Finite Groups of Automorphisms: Course Given at the University of Southampton, October-December 1969 0521082153, 9780521082150

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Moyatefoyam\it-laatclaar-haler=] mexeyer(=18\ Lecture Note Series

|

.

Finite Groups of Automorphisms ~ NORMAN BIGGS

CAMBRIDGE

UNIVERSITY

'

PRES

see NET IN U.K.

i IN U.S.A

London Mathematical Society Lecture Note Series.

Finite Groups of Automorphisms Course given at the University of Southampton, October-December

NORMAN

1969.

BIGGS

Royal Holloway College University of London

CAMBRIDGE AT THE

UNIVERSITY

PRESS

1971

6

Published by The Syndics of the Cambridge University Press Bentley House,

200 Euston Road,

American Branch:

©

London N. W. 1

32 East 57th Street,

New York,

Cambridge University Press 1971

Library of Congress Catalogue Card No. : 74-154510 ISBN:

0 521 08215 3

Printed offset in Great Britain by Alden and Mowbray Ltd at the Alden Press,

Oxford

N. Y. 10022

-

Contents

Page Introduction

l.

Permutation groups se i

el at a i

2.

iii

Preliminary definitions

2

Counting principles

3

Transitivity

4

Applications to group theory

bs)

Primitivity

6

Regular normal subgroups

oy oO We

dd

Geometry of finite spaces

19

Ze 1

Introduction

1

ae

Finite fields

21

le?

Finite vector spaces

22

2.-4

The structure of GL(V) and

205

Projective spaces and projective groups

oe

2.6

Miscellaneous

om

SL(V)

results on projective spaces

rae | The classical simple groups

3.

26

42

48

Designs Four fundamental problems

48

Designs

a

The type of a design

55

Symmetric

SS)

designs

Automorphisms

and extensions of designs

60 64

The Mathieu groups INS ee HS (Gy) os eS} 5 The transitivity problem SOS Noe SESS ROSS

i

69

4.

Linear graphs

71

4.1

Definitions and examples

Vi

4,2

Regular graphs

74

4.3

Graphs and permutation groups

82

4.4

Distance-transitive graphs

4.5

Distance-transitive graphs of diameter 2

—

86

4.6

Graphs with no triangles

102

4.7

Conclusion

109

96

Appendix

110

Guide to the literature

LTS

Index

116

ii

a

Introduction

These are the notes of lectures given at the University of Southampton,

October-December

1969.

The lectures were intended

for research students working in areas related to the topics discussed,

and for mathematicians working in other fields who

were interested in hearing a survey of some current problems and their background. I have tried to retain the character introducing occasional

of the lectures by

'philosophical' remarks,

the text free of references,

as far as possible.

and I have kept There is a short

survey of the literature at the end, and all the main sources are mentioned there. I thank David Sands who took the original notes,

and

Dr David Kirby who read the manuscript with great care and pointed out several errors and omissions.

Southampton

Norman

January 1970

iii

Biggs

St sates iat om nts‘hae mo

bith v180 feere Hitw

|

4 :

1. Permutation Groups

',.. it will afford me much satisfaction if, by means of this book, I shall succeed in arousing interest among English mathematicians in a branch of pure mathematics which becomes the more fascinating the more it is studied. W. Burnside, in his preface to Theory of groups of finite order, 1897.

11

Preliminary definitions

It is presumed that the reader will already be familiar with the contents of this section;

he is advised to read it quickly in

order to accustom himself to the notation which will be used hereafter. If X a:X > X;

is afinite set, a permutation of X

two such permutations

give the permutation (x)@£ = ((x)a@)8.

S,,,

Under this operation the permutations the symmetric group on

5 Py ieee aesoe cere let S.= Sy; so that

A permutation group is atriple set,

f can be composed to

That is, we write functions on the right and com-

form a group

oe

and

af:X ~ X, which here will be defined by

pose in the natural order. of X

@

is a bijection

G

(X, G, i) where

X.

If

Is. =n!

X is a finite

is an abstract finite group and

i:G —- Sy is a homomor-

phism.

In practice we usually identify

G

we say

G acts on

X, or,

G

with its image in Sy

has a permutation representation

in X, and we refer to the permutation group important case when

i:G > S,, is xX

(X, G).

a monomorphism

In the

we speak of a

(X, G)

The degree of

faithful permutation representation.

is

[x|. a [F@ly Zlel=" £66

1.3.6.

Proposition.

is divisible by

Proof.

n(n- 1)...

|G|

FY TF@!*.

geG

If (X, G) is k-transitive, then (n-k+1)

where

n=

|G|

|x|.

=n/|G_| xX n(n - 1)|(G, )§ liane ee

n(n

ries

(n-k+1)|G

pihiy

eo //

if |G| =n(n- 1)... (n- k +1) then we say that (X, G) is exactly (or sharply) k-transitive; (X, G)

is regular.

X

G

and

x, € X,

if

Ic| =n=

|x|

we say that

In the latter case it is often useful to identify

under the bijection

x +> 8

where for some fixed

(x 8, ae Our first examples of k-transitive groups with

the alternating and symmetric

1.3.7.

(ii)

Proposition.

groups.

(i)

AL is (n-2)-transitive.

s, is n-transitive.

k > 1 are

Proof.

when

The first part is obvious.

In the alternating case,

n= 3, A,= (Ai ZB), (132) }, which is 1-transitive, so

we can formulate a proof by recursion, 1. 3.5 and the fact that the stabilizer

using the remark after (An

is ALi

that AL cannot be more than (n-2)-transitive, permutation of tuple?

{1, 2, ...,

n}

To show

notice that any

which takes the ordered (n-1)-

(1, 02, 0s4 4,!ms2) n-L)yto >(1,2, cee, n=2, n)omustitake

to n- 1 and so is the odd permutation

(n - 1, n), which is not

in A.

// Examples

abundant; for

n

of k-transitive groups with

k>

2 are not

in fact it is thought that there may be an upper bound

k, if we exclude the symmetric

this is a very deep subject,

and alternating groups.

But

and almost nothing is known for certain.

The most highly transitive groups constructed so far which are not symmetric

or alternating are the Mathieu groups

which are 5-transitive; stabilizers

M,,

and

M,,

and

M,,

apart from these and their respective M,,

which are 4-transitive,

permutation groups are not more than 3-transitive.

all other known One of the

incidental aims of this course of lectures is to construct these highly transitive groups.

1.4

Applications to group theory

An abstract group

G

acts on the set

26 of subsets of G

in two especially important ways:

(i) (ii)

(K)g= Keg, (K)g=¢ Ke;

where

geG,

KCG.

Many notions and theorems

quotient being the index

of cosets.

or the number

|G: H|

lal, the

divides

|H|

1. 2. 3 gives Lagrange's theorem that

Thus

itself.

H

is

H

in G, and the stabilizer of

cosets of H

consists of the right

H

then the orbit of

(H a

is a prime

Now the set

re a, € ZW has an additive

ah a4)»

structure in which addition i performed coordinate-wise, when

n

is aprime

ture which makes

(a; Ay sees in the ring

p, this set may be given a multiplicative struc-

it a field.

a7)

and again

This is done by identifying the sequence

with the polynomial

a tajtt...

Zt), choosing a polynomial

which is irreducible over

Z_

f(t) € Zt]

Bo

Sito

of degree

r

(that is, has no zeros in Z)) and

defining multiplication to be polynomial multiplication in Zt] followed by reduction modulo

sequence

(0, 1, 0, ...,

f(t).

The polynomial

t (or the

0)) has the property that the elements of

the field, which is called GF(p'), can be written

For example,

irreducible over

to construct

Z, =

GF(3°)

1Oci1,243-thus

note that

GF(3")

t° + 2t+2

is

may be listed as

follows:

reduction:

Sequencese

0

t

t

0

t

to

t3 Lee

t*

ie & te

2

ots

Y2tsh2

t!

1°

mut ¢,

of

Let F

be afinite field.

That

in, ¢ ¢-F

A

m0,

c=x,

x=>m.0+c,

m=y-x

formation taking

23

in F.

F.

is a group (under composition) is immediate

It acts exactly 2-transitively on equations

A

F

is a group which acts exactly 2-transitively on

Proof.

The set

F

y=m.1+c Thatis,

0, 1 to x, y.

since,

given

havea

x, y in

F

the

unique solution

there is a unique affine trans//

Finite vector spaces

We write

V=

V(n, q) for a vector space of dimension

22

n

a over

GF(q);

V

has

q° elements.

Our aim is to use standard

techniques of linear algebra to define structures of geometrical and combinatorial

2.3.1. is apair s:V - V

interest.

Definition.

(s, @) where

A semilinear automorphism of V(n, q)

a

is an automorphism

F = GF(q)

and

alone as the semilinear automorphism,

and

isa permutation of V

satisfying:

(x + y)s = (x)s + (y)s, (A.x)s = (A)a. (x)s,

We usually refer to

s

forall

forall

call

q@ its companion automorphism.

that

s_

x,

yeV;

Ae F,

xeV.

If a

is the identity we say

is linear.

Choosing a basis points

of

le, _ o. e,} for V we can identify n

x of V with row vectors

(x,, —.

x)» where

x= ) Xe. , =]!

and each

x, € F;

similarly a linear automorphism

specified by its action on the basis,

nXn

matrix

L= (255) over

and we identify

J of V

is

2 with the

F, where

n (e.)2 =

) l..e

j=1 V4

If we had chosen a different basis is a unique linear automorphism

oa lft re

if, , bas i. for V_ then there h of V_ such that

(e,)h = f. :

it aS (h,.) is the matrix representing

respect to the basis

le, oe

respect to the new basis

ae e

then

if, , epee i

23

h with

2 is represented with

by the matrix

HLH™?.

Thus the determinant of a linear automorphism

2

is unimodular.

q),

TL(n,

GL(n,

q),

linear,

TL(V),

are denoted by

If V = V(n, q) we also use the nota-

respectively.

GL(V), SL(V)

of V

automorphisms

and unimodular

TL(V);

The groups of all semilinear,

Definition.

2.3.2.

tions

we say

if det / =1

the determinant of any representative matrix; that

may be defined as

SL(n, q).

SL(V) g

We show

GL(V)

j IL(V)

>

;

AutF

1 J SL(V) > GL(V)«> F*, is the multiplicative group of

F.

i Proof.

(A short exact sequence

another way of stating that consequently

i:A—~B

>

j B

companion automorphism

j:TL(V) > Aut F of s.

A;

by letting

(s)j be the

It is routine to check that

j is an

and its kernel is the group of semilinear automor-

Part (ii) is similar,

Ww!:

is just

j is an epimorphism with kernel

phisms with the identity companion automorphism,

2.3.4.

=> C

is the inclusion monomorphism. )

In part (i) define

epimorphism,

A

defining

that is,

(2)j = det l.

Proposition.

Sener

tage gtaleyigh wa = 4M. @h- a = qrin- P72

n+, i (aq a resail|

24

GL/(V). pd

(ii)

|SL(, a)| =4

ie 2 n(n-1) /2 tl Pe Pe i=2

(iii)

|TL(, p™)| = rl/GL@, p*)] .

Proof. if, , rari i

For each pair of ordered bases le,, n. ew} of V=V(n, q) there is a unique 2 € GL(V) such

that (e)2 =f, i=1, .1 of ordered bases

of

V.

,» n.

1 ways;

GL(V)

is equal to the number

Now the first member

may be any element of V q” -

Thus

except

of an ordered basis

0 and so may be chosen in

the second element must not be linearly dependent

on the first and so can be chosen in q: - q ways, the formula for

GL(V).

This gives

The formulae in parts (ii) and (iii) follow

by applying 2. 3. 2 and recalling that

when

etc.

|Fx | =q =I

| Aut F| a od

F = GF(p’).

Tf

Before proceeding to a detailed study of the structure of GL(V)

and

SL(V)

we make some informal remarks.

linear automorphism

of V

properties of GL(V)

is confined to its actionon

GL(V)

is transitive on

V*

fixes

0

Since every

our interest in the transitivity

V*=V-

since any two elements of V*

chosen as the initial members

of two ordered bases;

{0}. may be

it is not in

general 2-transitive because there is no linear automorphism taking an independent pair to a dependent pair. p prime,

we can see that linear automorphisms

with group automorphisms that is,

(Z,)”.

results 1. 6. 3-5:

When of V

V = V(n, p), are identical

of the underlying abelian group of V,

Thus we have a geometrical

in particular,

over the field

interpretation

of the

GF(2) = Z, two

different nonzero points in a vector space are always independent, so that in this case we may expect doubly transitive groups of automorphisms.

25

2.4

The structure of GL(V)

and

SL(V)

We shall suppose for the remainder dim

V=n=

2. A

of this chapter that

(linear) subspace of V(n, q) of dimension

contains

Gea points;

in particular a subspace of dimension

contains

fina points and is called a hyperplane.

trivial linear functional

U=

on

V

has equation

Wewrite

(x, u) for

(x, u) = 0; if (x, u)=0

(x, u) = A(x, v) forall (U)Z

x eV.

II {xev

2.4.1.

and

xu

(x, v)=0

U

are both

2X € F*

we have

If 2 is a linear automorphism of

where

and

(y, u) —0}

| («i7', w=0)}.

Definition.

A linear automorphism

called a transvection if for some hyperplane

Gia

Xtee x

if. x. €.U..

(ii)

xT-x€U

1S

U

in V

only

these parallel to the hyperplane

we might just as well write the second condition as Transvections

7 of V we have

qn - aos U;

of course,

xtT- x €U

are useful because in fact they move

every point in a fixed direction,

and so may be specified by a

26

is

UL

Thus a transvection in V(n, q) moves and it moves

and say that

then for some

is also a hyperplane in V

(U)t = {x eV |x=yl,

all x¢«V.

is a non-

then the set

equations of the same hyperplane,

points,

n - 1

{xeV|xu=0}

is a hyperplane in V.

V, the set

If u

m

for

re simple formula;

this is a consequence

of linearity,

as we now

show.

2.4.2.

Lemma.

U_ which has equation

If 7 isa transvection with hyperplane

(x, u) = 0 then there is some

ae€U=

such

that xT=x-

(x, ula

Proof.

Choose

forall

b ¢U

c= (b, u) b;

xeV.

and let

a=cr-c.

Then for all x € V we have

(x %

(x,

so that, since x - (x, ujc,

u)(b,

u)’b,

u) en)

7 fixes all points of U and x - (x, u)(b, u) 7 fixes

xT - x= =

x - (x, u)c.

b=

Thus

(x, u) [cT- ¢] (x, uja :

We shall write

7=

//

ee

for the transvection given by the

formula

Xt~>x - (x, uja and remember different pairs

that the same transvection can be given by many u, a.

We notice that since

at

a ¢«U

we have

(a, u) = 0, and also we have the following identities;

(Gj)

7? a

ii)

Tu,a'u,b~ ‘u,atb’

U8

ae A

The set of all transvections

Proposition.

2.4.4.

provided

a+b

V

of

#0.

is

a complete conjugacy class in GL(V).

Proof.

If T=

es,

and

h € GL(V)

then

(x)h +th= [xh”* - (xh™*, wah =

Thus

hth

(xh,

is a transvection

equation of the hyperplane suppose

T=

hyperplanes

u)ah

Ta U

and

U'

ea

(U)h

ange

.

where

and

Ty" a'

(x, v)=0

a'= ah.

isan

Conversely,

are two transvections,

respectively.

Choose

with

f ¢U, f' €U',

such that (f, u) = (f', u')=1, and select bases {a, b, ..., e} of U- and*-{a', Botte’ | “of OUT Thenetiaminem 2 te. Feed {a', b', ..., e', f'} are bases for V; consequently we may find h e GL(V)

such that

ah=a',

bh=b',...,

fh=f'.

is a transvection by the first part of the proof, been given.

(xh? ,

In order to show that

u) = (x, u') forall

has been arranged, and

(x, u')=0

x=f

gives

and its formula has

ah=a'.

The latter condition

(U)h= U' we know that

are both equations of the hyperplane

(xh”*, u) = A(x, u') for some }=1

A € F*

as required.

ho th

hth = T' we require that

x eV, and

and since

Now

andall

(xh *, u)=0 U'.

Thus

x« V, and putting is

28

This explicit knowledge of one complete conjugacy class in

GL(V)

enables us to deduce many properties of GL(V)

and

SL(V), especially properties connected with the normal structure. In what follows we shall have to assume for some results that dim V = 3; this assumption is often necessary only because of the particular mode of proof we present,

and an exact statement of

the conditions required for our results will be found later in the chapter.

2.4.5.

Proposition.

All transvections belong to SL(V),

and if dim V = 3 they form a complete conjugacy class in SL(V). Proof.

Since all transvections are conjugate in GL(V)

they all have the same determinant of the result 2. 4. 3(ii) we find

} ¢ F*.

» = 1.

Taking the determinant

Suppose now that

and refer to the proof in 2. 4.4 that if 7 and

in V they are conjugate in GL(V). has

deth=

w#1,

we obtain an h ¢€ GL(V) that is,

h « SL(V).

7T' are transvections

If the h € GL(V)

change the element

b

dim V = 3

found there

of the first basis by

such that (h) ‘th=7'

and

Notice that since two elements

the basis are prescribed by the proof of 2.4.4,

deth= Ty (a and

it must,

anormal subgroup by 2.4.5,

N 3 (that is, it has no triangles) and each two nonadjacent points are joined by precisely Further,

the graph has

pe athe 4.6.1.

n vertices,

c paths of length

2.

where

Ree. c

Proposition.

(i)

If c ¢ {2, 4, 6}

then

B(k, 0, c) is feasible if and only if k is one of a finite list of values for each given

c.

(ii) B(k, 0, 2) is feasible if and only if k=t* +1 where t is aninteger

# 0 (mod 4).

(iii) B(k, 0, 4) is feasible if and only if k=t* where t is any integer.

(iv)

B(k, 0, 6) is feasible if and only if k=t? - 3 where

t is aninteger

Proof. if

+ 0 (mod 4),

Applying 4.5.2,

s= (c? - 4c + 4k)?

B(k, 0, c) is feasible if and only

is an integer and

102

me th

Si b|-

k|

is an integer.

s

dag led ea Cc

Eliminating

k from these two expressions and

writing the result as a polynomial equation in s we find

6 Thus,

ee

if c #2,

= Zoe Ac = 6) = 0,

4, 6 the integer

s

must be a divisor of

e*(c - 2)(c - 4)(c - 6) andfor each c the list of possibilities for s isfinite.

Since

result (i) for

k=

(s? - ce? + 4c)/4

k.

we have the corresponding

:

If c= 2, then s= 2(k- 1)”, sothat k=1+t’, and m=

(t? + 1)(t? + t+ 2)/4

which is integral only when

t #0 (mod 4).

If

4

c=4, then s= 2k”, sothat k—t* and

m=tit + 1)(t? + t+ 2)/8, which is always integral.

If c= 6, then

s = 2(k+ 3)”, so that k=

t7-°3* and

m = (t + 1)(t + 2)(t® - 3)/12 which is integral only when t 7 0 (mod 4),

//

The case

c= 1

of the preceding proposition is just the

result of 4.2.6 ina slightly weaker form, graph of valency have diameter

k and girth

5, with

For the case

divide 27 and so

3, 21 or 183.

that

k=

c=

either trivial or meaningless.) as may be seen by calculating

Here m,

B(k, 0, 1) if itis 3 we find that

s

must

lesser values of k are k=

183

is actually impossible

and the realizability of the

B(21, 0, 3) is undecided.

103

must

(It will be convenient to suppose

k = 3 throughout this discussion;

feasible matrix

1+ k? vertices,

2 and intersection matrix

distance-transitive.

for it is clear that a

Of course,

B(3, 0, 3)

is realized by oa 3°

3

Apart from the graphs

Ke ic which always realize

B(k, 0, k), the only known example of graphs with intersection matrix

B(k, 0, c) are the two graphs with

c=1

and

k= 3, 7

mentioned after 4. 2.6, and afew isolated examples with ce

{2, 4, 6}.

Before reviewing these examples we collect a

few simple facts about the structure of such graphs.

4.6.2.

Lemma.

with intersection matrix and let X = A (0),

Y=

Let

I be a distance-transitive graph

B(k, 0, c).

Each vertex in X

(ii)

Each two vertices in X

in

is joinedto

k-1

vertices in Y;

are joined to

c -

then

(i) is immediate.

For

(ii), notice that if

d(x, , x.) = 2 and so there are

I’ adjacent to both

x, and

X,.

Since

c vertices of

0 is one such vertex and

the rest must be in Y, we have (ii). If we label the vertices

Y

b=k(k-1)/c

blocks and

possible that two vertices in Y

/

by the vertices of X

they are joined then 4. 6. 2 says that design with

1 common

Y.

Proof.

X,, X, €X

0 of T

A, (0). Then:

(i)

vertices

Choose one vertex

(X, Y) isa r =k

to which

2-(k, c, c-1)

- 1, except that it is

have the same label, and so we do not necessarily have a design in the strict sense of definition 3.2.1. It is also worth noticing that two vertices in Y can be joined by an edge only if their labels are disjoint, for otherwise the graph

I would have triangles.

104

In the case

_c = 2 no two vertices in Y can have the same

label, for a configuration

can be rearranged as

p

—

=>

q

pq contradicting the fact that

c= 2.

In this case

(X, Y)

2-(k, 2, 1) design and must be the complete design, Lee Pe,€ ‘ .

Each vertex

pq € Y

is joinedto

that is

k - 2. other vertices

in Y, each of which is labelled by a pair disjoint from there are

oe =

such vertices to choose from.

4, 6.1 (ii) we see that the first feasible value for oN

ie ea

aa ps = and so

conditions.

I

isa

pq, and

Returning to k is

5, and here

is completely determined by these

It is not hard to check that this graph does satisfy all

the relevant conditions, next feasible case is k=

and so 10

B(5, 0, 2) is realizable.

The

and here too there is a graph,

has a group of automorphisms

isomorphic with

vertex stabilizer

See the appendix for references.

PSL(2,

9).

105

PSL(3,

which

4), and No

other realizations are known when c = 2, the next feasible matrix

2

being B(26, 0, 2).

and

not realizable,

The first three feasible

is realizable.

16

k=

k=, 94 is

7

4, 9, 16; k=4 is realized by

k are

values for

4.

c=

We pass on to the case

We sketch the justifica-

tion of the last two statements.

4.6.3.

Proposition.

Proof.

Suppose there is a graph with this intersection

matrix; 4.6.2.

choose one vertex For each

x € X

which are joined to x. we have

B(9, 0, 4) is not realizable.

0 and label the other vertices as in

let

(x) denote the set of vertices in Y

Then by 4. 6. 2, |(x) | = 8 andif

|(w) n (x) | = 3, so that

|(w) U (x) | = 13.

is the label of some vertex in Y;

it is joined to

vertices in Y, none of which contains But there are only 18 vertices in Y contain of u

w

and

or v

vertices in Y and

x;

in Y 8- 5=

also.

9- 4= 5 or

x

uvwx other

in its label.

and 13 of them

Now we have a contradiction,

containing

containing

v

and

3 vertices in Y

w

w

and

Proposition.

for there are 3

w, and 3 vertices containing

v

and so there are at most 5 vertices or

x.

Thus there are at least

containing

on which the Mathieu group

Proof.

altogether,

Suppose

x, so that these 13 must contain all the occurrences

one of these is uvwx

4.6.4.

u, v, w,

w#x

v

and neither

w

nor

x.

Vf

B(16, 0, 4) is realized by a graph M,,

acts as a group of automorphisms.

Recall that in 3. 6.4 we constructed a 3 - (22, 6, 1)

design with automorphism group

M, >> our graph will have the 77

blocks of this design as vertices,

two vertices being joined by an

106

edge whenever the corresponding blocks are disjoint.

To check

that this graph has the required. properties recall that the parameters of the design are, r, = 5 and

in the notation of chapter 3,

r, =2X=1.

0, 1, or 2 points,

J =a

Two distinct blocks could intersect in

but if they have one common point,

this point and the blocks not containing it gives a design which is the projective plane blocks (lines) intersect; they have two.

21,

PG(2,

then removing

2 - (21, 5, 1)

4) in which every two

thus if two blocks have one common point

The number of blocks meeting a given block in two

points is Ge (5 - 1) = 60, so that the number of blocks disjoint from a given block is correct valency.

77-1-60=

16.

Thus our graph has the

The remaining conditions can be checked by

using counting techniques and applying the known transitivity properties of

M,,

on the points and blocks of the design.

No other realizations of

B(k, 0, 4) are known.

We pass to the case

6.

c=

//

Here the only known graphs are

for@iv=t6 (Ky 4)? and k= 22, the graph due to D. G. Higman and C. C. Sims,

which we now construct.

k, = 77 and that the 'design' parameters familiar

First we notice that

k, 22

(X, Y) which follows from 4. 6. 2 has

2 - (22, 6, 5), which are the parameters of the now

3 - (22, 6, 1) design when considered as a 2-design.

This motivates following construction.

3 - (22, 6, 1) design and let V=

Let

(X, B) be our

{0} UXUB;

define the set E

of edges thus:

{0, x} €E

for all

{x, B} €E

eee

{f,, 8B,} €E

if 2, 6, €B

107

x eX, Pee

ean

x ih,

and pe

=

I= (V, E) is a distance-transitive graph realizing B(22, 0, 6). The proof requires us to show that I’ is vertex-

Then

transitive only, for then the fact that the stabilizer of the vertex M, Be which is transitive on

0 contains

and

B, implies

To prove vertex-transitivity we pick a

distance-transitivity.

x € X, show by counting that

vertex

X

3 - (22, 6, 1) design,

(A, (x), A, (x))

isa

and then use the known result that there is

a unique design with these parameters to construct an automorphism of

0 to x.

IT taking

The automorphism group of this graph contains a subgroup HS

(actually of index 2) such that

4.6.5.

Theorem.

Proof.

By 4.5.3

The group

HS

IT, and since the stabilizer that either

HS

solvable, N’ 4

N

N

2°.5*

andso

is simple we know from 1. 5. 7

and so by a famous

1.

HS.

Then

result of Burnside, subgroup

which would mean

Now we have a contradiction,

N

N’ #N.

is But

N=

N.

Thus in

for the subgroup

eN}

is normal in HS

must be

M,,

N’ = 1, since any nontrivial normal subgroup

is abelian.

{n? ln of N

acts primitively on the vertices of

and in particular its commutator

HS

is simple.

is a regular normal subgroup of

must be transitive by 1.5.6, fact

HS

is simple or it contains a regular normal subgroup.

Suppose IN| = 100=

(HS) | = M,,.

But

N

and so by the same argument as above,

contains elements

We conclude that

HS

of order

it

5.

has no regular normal subgroup and so

it must be simple.

Wi.

108

4.7

Conclusion

We conclude these lectures by remarking that the four problems introduced in 3.1 each has relevance in the study of distance-transitive graphs, of diameter

2.

in particular,

in the study of such graphs

The major part of the work presented here has been

concerned with the existence problem for which we have a partial theory;

the fact that the other problems have hardly been mentioned

reflects the current lack of knowledge about them. problem seems

The extension

especially worthy of study, for we have seen that

the new Higman-Sims

graph is an extension of a graph based on

M,.»

and several other new simple groups have been constructed in this way.

Inparticular,

Fischer has found three new simple groups by

successively extending a distance-transitive graph with intersection

matrix B(180, 51, 45) and automorphism group PSU(6, 2”). A list of the parameters groups now follows,

of graphs with new simple automorphism

together with a list of feasible parameters

for distance-transitive graphs of diameter 2 and 'small valency’.

109

Appendix

This is a list of triples

(k, a, c) which are realized by a

distance-transitive graph possessing a 'sporadic' simple group of automorphisms.

The word

'sporadic' is interpreted to meana

group which is not one of a known infinite family.

The names

of

the discoverers are given together with a page reference to the

Harvard Symposium (Brauer and Sah, 1969) where details may be found.

Fischer's groups are described in a set of duplicated notes

published by the Mathematical Institute of the University of Warwick.

k

a

22

0

6

36

14

416

Higman and Sims

p. 203

12

Hall and Janko

DT

100

96

Suzuki

p. 165

162

105

81

Maclaughlin

p. 109

693

180

126

3510

693

351

31671

LO

Fischer

3240

Additional information on sporadic groups is given in a paper by

J. Tits, Groupes finis simples sporadiques,

22 (1969-70),

375.

110

Seminaire Bourbaki,

This is a list of the triples which

B(k, a, c) is feasible,

(k, a, c) with

k=

16 for

with the exception of those parameters

which can be dealt with by standard constructions.

That is, we

exclude:

(i)

the case

(ii)

thetriples

(2t - 2,

t- 2, 2),

c=k,

dealt with in 4.5. 4;

(2t- 4, t- 2, 4), t

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Biases

LONDON MATHEMATICAL LECTURE NOTE SERIES Editor: PROFEssoR G, C. SHEPHARD,

SOCIETY

~

University of East Anglia

The purpose of the series is to publish records of lectures and seminars on advanced topics in mathematics held at universities throughout the world. For the most part, these are at postgraduate

level, either presenting new material or describing older material in a new way. Exceptionally, topics at the undergraduate level are published if the treatment is sufficiently original. The notes are normally 80 to 200 pages in length, printed by offset lithography from typescripts.

Finite Groups of Automorphisms NORMAN BIGGS In the study of finite groups it is often illuminating to consider structures of geometric or algebraic origin and their associated groups of automorphisms. These lecture notes deal with those

structures which have been found to give rise to groups of auto-

morphisms

with especially interesting properties,

in particular,

those which lead to finite simple groups. They are intended for research students in the particular topics covered by the text, and mathematicians in other fields who want a survey of some current problems. Also published in this series 1. General Cohomology Theory and K-Theory

PETER HILTON

2. Numerical Ranges of Operators Elements of Normed Algebras

on

Normed

Spaces

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F. F. BONSALL and J. DUNCAN 3. Convex Polytopes and the Upper Bound Conjecture

P. MCMULLEN

and G. C. SHEPHARD

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J. F. ADAMS

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