering's "Lenz" clas
sification of finite groups which contain elations can be refined to a "LenzBarlotti" classification of finite groups which contain elations and homologies (see Theorem 3.1 and [5], p . 126).
This should be feasible; the subdivision of
type I would presumably include all the analogues of the Barlotti subclasses of Lenz class I, and perhaps others. The following theorems are fundamental in the study of collineation groups which contain involutory homologies. THEOREM 4.3 (see [5], p.120). (P 1 ,
(P 2
R 1 ) ,
=
either P 1 p
2
E
,
Let a
a 2 be nontrivial
1,
homologies such that a 1 a 2
R 2 )
P 2 and
=
R 1
£E_ P 1 ~ P 2
R 2 ,
,
R 1
~
1
1
(P 2
= a2a 1 •
Then
and P 1
R 2
R 2 ,
E
,Q, 1 •
PROOF. dually.
If
R 1
If P 1
= R 2 but P 1 ~
P2
,
R 1 ~
P2
~
R 2 ,
,
a 1a 2 a 1 a 2
then a 1 a 2 a
1
1
)
~
P2
;
and
= a 2 implies
THEOREM 4.4 (Ostrom, Llineburg; see Dembowski [5], p . 120). Let cr 1 , cr 2 be involutory (P 1 , respectively, where P 1 Then (a) cr 3
~
P2
,
R 1 ) R 1 ~
and (P 2 R 2 ,
P1
= cr 1 cr 2 is an involutory (P 3 ,
,
R 2 )
homologies
E
R 2 ,
and P 2
R 3 )
l.
R.) l.
homology, i = 1, 2, 3.
325
R 1 •
homology, where (c)
only involutory (P . ,
E
cr . is the l.
The situation considered in the next theorem probably cannot occur: if it does, the theorem tells us a good deal about the group. THEOREM 4.5 (Hering (14]).
Let
TI
be a plane
(not neces
sarily finite), and let 6 be a finite collineation group of
Let Z
TI.
=
{Z I there exists a E Z such that there is
more than one involutory (Z, a) homology}.
Let Ebe the
subgroup of 6 generated by the perspectivities with centre
z
E
z.
1z1
If
> 1, then E "'SL(2, q), Sz(q), SU(3, q),
or PSU(3, q), where q PROOF.
= 2r
4.
~
(1) For each z E ~, there exists exactly one line
a / Z such that
admits a nontrivial (Z, a) homology.
TI
Assume that there exist 2 distinct involutory (Z, a) homologies a 1 , a 2
a~
logy, with
a.
,
and that a is a nontrivial (Z, Then must act
not on Z, with r
a
a)
homo
as a Frobenius
as a Frobenius
must contain only one involution,
a contradiction. (2)
Let Z E ~, and let N be the group of all (Z, a)
homologies in 6.
Then N ~ 6 2
of N fixes Z' E Z with Z'
~
,
INI is even, and no element
z.
The first two statements are obvious. Z' E ~, where Z' ~Zand a~ 1, then Z' Ea.
the unique homology axis a' of Z' must be on contradicts Theorem 4.4(c).
326
If a EN fixes Moreover,
z.
But this
(The group N is in fact the
group of all perspectivities inn with centre Z, since a nontrivial elation with centre Z would move the axis a.)
(3)
The group I is transitive on~, and is the normal
closure of Nin n. By (2), for each Z
E
~,
there exists an involution
which fixes Z but no other point of~tive on~ by a theorem of Gleason, 1956.
Hence I is transiIt follows that
I is the normal closure of Nin~, i.e. the group generated by N and all its conjugates in~ . We can now apply Theorem 3.3.
The case I= N.O(I)
does not occur, since N contains more than one involution. (Hering also shows that if~ fixes no point or line, then I "' SU ( 3, q) or PSU ( 3, q) • )
We shall conclude by giving a brief account of some recent results of Kantor (23).
The starting point is the
following theorem. THEOREM 4.6 (Kantor (23),
(21)).
Suppose that the plane n
(not necessarily finite) admits a finite collineation group ~
which contains commuting involutory homologies with dis
tinct axes.
Then (i) for each (P, 2) En with P /
exists at most one involutory (P, 2) homology in~, ~
2, there (ii)
contains no elementary abelian group of order 8 which is
generated by 3 involutory homologies. PROOF.
327
involutory homologies respectively, where £ 1 by Theorems 4.3, 4.4, P 1
P2
~
£1
,
£2 , P 1
~
E
~
£2
•
Then,
£2
,
P2
E
=
and o. is the only involutory (P., £ . ) homology, i 1
1
1
£1 , l, 2.
Suppose that for some (P, £), there exist 2 distinct involutory (P, £) homologies
der .
which is not the case.
T1,
T 2
0
1
T1•
Hence, by Theorems 4.3
is an involutory (Q, m) homology,
0 1
a commutes with o 1 but o f o 1 •
£ 1 because
By Theorem 4.4 applied too and o 1 , o is the only involutory
(P, £), and by Theorems
Hence (Q, m) f
(Q, m) homology.
4.3, 4.4 applied too and
T1,
must be the only involutory
T1
(P, £) homology, a contradiction. (ii)
Let K be a Klein 4group generated by involutory a ,
a
1
a2
f
u A2
,
A1
E
a 2 , and A 2
Suppose that a
•
which commutes with fixes Ai' a . , i 1 some i
=
~, i = l, 2.
E
1
=
E
E
~
al,
a
1•
=
Let A 3
a
1
n
a2
a
,
3
,
=
A1
is an involutory (A, a) homology 2
1, 2, 3.
1, 2, or 3.
a
By ( i) , A 1 f A 2
(hence with
=
Hence A
This implies
a
a3
=
A. and a
=
Then
a1a2).
1
ai, i.e.
= a
a. for 1
E
K.
The importance of Theorem 4.6 lies in the fact that, with certain additional restrictions on~, it permits the 328
a
application of the following deep characterization theorem . THEOREM 4.7 {Alperin, Brauer, and Gorenstein (1)).
r
Let
be a finite simple (nonabelian) group which contains no elementary abelian 2group of order 8. PSL{3, q), PSU{3, q), M11 odd prime power .
,
Then
r
~
PSL{2, q),
A 7 , or PSU{3, 4), where q is an
{Here M11 is the Matthieu group of degree
11, and A 7 the alternating group of degree 7.) Before describing Kantor's main results, we mention that (23) contains a number of lemmas which are certainly of independent interest, one of which is the following. THEOREM 4.7.
Let~ be a collineation group of a finite
plane of odd order.
If~ contains an involutory homology,
then so does the centre of a Sylow 2group of~. PROOF OUTLINE.
Leto be an involutory homology in ~ such
that a Sylow 2group Since
r
r
of C~ (o) is as large as possible.
is not a Sylow 2group of~, we can find~
that rs~ and r is of index 2 in~. involution
TE
s
~
such
We can construct an
Z~ which is the product of commuting invo
lutory homologies, and hence is a homology, by Theorems 4.3, 4.4.
This contradicts the maximality of
r.
The paper also contains useful results on groups which are generated by 2 involutory homologies o, T with OT of primepower order,and on collineation groups of primepower
329
order which are inverted by an involutory homology.
The
following lemma will be used here in outlining the proof of Theorem 4.l0(i).
The proof is straightforward, and uses
Theorem 4.4. THEOREM 4.8. where
r 1 , r2
Suppose that
r1
x
r2
is a collineation group,
are dihedral groups of order 2p with pan
odd prime, and suppose further that all involutions in
r 2 are homologies.
Then all elements of
r1,
r 1 are perspec
tivities with the same centre or axis. The main object of [23] is to study the following situation.
(*)
Let TI be a finite projective plane of odd order
n, and let~ be a collineation group of TI which is generated by involutory homologies.
Assume that~ contains commuting
involutory homologies with different axes, and that there is no involutory homology cr such that O(~)
E
Z(~/0(~)).
The assumption on Z(~/0(~)) excludes, for instance, the group of all projective collineations which fix a line in a desarguesian plane of odd order; it also plays an important role in the grouptheoretic parts of the proofs. We omit the statement of the first main theorem (Theorem A) of [23], since it is rather technical.
The other
main theorems (Theorems Band C) are stated in Theorems 4.9, 4.10.
330
THEOREM 4.9.
Let n,
~
satisfy (*), and suppose that~
contains no Baer involutions. holds:
(a)
~
Then one of the following
" PSL(2, q), PGL(2, q), PSLA (2, 9), PSL(3, q),
SL(3, q), PSU(3, q), SU(3, q), A 7
,
or PSU(3, 4); here q is
an odd primepower and PSLA(2, 9) is a nonsplit central extension of PSL(2, 9) by a group of order 3. a 2subgroup
N
=
x
with 1¢ 1 1
O(~) ~~and ~/N" S 3 ;
= ~
(b)
~
has
1¢ 2 1 ~ 2 such that induces S 3 on the
set {X, Y, Z} of centres of the involutions in the Klein group E ofand~ 1
2

xyz
= N
~
O(~) x E.
The proof of Theorem 4.9 uses some very sophisticated group theory.
The basic geometric distinction between
cases (a) and (b) is that in (b) there exists a triangle whose vertexset is fixed by~, while in (a) the case.
this is not
(In (22], §2, case (b) does not arise, since
it is assumed there that~ contains no proper normal subgroup of even order.)
Essentially, the fact that n and~
satisfy the hypotheses of Theorem 4.6 permits the application of Theorem 4.7 (though of course~ is not assumed to be simple here).
The case corresponding t o r " M11 in
Theorem 4.7 cannot occur here, and the groups which appear in Theorem 4.9(a) but not in Theorem 4.7 mostly arise as Schur extensions of the related groups in Theorem 4.7. The apparent "odd men out" in Theorem 4.9(a) are PSU(3, 4) and A7 •
While PSU(3, 4) probably cannot occur, A7 is in
fact a subgroup of PSU(3, 5). 331
THEOREM 4.10. (i)
Let TI,~ be as in Theorem 4.9.
If~~ PSL(3, q) or SL(3, q), then TI contains a sub
plane TI' of order q which is fixed by~ and on which~ induces PSL(3, q). of TI.
Also q
(ii)
I
All elations of TI' are induced by elations n,
(n  1), and (q + 1)
(q  1)
(n 2  1).
If~~ PGL(2, q) with (q, n) > 1, then TI contains a
subplane TI' of order q which has an orthogonal polarity Moreover q
preserved by~
+
(q
1).
(n 2
1)
(iii)
with (q, n) > 1, then
TI
I
n,
(q  1)
( n  1) , and
If~~ PSU(3, q) or SU(3, q)
contains a subplane
which has a unitary polarity preserved by~. q
I
n2
,
(
I (n if 5 I
q  1)
If~~ A 7 and
 1) , and (q + 1)
of order q 2
TI'
(n 2
Moreover 1).

(iv)
n, then TI contains a subplane of or
der 5 2 which is fixed by~ and on which~ induces A 7 • (Statement (iv) is not contained in the preprint of (23), but was mentioned to me by Kantor in a letter.) Theorem 4.10 (i),
(iii) give very strong generaliza
tions of some of the results described here in sections 1 and 2.
(For the corresponding analogue of (ii), see Lune
burg, 1964c.)
Note that in (i) there is no restriction
whatever on the relation between n and q, and that the relations demanded in (ii) to (iv) are quite slight: essentially they ensure that, with pr= q in (ii),
(iii) and
p = 5 in (iv), a Sylow pgroup of~ fixes at least one point.
It is, of course, assumed here that~ contains no
Baer involution. 332
The proofs of (i) to (iii) use the lemmas mentioned (Theorem
earlier, and do not involve deep grouptheory.
4.10 is a "Type I" theorem, in the sense of the IntroducWe illustrate briefly by giving the proof of the
tion.)
first part of (i) for ti " PSL(3, q). r By considering PSL(3, q) acting on the Let q = p
.
desarguesian plane of order q, we can find a subgroup r
1
x
r
2
of ti, where r
1 ,
r
2
are dihedral of order 2p.
By
Theorem 4.8 we can assume without loss of generality that r
1
consists of perspectivities with centre X (otherwise
dualize).
Suppose ti fixes X.
Then since all involutions
are conjugate in ti and ti is generated by its involutions, ti consists of perspectivities with centre X: this contra
dicts (*).
Leto be an involution and Tan element of
order pin r
1•
Then Cti(o)
~
tix, and Tis centralized by
some Sylow pgroup E of ti, whence E £ tix.
Since Cti(o) is
of index g 2 (q 2 + q + 1) in ti, the group generated by Cti(o) and Eis of index at most (q 2 + q + 1) in ti.
But since
ti~ tiX, this in fact implies that tix is of index precisely
q 2 + q + 1 in ti, and is isomorphic to the stabilizer in PSL(3, q) of a point in the desarguesian plane Thus lxrl
= q 2 + q + 1, and
on the points of
TI.
rr
of order q.
ti acts on Xr as does PSL(3, q)
The involution o therefore fixes
q + 2 points of Xr, and Cti(o) permutes these in orbits of lengths 1 and q + 1.
It follows that the axis of o con
tains q + 1 points of xr.
Now ti is 2transitive on Xr, and
333
it is easily deduced that the points of
Xr
and the lines
which meet it in at least 2 points form a subplane TI' of order q ((SJ, p.138).
Finally, since~ is faithful on
xr
and~~ PSL(3, q), TI' is desarguesian, by Dembowski, 1965c. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
J. Alperin, R. Brauer, and D. Gorenstein. Finite simple groups of 2rank two (to appear). H. Bender. Transitive Gruppen gerader Ordnung denen jede Involution genau einen Punktfestlasst. J. Algebra 17 (1971): 527554. J. Brown. Homologies in collineation groups of finite projective planes I. Math. Z. 124 (1972): 133140. Homologies in collineation groups of finite projective planes II. Math. Z. 125 (1972): 338348. P. Dembowski. Finite Geometries. Springer, New York (1968).   Zur Geometrie der Gruppen PSL 3 (q). Math. Z. 117 (1970): 125134. Generalized Hughes planes. Canad. J. Math. 23 (1971): 481494. Gruppen erhaltende quadratische Erweiterungen endlicher desarguesscher projektiven Ebenen. Arch. Math. 22 (1971): 214220. W. Feit. Finite simple groups, Proc. International Congress, Nice, 1970, vol. 1: 5593. M. Ganley. Polarities in translation planes. Geom. Dedicata 1 (1972): 103116. D. Goldschmidt. 2fusion in finite groups. Ann. Math. 99 (1974): 70117. D. Gorenstein. Finite groups. Harper, New York (1968). C. Hering. On subgroups with trivial normalizer intersection. J. Algebra 20 (1972): 622629. Eine Bemerkung iiber Streckungsgruppen. Arch. Math. 23 (1972): 348350. On shears of translation planes. Abh. Univ. Hamburg 37 (1972): 258268. On involutorial elations of projective planes. Math. Z. 132 (1973): 9197. A. Hoffer. On unitary collineation groups. J. Algebra 22 (1972): 211218. D.R. Hughes and F. Piper. Projective Planes. Springer, New York (1973). B. Huppert. Endliche Gruppen I. Springer, Berlin (1967). 334
20. 21. 22. 23. 24. 25. 26.
C. Hering and W. Kantor. On the LenzBarlotti classification of projective planes. Arch. Math. 22 (1971): 221224. W. Kantor. On unitary polarities of finite projective planes. Canad. J. Math. 23 (1971): 10601077. Those nasty Baer involutions. Proc. Internat. Conf. on Projective Planes at Washington State Univ. (1973): 145155. On the structure of collineation groups of finite projective planes (to appear) . E. Shult. On the fusion of an involution in its centralizer (to appear). M. Seib. Unitare Polaritaten endlichen projektiven Ebenen. Arch. Math. 21 (1970): 103112. H. Unkelbach. Eine Charakterisierung der endlichen HughesEbenen. Goem. Dedicata 1 (1973): 148159.
ADDITIONAL REFERENCES R.C. Bose. On a representation of Hughes planes. Proc. Internat. Conf. on Projective Planes at Washington State Univ. (1973): 2757. T. Czewinski. Collineation groups containing no Baer involutions. Ibid. 7175. On collineation groups that fix a line of a finite projective plane (to appear). W. Kantor. On homologies of finite projective planes. Israel J. Math. 16 (1973): 351361. M. O'Nan. Characterization of u 3 (q). J. Algebra 22 (1972): 254296.
NOTES ADDED IN PROOF
1.
In Theorem 1.5, hypothesis (1) should be replaced
by the following condition.  {(0,0)}, the matrix [u(~,y)
(1)'
For each (x,y) EK
x K
v(~,y)] is nonsingular.
(Condition (1)' is necessary for the existence of multiplicative inverses in F, but it is not obvious that (1)' follows from (1)  (4)).
"Nearfield functions", as de
fined on page 288, clearly satisfy (1) ', and so do the
335
functions which occur in part (6) of the proof of Theorem 1.7.
Thus the result of Theorem 1.7 is not affected. 2.
When these notes were written, I was unaware
that the special Hughes planes were already known to be selfdual.
(L.A. Rosati:
Sui piani di Hughes general
izzati e i loro derivati. (1967)).
Le Mathematiche 22, 289302
By using this result, the proof of (6) in
Theorem 1.7 can be simplified.
(See the remark after
Theorem 1.7). 3.
It was pointed out by Professor Luneburg that
there is a mistake in the published proof that the case q =19 cannot occur in part (2) of the proof of Theorem 1.9.
However, this can be rectified, and the result of
Theorem 1.9 is correct. 4.
A unified coordinatefree proof of Theorem 1.2
has now been given by Luneburg generalized Hughes planes".
("Characterizations of
To appear in Canad. J. Math.).
He also shows there that in Theorem 1.1 "every perspectivity" can indeed be replaced by "every elation" when n
is finite.
336