Groups: A Path of Geometry 9780521300377, 0521300371

Table of contents :
Contents
Preface
1. Functions
Injections (one-one)
Surjections (onto)
Composition of functions
Closure
The symmetric group
Historical note
Answers to chapter 1
2. Permutations of a finite set
Cycle notation
Signature
Alternating group
Subgroups of S_n
Cross-ratio
Summary
Answers to chapter 2
3. Groups of permutations of R and C -- lsometries of the plane
The real line, R
The real plane R2, the Gauss plane or Argand diagram C
Isometries or symmetries of the plane
Cyclic and dihedral groups
Similarities
Transitive groups of transformations
Summary
Answers to chapter 3
4. The Möbius group
Cross-ratio
Triple transitivity
The tnmsformation z --> 1/z
The inversion z --> 1/\bar{z}
Stabilisers
Subgroup fixing the unit circle
Historical note
Answers to chapter 4
5. The regular solids
Summary
Answers to chapter 5
6. Abstract groups
Subgroups
Order
Groups generated by two elements
Dihedral groups
Groups generated by larger sets of elements
Summary
Answers to chapter 6
7. Inversions of the Mobius plane and stereographicprojection
Stereographic projection
Riemann sphere
Historical note
Answers to chapter 7
8. Equivalence relations
Summary
Answers to chapter 8
9. Cosets
Summary
Historical note
Answers to chapter 9
10. Direct product
Summary
Answers to chapter 10
11. Fields and vector spaces
Vector spaces
Summary
Answers to chapter 11
12. Linear transformations
Summary
Historical note
Answers to chapter 12
13. The general linear group GL(2, F)
The group of nonsingular transformations
The centre of the general linear group
More subgroups of GL(2, F)
The orthogonal group
Historical note
Answers to chapter 13
14. The vector space V_3(F)
Vector products
Determinants
Singular and nonsingular transformations
Summary
Historical note
Answers to chapter 14
15. Eigenvectors and eigenvalues
Characteristic equation
Similar matrices
Change of basis
Shears
Summary
Answers to chapter 15
16. Homomorphisms
The kernel of a homomorphism
Quotient groups
The field Z_p
Summary
Answers to chapter 16
17. Conjugacy
Conjugacy classes
Normal subgroups and conjugacy classes
Direct products
Normal subgroups of A_4
Conjugate rotations in three dimensions
Automorphisms
Summary
Answers to chapter 17
18. Linear fractional groups
The homomorphism GL(2, F) -> LF(F)
Projective special linear group PSL(2, F)
Summary
Historical note
Answers to chapter 18
19. Quaternions and rotations
Algebra of quaternions
The transformation X -> R^{-1}XR
Summary
Historical note
Answers to chapter 19
20. Affine groups
Affine group
Lines in a vector space
Summary
Answers to chapter 20
21. Orthogonal groups
The special orthogonal group SO(3)
The orthogonal group O(3)
Finite subgroups of SO(3)
Finite subgroups of O(3)
Summary
Historical note
Answers to chapter 21
22. Discrete groups fixing a line
A homomorphic image: the point group
Discrete groups of transformations
Classification of frieze groups
Summary
Historical note
Answers to chapter 22
23. Wallpaper groups
Possible point groups
Point group C_3
Point group C_4
Point group D_1
Point group D_2
Point group D_3
Point group D_4
Point group D_6
Each of the 17 types is unique up to isomorphism
Answers to chapter 23
Bibliography
Index

Citation preview

R.P.BURN

R. P. BURN

GROUPS A. PA.TH TO GEOMETRY

--

R. P. BURN

GROUPS A PATH TO GEOMETRY

71,f,,..,,,/,,., IIIH-1111

f>/ C--,•

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CAMBRIDGE UNIVERSITY PRESS Cambridge London New York New Rochelle Melbourne Sydney

Published by the Press Syndicate of the University of Cambridge The Pitt Building, Trumpington Street, Cambridge CB2 I RP 32 East 57th Street, New York, NY i0022, USA 10 Stamford Road, Oakleigh, Melbourne 3166, Australia

© Cambridge University Press 1985 First published 1985 Printed in Great Britain at the University Press, Cambridge Library of Congress catalogue card number: 84-21354 British Library cataloguing in publication data

Burn, R. P. Groups: a path to geometry. I . Groups, Thc-ory of I. Title 512'.2 QA! 71

ISBN O 521 30037 I

APL

Contents

Preface xi Acknowledgements

1 Functions

xii

I

Those properties of functions (or mappings) are established by virtue of which certain sets of functions form groups under composition. Summary 7 Historical note 8 Answers 9

2 Pennutations of a finite set 11 The cycle notation is developed and even and odd permutations are distinguished. Summary 19 Historical note 19 Answers 20 3 Groups of permutations of R and C 23 Every isometry of the plane is shown to be representable as a transformation of C of the form z 1--+ ei9z + c or of the form z 1--+ erni + c. Summary 33 Historical note 33 Answers 35 4 The Mobius group 40 The group of cross-ratio preserving transformations is shown to be the set of transformations of the form

az + b z 1--+ - - 0 where ad - be ,t, 0. CZ+

Each such transformation is shown to preserve the set of lines and circles in the plane. Summary 50

Contents Historical note Answers S3

vi S2

5 The regular solids 57 The groups of rotations of the regular solids are identified with the permutation groups A 4 , S4 and A 5• Summary 60 Historical note 60 Answers 61 6 Abstract groups 62 Groups are defined by four axioms. Some groups are seen to be generated by particular sets of elements. Isomorphism of groups is' defined. Summary 72 Historical note 72 Answers 74

7 Inversions of the Mobius plane and stereographic projection

77

Stereographic projection is used to show that the group generated b)l inversions of the Mobius plane is isomorphic to the group of circle preserving transformations of a sphere. Summary 84 Historical note 84 Amwers 85

8 Equivalence relations 88 The process of classification is given a formal analysis. Summary 91 Historical note 91 Answers 92

9 Cosets 93 A subgroup is used to partition the elements of a group. Lagrange's theorem for finite groups, that the order of a subgroup divides the order of the group, is obtained. A one,-one correspondence is esta_blished between the cosets of a stabiliser and the orbit of the point stablised. Summary 97 Historical note 98 Answers 99

10 Direct product 10 I A simple-method of using-two·given groups to construct one new group is defined. Summary 103 Historical note 103 Answers 104 11 Fields a~ vector spaces105 When the elements of an additive group (with identity 0) are also the-clements of a multiplicative group (without 0) and the operations are linked by dis-

Contents

vii

tributive laws, the set is called a field when the multiplicative ;i,otip is commutative. When a multip~direct product is formed with the same additive group of a field as each component, -and this direct product is supplied with a scalar multiplication from the field, the direct product is called a vector space. Summary 110 Historical note 110 Answers 112

12 Linear transformations

114

When two vector spaces have the same field, a structure preserving function of one to the other ·can be described with a matrix. · Summary 116 Historical note 117 Answers 118

13 The general linear group GL(_2, F)

119

The structure preserving permutations of a 2-dimensional vector space are analysed. Summary. 123 Historical note 124 Answers 126

14 The vector space VJCF) 128 Scalar and vector products are defined in three dimensions. The meaning and properties of determinants of 3 x 3 matrices are explored. · Summary 132 Historical note 133 , Answers 134

15 Eigenvectors and eigenvalues 136 Vectors mapped onto scalar multiples of themselves under a ·linear transformation are found and used to construct a diagonal matrix to describe the same transformation, where possible. Summary· '143 · Historical note -143 · Answers 145

16 Homomorphisms

148

...

;

.... , ..

.,

Those functions of a group to a group which preserve the multiplica_tive • structure are analysed. The subset of elements mapped to the identity is a normal subgroup. Each coset of that normal subgroup has a singleton" . image. Summary 152 Historical• note 152 Answers 153 17 Conjugacy 15 5 When x and g belong to the same group, the elements x and g- 1xg are said to be conjugate. Conjugate permutations have the same cycle structure.·

Contents

viii

Conjugate geometric transformations have the same geometric structure. Normal subgroups are formed from a union of conjugacy classes. Summary 162 Historical note 162 Answers 163

.18 Linear fractional groups 167 The set of transformations of the form ax+ b x ...,.. - - where ad - be 'f' 0, ex+ 0 is a homomorphic image of the group GL(2, F). Summary 173 Historical note 174 Answers 175 19 Quaternions and rotations Matrices of the form (

178

: ~). with complex entries, are called quak:rnions.

- w z

The set of quaternions satisfies all the conditions for a field except that multiplication is not commutative. The mapping of quaternions given by x ...... R- 1xR acts like a rotation on 3•dimensional real spaces. Summary 182 Historical note 183 Answers 184

20 Affine groups 185 Line and ratio preserving transformations are shown to be combinations of translations and linear transformations. Summary 188 Historical note 188 Answers 190 21 Orthogonal groups 191 Isometries are shown to be combinations of translations and linear transformations with matrices A such that A· AT = I. Finite groups of rotations in three dimensions are shown to be cyclic, dihedral or the groups of the regular solids. Summary 199 Historical note 200 Answers 201 22 Discrete groups fixing a line 205 If G is a group of isometries and T is its group of translations, the quotient group G/-t:. is isomorphic to a group of isometries fixing a point, called the point group of G. If G fixes a line, its point group is either C 1, C2, D 1 or D 2 • This provides a basis for identifying the seven groups of this type. Summary 209 Historical note 210 Answers 211

Contents

23 Wallpaper groups

213 Groups of isometries not fixing a point or a line are shown to contain translations. If there are no arbitrarily short translations, the translation group has two generators. If such a group contains rotations, their order may only be 2, 3, 4 or 6. The possible point groups are then C 1, C2 , C3, C4 , C6 , D 1, D 2 , D 3, D4 or D 6• This provides a basis for classifying the seventeen possible groups of this type. Summary 225 Historical note 225 Answers 227 Bibliography 236 Index 238

1x

Preface

This book contains a first course in group theory, pursued with conventional rigour. There are three unusual aspects of the presentation. Firstly, the book consists of a sequence of over 800 problems. This is to enable the course to proceed by seminar rather than by lecture. Mathematics is something we do rather than something we learn, and, all too otten, lectures give the opposite impression. Secondly, at the outset, the groups under discussion are groups of transformations. This is faithful to the historical origins of the theory. It provides the one context in which the proof of the associative law is immediate, and it makes the study of sets with only a single defined operation obviously worthwhile. For Galois (1830), Jordan (1870) and even in Klein's 'Lectures on the Icosahedron' {1884), groups were defined by the one axiom of clvsure. The other axioms were implicit in the context of their discussions -- finite groups of transformations. Our work on abstract groups starts in chapter 6. Thirdly, the geometry of two and three dimensions is the context in which most of the groups in this book are constructed, and is also the major field of application of group theory in chapters 7, and 17-23. Geometry is the best context in which to understand conjugacy, and linear and affine groups are some of the easiest in which to put homomorphisms to work to good effect. The geometrical bias of the book links group theory with complex analysis, linear algebra and crystallography and provides a useful background for anyone about to study these fields. The mathematical knowledge assumed for this course is a confident familiarity with high school mathematics. In a few places an argument by induction is needed and in one place the cosine rule is required. Some experience at school level with groups, with matrices and with complex numbers, would certainly be helpful, though the text is self-

Preface

XU

contained in these respects. Although this book assumes less previous knowledge than my earlier Pathway to Number Theory, the sequence of questions on geometrical groups may cause more difficulty to the solitary student. The cumulative development of concepts is heavier in this area. Partly to compensate for this, ideas required from outside group theory are developed concretely and with the minimum of abstraction. For example, abstract vector spaces are not defined in this book: only spaces of n-tuples are used. It may be reassuring to the student to know that the results of chapters 7, 18 and 19 are not used in the rest of the book. A great many of the results obtained in this book have, been claimed and quoted by schoolteachers during the last 20 years. but many of the proofs have been inaccessible. I hope I have prO\idcd here, not just proofs, but also insight into transformation geometry as it is now done in British schools. For the undergraduate, there is a broad concrete base for generalisation and abstraction in his further studies; Acknowle«!gements I am very grateful to Dr Alan Beardon for partnership in designing the course for which this book is a text and to my colleague, Bob Hall, for many hours of discussion about the actual questions and their solutions. I am grateful too for correspondence on historical matters with Dr P. Neumann and Prof. B. L. van der Waerden. The faults that remain are my own.

Homerton College, Cambridge July 1984

R. P. Burn

1 Functions

Throughout the nineteenth century, group theory was a study of permutations and substitutions. Group elements were generally referred to as 'operations', being what we would now call transformations (bijections) of a set to itself. This view of groups suits geometers very well, and it is the view that we adopt f~r most of this book. In this first chapter we establish those properties of transformations· which make sets of transformations form groups under composition and we do this with a set-theoretic rigour unknown in the nineteenth century. This chapter is the most abstract chapter in the book, and the student who finds this uncomfortable may start at chapter 2 provided that he accepts the results of the last question of chapter I: Because we will be establishing a formal definition of a function in this chapter we will also be providing a background for the terms isomorphism, homomorphism and one-one correspondence, all of which describe special kinds of functions which are of use in group theory, which are not usually thought of as group elements themselves. Concurrent reading: Green, cha;-iter 3.

1 Functions FUNCTIONS

2 N➔ L

Arrow diagram

Graph

N = numbers

L = letters

1·•----

c--+-----+--,.,____

L{: ------------V=b 4/= ---------+--lf=c

2•----•a f

·t::

3•----•b

C

4• - - - - - - •c

5• 2 f

= {(l, a), (2, a), (3, b), (4, c), (5, c)}

I•---j :-----:::::.:: N

g

3

L

~

4•---:?' •c

4

5

N

r::: c _ _ _ _ _ _ _ _ _ _ lg= a

2g= a lg=a

·;

·1 8

§b "8

4g

5•~ I

2

g ={(I, a), (2, a), (3, a), (4, b), (5, b)}

Domain N

3 4 domain

5

= {I, 2, 3, 4, 5}, Codomain L = {a, b, c}.

Both f and g are examples of functions from N to L, f:N ➔ Landg:N ➔ L

NOT a FUNCTION N ➔ L

1• 2•----•a 3•----•b

s•-----

4•----•c NOT a FUNCTION

c-------------

b

--+-+----11---+--+--

a --+-+----l--+--~2

3

4

5

N➔ L

--+-+--.---4----

I•~

C

2•~•a

b---------+--

3•~•b 4• _. •c

5•-----

=b

5g = b

u a

a--+-+---+----+--

2

3

4

5

Injections (one-one)

3

1 Use the diagrams on page 2, with their implied rules to complete the following sentence. A function/: N-+ Lis defined when for each element n of the set N there is . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Use the diagrams on page 2, with their implied rules to complete the last sentence. The rectangular array used for the graphs of the functions on page 2 represents the so-called cartesian product N x L. Each element of N x Lis an ordered pair (n, x) with n from N and x from L. The graph of a function N-+ L consists of a subset of N x L with exactly one element (n, x) for each . . . . . . . . . 3 Which of these define functions of the real numbers IR (i)

IR?

X r-+ X 2 ,

(ii) x 2 r-+

(iii)

-+

X H

X,

1/x,

(iv) x

H

sin x,

(v) x

H

tan x.

4 If A = {O, 1}, how many different functions, A

-+

A, are there?

Injections (one-one)

5 Draw sketch graphs of the functions IR (i) x

r-+

x 3 and by (ii) x

1--+

= {1,

IR given by

x 2•

If x 3 = /, does it follow that If x 2 = v2, does it follow that The distinction here leads IX: x 1--+ x 3, a one-one function not one-one on IR. 6 Let A

-+

x x

= y? = y?

us to call IX : IR -+ JR defined by or injection. We say that f3: x r-+ x 2 is

2, 3, 4}.

(i) Exhibit the graph of a one---one function A

-+

A.

(ii) Exhibit the graph of a function A ...... A which is not one-one. (iii) How many functions A ...... A exist? 7 Let N denote the set of natural numbers {l, 2, 3, ... }. Draw part of the graph of the function N -+ N defined by n r-+ n2• Is this a one-one function?

8 What can be said about the rows of the graph of a function if that function is known to be one---one?

1 Functions

4

Surjections (onto) 9 Draw sketch graphs of the functions IR _. IR given by

(i) ex: x

H

e' and (ii)

/J: x

H

x

+

1.

Can you always find a real number x, such that ex: x Hy for any choice of the real number y? Can you always find a real number x, such that f3 :x H y for any choice of the real number y? The distinction here leads us to call f3 an onto function or surjection and to say that ex is not onto. 10 Let A = {I, 2, 3, 4}. Exhibit the graph of a function A_. A (i) which is onto, (ii) which is not onto. 11 Illustrate part of the graph of the function N _. N defined by

the even number 2n 1----> n, the odd number 2n - 1 1---> n. Is this a one-one function? Is this an onto function? 12 What can be said about the rows of the graph of an onto function? 13 Let A = {l, 2, 3, 4}. Can you construct a function A _. A which is one-one but not onto? Can you construct a function A _. A which is onto but not one-one? 14 Can you construct a function N _. N which is one-one but not onto? Can you construct a function N _. N which is onto but not one-one? 15 Conjecture a condition on an arbitrary set A such that every one-one function A _. A must be onto, and every onto function A _. A must be one-one. Justify your conjecture with the help of qn 8 and qn 12.

An injection which is also a surjection is called a bijection. In the context of functions with the same finite set as domain and codomain, injections, surjections and bijections are in fact indistinguishable. 16 Give examples of functions IR _. IR which are (i) one-one and onto (bijections), (ii) one-one but not onto (injections), (iii) onto but not one-one (surjections), (iv) neither.one-one nor onto.

Inverse functions

5

17 If there exists a one--0ne function A be said about the set A?

A which is not onto, what can

-+

Composition of functions

18 If Y. and /J are functions IR Y. :

x ,-.., 2x

and

-+

IR defined by

+ I, ,-.., 2x + I .

fJ : x ,-.., x

then we define :1.{J : x x

..:+ 2x ~ 2x +

I.

We write this (x)Y.{J = (2x)/3 = 2x + 1. Determine (x)/J:1. under a similar definition. 19 If 1:1.: A -+ Band fJ: B -+ Care functions, give a formal definition of 1:1./3: A -+ C by determining (x)1:1./J (the image of x under the function r:x./3: first r:x., then /3).

The function 1:1./3 is called the composite of the function r:x. and /3. a

20 If

and

2............... 2 3~ 3 4---4

5

exhibit 1:1.{J.

21 What formal condition makes r:x.: A tion? 22 If r:x.: A r:x./3: A

-+ -+

B and fJ: B

-+

-+

Ba one--0ne function or injec-

C are injections, what can be said about

C?

23 Let A = {I, 2, 3} and B = {I, 2, 3, 4} and let r:x.: A -+ B be an injection. Can you construct a function fJ: B -+ A such that (i) /h is the identity function B -+ B, (ii) :1./J is the identity function A -+ A? (Under the identity function on the set, each point is its own image.) Inverse functions

24 Let A and B be arbitrary sets, with r:x.: A -+ B an injection. Show how to define fJ: B -+ A such that r:x.{J is the identity function en A.

1 Functions

6

The function /3 is then called a right inverse for a. So injections have right inverses. 25 What formal condition makes a: A tion?

-+

B an onto function or surjec-

26 If a: A -+ B and {3: B -+ Care surjections, what can be said about the composite function a{J: A -+ C?

= {I, 2, 3, 4} and B = {l , 2, 3} and let a : A ...... B be a surjection. (i) Can you construct a function {3: B -+ A such that a{J is the identity function on A? (ii) Can you construct a function {3: B -+ A such that /Ja: B -+ B is the identity function on B?

27 Let A

28 Let A and B be arbitrary sets, with a: A -+ B a surjection. Show how to define a function f3: B -> A such that {Ja is the identity func~ion on B.

The function f3 is then called a left inverse for a. So surjections have left inverses. 29 Let A = {l, 2, 3}. Make a list of the bijections A words of the permutations of A. Find a left inverse for each bijection. Find a right inverse for each bijection.

-+

A, or in other

30 For any bijection a: A -+ B, define a bijection /3: B -+ A such that a{J is the identity function /: A -+ A and {Ja is the identity function B-+ B . .

Prove that either a{J = I: A ➔ A or {Ja = l: B uniquely. So bijections have two-sided inverses. Closure 31 If a and /J are both bijections A a{J:A -+ A?

➔

➔

B, determines

/3

A, what can be said about

Associativity 32 Let a: A ➔ B, f3: B ➔ Candy: C ➔ D be functions. Use your definition of qn 19 to show that for each point x of A, (x)[(a{J)y] = (x)[a({Jy)],

so that the two functions A guishable.

➔

D, (a{J)y and a({Jy) are indistin-

Summary

7

The theorem (1X/3)y = 1X(/3y) is called the associative law for functions under composition. 33 Use the associative law to prove for four functions IX: A ---. B, /3: B ---. C, y: C ---. D, c5: D ..... E that 1X[/3(yc5)] = [(1X/3)y]c5. The symmetric group

34 Let S be the set of all bijections A ---. A.

Justify each of the following claims. (i) If IX and /3 are in S, then 1Y./3 is in S (closure). (ii) If IX, /3 and y are in S, then 1X(/3y) = (1X/3)y (associativity). (iii) I: A ..... A, defined by I: x H x for all x in A, is in S (identity). Ia = 1Xl = IX for all IX in S. (iv) For each a: in S there is a /3 in S such that IX/3 = f3a = /. (inverses) These four theorems are summarised in the statement that the bijections of a set to itself form a group under composition. The group in this case is called the symmetric group on A and is denoted by SA. Summary

Definition A function a: A ..... B, with domain A and codomain Bis qn l defined when, for each a E A, there is a unique b E B such that a1X = a(a) = b. Definition A function IX: A ..... B is an injection when, for any qn 8 a,, a2 E A, a,o: = a20: a, = a2. Definition A function IX : A ..... B is a surjection when, for each qn 12 be B, there exists an a E A such that arx = b. Definition The composite (or product) •IY./3 of two functions qn 19 IX: A ---. Band f3: B ---. C is defined by (a)1X/3 = (a1X)/3. Theorem The product of two composable injections is an qn 22 injection. Theorem An injection has a right inverse. qn 24 Theorem The product of two composable surjections is a qn 26 surjection. Theorem A surjection has a left inverse. qn 28 Theorem A bijection (an injection which is also a surjection) has

=

qn 30 a unique two-sided inverse. The composition of functions is associative. qn 32 Theorem The set of bijections of a set to itself forms a group qn 34 under composition. If the set is A, the group is called the symmetric group on A and is denoted by SA. Theorem

1 Functions Historical note

The modern notion of a function, with domain, and codomain, is essentially that of P. G. L. Dirichlet (1837). The language and style in which functions are discussed today owes much to the corporate twentieth-century French mathematician N. Bourbaki.

8

Answers to chapter 1

9

Armrers to chapter 1 1 A function/: N ➔ 1 is defin9d when for each element n e Nthere is a unique l e L with nf = I. j 2 Exactly one element (n, x) for each n e N

(ii) I has no unique image, (v) ½n has no image.

3 (i) yes,

(iv) yes,

(iii) 0 has no image,

4 Four.

5 ./

=/

6 (i)

2

= y.

=> X

X

= y 2 => X = ± y.

4 _ ......_J..___J~_.__ 3 _...__

(ii)

4 --'---'----1---+--

___J~_.__

3

4".

3 --'---'----111---2 _ ......_---I___+-_

2 -....1...-l.----ill....._.__

2

(iii)

4

2

3

4

7 Yes. 8 Each row contains at most one entry. 9 (i) If ex 10 (i)

= y, y must be positive. (ii) If x + = y, every y is possible. 4

(ii)

--------1---+--

3 --'----~~-1--

2

2----------

--1--..---111--+...-

2 3

4 --+--+--

C by (x)a{J = (xa){J.

= y.

22 xa{J = ya{J => xa = ya since /J is an injection, tion. Thus a{J is an injection.

=>

x

=

y since a is an injec-

23 (i) No, because there is one point of B which is not an image under a. (ii) Yes. I a, 2a and 3a are well defined and distinct. Let b be the fourth element of B. Defii'le (aa)fl = a for a = I, 2, 3 and define bf] = I. 24 For a

E

25 Given b

A define (aa){J E

=

a. For b

B, there exists a

E

E

B, b ¥- aa for any a, define bf]

A such that aa

=

=

ai E

A.

b.

26 Given c E C, there exists b E B such that bf] =;= c and there exists a that aa = b so aafJ = c and afJ is a surjection.

E

A such

27 (i) No, because one point of A is not an image under /J. (ii) For each b eB there is at least one a EA such that aa = b . .For each b E B, define bfl to be one element a E A such that aa = b.

28 As in the second part of qn 27. 29 Images of(!, 2, 3) are (I, 2, 3), (2, 3, 1), (3, I, 2), (!, 3, 2), (3, 2, I), (2, l, 3). First, fourth, fifth and sixth bijections are self-inverse. Second and third bijections are inverses of each other. 30 For each b E B there is a unique a E A such that aa = b. Define bf] = a. If a{J = I, then aa{J = a so (aa){J = a. If {Ja = I, then b{Ja = b = aa for a unique a. Now a is an injection so bf] = a, and, as before, aafl = a. 31 From qn 22 and qn 26, a{J is a bijection.

= [(xa){J]y = (xa) (/Jy) = x[~/Jy)]. = (a{J) (yo) = [(afJ)y]o. This kind of argument can be extended to

32 x[(afJ)y] = [x(a{J)]y

33 a[fJ(yo)] any finite product, to show that its value is independent of the position of the brackets. 34 (i) from qn 31, (ii) from qn 32, (iii) obvious, (iv) from qn 30.

2 Permutations of a finite set

Concurrent reading: Fraleigh, sections 4 and 5.

1 Let A = {I, 2, 3} and let B = {l, 2, 3, 4}. The bijection a:A-+ A a • with arrow dia_gram

~ >(~ can be denoted by

Gi ~),

and the bi-

3- 3 jection

/J:

B

-+

B with arrow diagram

:~: 3~3

4

4

3 4) can be denoted by ( 31 2 2 4 1 . Write down the six elements of SA using this notation. Also write down the six elements of SB each of which maps l to 2 using this notation. A bijection A of A.

-+

A (for any set A) is often called. a permutation

= {l, 2, 3, ... , n}, the group SA is usually denoted by Sn. When n = 3, a geometrical description of the elements of S3 is available if we label the vertices of an equilateral triangle with l, 2

2 When A

and 3. The permutation

C; ;)

then corresponds to a reflection

symmetry of the triangle and the permutation

G; D

to a rota-

2 Permutations of a finite set

12

tional symmetry through 120°. Describe the geometrical analogues of the other permutations in S3• 1

263 Cycle ootation 3 If a

=

G; ! ~ i).

write down the permutati~? a • a which we

denote by r:i, the permutation a.2 • a which we denote by ri, the permutation a3 · a which we denote by a4, the permutation a4 • a which we denote by a5 and the permutation c: 5 • a which we denote 6 . by a. Suggest a simple description of a5 and of a6. Write down the images of 1 under the permutations~oc, a.2, oc3, ... , and thereby record the sequence I, loc, lei, loc 3, . • . • . Record also the sequence 2. 2a. 2a2, 2oc 3, ••• and the sequence 3, 3a, 3a 2, 3oc 3, . • • . Because each of these sequences is periodic and runs through the five digits in the same order, the permutation ex is usually represented by (12345), or equally well by (23451), or (34512) or (45123), or (51234), in what is called cycle notation. ,..... I ~

5

2

(

a

)

. •

3~4

4 With a

= (12345), write down the permutations a.2, a 3 and a4 in cycle

notation. Although this notation only describes a rather special kind of permutation, its suggestiveness and compactness are useful in more general settings. 4 5 Fmd a, b, c, d and e when (134)(25) = a b c d e · 6

. (1 23 5) . a, b and c when (l5 21 4343 25) = (lab) (3c). Fmd

7 For·any permutation et, we define oc = oc 1 , oc • oc = ri, oc 2 • oc = oc 3 and foi:' any positive integer n, rx:+ I = a!' • a. By an induction on m we can prove that oc•+m = rx' • am and also prove that (rx'r = etnm for all positive integers n and m. Now we define oc0 to be the identity permutation, oc- 1 to be the (two-sided) inverse of oc, and we define

Cycle notation

13

:x-n = (ix- 1)" for positive integers n. With these additional definitions, we can prove by induction that ix"+m = ix"·~ and (ix"r = ix"m for any integers n and m. Can you prove ix" • ixm = ixm • ix" directly from associativity? 8 If ix

=

G; ! ~ ;), exhibit the sequences

1, lix, lix 2, lix 3,

•••

2, 2ix, 2ix2, 2ix3, .. . 3, 3:t, 3ix2, 3ix3, .. . 4. 4:t., 4ix2, 4ix 3,

.•.

5, Six, 5ix2, 5ix and determine a, b, c, d, e, f, p, q, r if :x = (Iab)(2c) = (3de)(5j) = (2p)(Iqr). Because the two cycles (134) and (25) have no digit in common they are said to be disjoint. 3, •••

9 Express the permutations

. (1 2 3 4 5)

(l) 3 4 5 1 2 '

.. (1 2 3 4 5 6 7)

(l;) 7 6 1 2 3 4 5 '

( .. ") ( 1 2 3 4 5 6 7 8) lll 62578134' as products of disjoint cycles. 10 Exhibit the six elements of S; in cycle notation.

When 1ix = 1, the cycle ( 1) is often dropped from the expression of the permutation ix as a product of cycles. 11 Exhibit the 24 elements of S4 in cycle notation.

= (123456), express the permutations ix2, ix3, ix4 and ix 5 as products of disjoint cycles.

12 If ix

13 Express the product (12)(13) as a single cycle. Express the product (13)(12) as a single cycle. Does (12)(13) = (13)(12)? 14 Does (12)(34) = (34)(12)? If a; ::/= bj for any i or j, can you be sure that (a1a2 ... a,)(b1b2 ... b,) = (b1b2 ... b,)(a1a2 ... a,)?

The property we have obtained here is usually dercribed by saying that disjoint cycles commute.

2 Pe,-mutations of a finite set

14

A cycle (a1a2 ... an) is called a cycle of length n.

Cydes of length I are often not written down. Cycles of length 2 are called transpositions. Cycles of length n are called n-cycles. 15 To show that every permutation in Sn can be written as a product or' disjoint cycles, we consider an arbitrary permutation ex e Sn, and try to find one of the cycles of which it is composed by looking at the sequence I, lex, Icx2, lex 3, Iex4, .... How do you know that the digits in this sequence cannot aUbe different? Can you be sure that some repetition occurs in this sequence by the term lexn at the latest? Suppose lexm is the first term which is a repetition of a preceding term in the sequence, and that in fact lit" = Icl with m > k ~ 0. Then lexm-I = lexk-J and this contradicts lexm being the first repetition, unless k = 0. Deduce that the sequence has exactly m different digits in it and that these ., digits are continually repeated in the saine order. These m digits form one of the cycles of ex.

1f. ' Jaa

,,..-1~ la Jam-1

\

a

'

s., the cycles formed as in qn 15 are either identical or disjoint. Let a and b be any two of the digits 1, 2, 3, ... , n, and consider the two sequences

16 The next step is to show that, for any ex e

a, aex, aex 2 , aex 3 ,

• • • ,

and

b, bex, bcx2, bex3, . . . . Suppose that these sequences are not disjoint, and that, in particular, aex; = bri. If i ~ j identify the second sequence within the first and, using qn 15, the first sequence within the second. 17 Use qn 15 and qn 16 to prove that eyery permutation ins. is a product of disjoint cycles. · · Signature We explore the expression of permutations as products of transpositions. ·

Signature

15

18 Write out the following products in cycle form:

(12)(13), (12)(13)(14), (12)(13)(14)(15). (12)(13) ... (In). 19 Express each of the following permutations as a product of trans-

positions: I 2 3 4 5 6 7 8 9) (234), (2468), (13579), (12) (345)(6789), (, 6 9 8 3 I 5 7 4 2 ·

20 How can you be sure that any permutation in Sn can be expressed as a product of transpositions? Can the identity be so expressed? 21 Express the 3-cycle (123) in three different ways as a product of two

transpositions. Express the same 3-cycle as a product of four transpositions. The questions that follow lead towards the result that a permutation is either the product of an even number of transpositions or the product of an odd number of transpositions but not both.

22 Evaluate the product ( 123456789)( 16). 23 Evaluate the product (a1a2 ... a,b1b: ... b,)(a1b1).

24 Evaluate the product (123)(45678)(14). 25 Evaluate the product (a1a: . .. a,)(b1b2 . .. b,)(a1b1).

26 If :x is a permutation in Sn which is a product of c disjoint cycles (singleton cycles are to be counted) and r is a transposition in Sn, prove that :xr is a product of either c + 1 or c - 1 disjoint cycles. 27 If r1 is a transposition in Sn, check that r1 consists of n - I cycles and that n - 1 is even or odd according as n + 1 is even or odd. If r1, r2, . . . , rm are transpositions in Sn, prove by induction on m that when the product r1r2 . . . Tm is expressed in disjoint cycles, the number of disjoint cycles is even or odd according as n + m is even or odd.

28 If T1,

T2, • . . , Tm

T1T2 . . . Tm

=

and

are all transpositions in s. and prove that m and k are both even or

171, 172, . . . , 17k

171172 . . . 17k,

both odd. 29 An element of s. which may be expressed as a product of an even number of transpositions is called an even permutation. An element of Sn which may be expressed as a product of an odd number of transpositions is called an odd permutation. The signature of a per- ' mutation is + 1 for even permutations and - 1 for odd permutations. Prove that if :x, /3 E Sn then the signature of :x/3 is equal to (signature of :x) • (signature of /3).

2 Permutations of a finite set

16

30 What is the signature of (1234) and of (12345)? What is the signature of (123 ... n)?

31 Determine the signature of each of the elements of S4 • 32 Write down the elements of the group generated by (123456) and determine the signature of each element. 33 Write down the elements of the group generated by (12345) and determine the signature of each element.

Alternating group 34 The set of even permutations in Sn is denoted by An. Prove that the product of two elements of An is in An. In the case of finite sets of permutations this is sufficient to ensure that all four of the properties in qn 1.34 hold for An,·which is called the alternating group. 35 List the elements of A3 and the elements of A4. 36 Denote the set of odd permutations in S" by Bn. Show that the function of Sn to itself given by a .--. a· (12) is a bijection, and deduce that the number of elements in An equals the number of elements in Bn. 37 Does(! 2

~

!1; ~) = 3

4

(34)(45)(23)(12)(56)(23)(45)(34)(23)? 5

6

(34) (45) (23) (12) (56) (23)

(45) (34) (23)

2

3

4

5

6

Subgroups of S"

17

s. Any subset of s,. which satisfies the four conditions of qn

Subgroups of

l.34 is called a group and is a subgroup of s,.. One way of finding subgroups is to choose some property or structure in A and to consider the subset of s,. which fixes this property. 38 List those elements of S4 for which 4 1-+ 4. This set may be defined as {ala E S4, 4a = 4} and is called the stabiliser of 4 in S4. Compare

this set with S3 and describe the similarities.

39 List the elements of SJ which stabilise 1, 2 and 3 respectively. These three lists form the sets

{ala E S 3, la = I}, {ajcx e SJ, 2a = 2} and {cxlcx E SJ, 3cx = 3}. Do each of these sets satisfy the four properties of a group as stated in qn 1.34? 40 Let a be an element of the set A and let

T = {alaESA,a,-:1. = a}. Does T satisfy the four properties of a group as stated in qn 1.34? The conclusion we draw is that Tis a group, a subgroup of s,., known as the stabiliser of a. 41 List and name the set of elements of S4 under which the set {1, 2, 3} is mapped onto itself.

42 Can you match the elements of the set T in qn 40 with the el;ments of SA-{u}? 43 List those functions in S4 which stabilise both 3 and 4, that is to say, the elements of the set {o:la E s4, 3ci = 3, 4a = 4}. Does this set satisfy the four properties of a group lisied in qn 1.34? How does this set relate to the stabiliser of 3 and the stabiliser of 4 in S4? 44 If Tis the stabiliser of a in SA and R is the stabiliser of b in SA, does the set T n R satisfy the conditions for a group as given in qn 1.34? Suggest a name for the set T n R. 45 Can you match the elements of the set T n R in qn 44 with the elements of SA-{a,bl? 46 List those elements of S4 which map the set {1, 3} onto itself; that is, those elements which either stabilise both 1 and 3 or else interchange these two digits.

"

2 Permutations of a finite set

18

By labelling the vertices of a rhombus 1, 2, 3, and 4, name geometric symmetries of the rhombus which correspond with these listed elements of S4. These elements of S4 are said to fix the set {l, 3}. It is important to note that they do not necessarily stabilise either 1 or 3. If B = {l, 3}, we may describe the~e elements as forming the set {cclcc

E

S4 , Bee = B}.

47 Let B be a subset of the set A and let

r = {etlcc E SA, Bt:1. = B}. Does T satisfy the four conditions for a group given in qn 1.34? The conclusion we draw is that Tis a group, the subgroup of SA, fixing B. 48 If cc = (1234), then {1, 3}cc = {lex, 3cx} = {2, 4}, and {2, 4}cx = {2cx, 4cc} = {3, 1} = {1, 3}. If fJ = (123), then {l, 3}/J = {l{J, 3/J} = {2, l} and {2, 4}/J = {2/J, 4/J} = {4, 3}. Thus cc preserves the partition of {l, 2, 3, 4} into the subsets {1, 3} and {2, 4} while fJ does not. List all the elements of S4 which preserves this partition. Do these permutations form a group? 49 By labelling the vertices of a square 1, 2, 3 and 4, name a geometric symmetry of a square which corresponds to each of the permutations in S4 which occurs in the list of qn 48 ..

50 If x1, x2,

X3 and x, are real numbers, for which elements ex of S4 must x1xi + X2X4 = xi,xi, + x2,X4, irrespective of the numbers assigned to x1, x2, xi and X4. These permutations are said to preserve the v.alue of the expression X1X3 + X2X4. Do they form a group?

51 Which elements ex

E

SJ have the property that

(x1 - x2)(x1 - xi)(x2 -

xi)

=

(x1, -

for distinct real numbers x1, x2 and

x2,)(x1, -

xi,)(x2, -

Xie;}

X3?

Cross-rapo 51 If x1, X2, X3 and (X E s4 such that X1 / X4 XJ - .r2/ X4 X3 -

-

~

X4

X1 X2

are distinct real numbers, find the four elements Xi, -- Xi, / Xi, - X2,/

X4c, -

X1,

X4c, -

X2,.

The function we are studying here is called the cross-ratio of the four numbers x1, x2, X3, X4. 53 By labelling the vertices of a rectangle 1, 2, 3, and 4, name a geometric symmetry of a rectangle which corresponds to each of the permutations you found in qn 52.

Historical note

19

Summary

Definition qn l Notation qn 2 Theorem qn 17 Theorem qns 20, 29

Definition qn 29

Definition qn 29 Definition qn 34 Definition qn 40

A bijection A

--+

A is called a permutation of A.

When A = {l, 2, ... , n},

SA is denoted

bys•.

Every permutation of a finite set can be expressed as a product of disjoint cycles. Every permutation of a finite set can be expressed as a product of transpositions. Neither the transpositions nor the number of them is unique. No permutation which can be expressed as a product of an even number of transpositions can be expressed as a product of an odd number of transpositions. A permutation which can be expressed as a product of an even number of transpositions is called an even permutation. A permutation which can be expressed as a product of an odd number of transpositions is called an odd permutation. The signature of a permutation which may be expressed as a product of n transpositions is ( - If. The set of even permutations in s. is denoted by A. and called the alternating group. If a E A, the subset of SA of elements which fix a is called the stabiliser of a. Any such stabiliser is a subgroup of SA.

Historical note

In 1770, J. L. Lagrange studied the groups S2, SJ and S4 in relation to the solutions of equations of degree 2, 3, and 4. In 1799, P. Ruffini proved by induction that there were n! permutations in S•. In his work on determinants, A. L. Cauchy had defined the signature of a permutation using the function II 1 ,.i i when c = c. The complex number c is then purely real and the translation is parallel to the axis of

the reflection. 30 (i) If a is not a reflection ewe + c =f. 0, from qn 25. So a2: z f-+ z + ei 0c + c is a translation. (ii) r: z ,_. z + ½(ei 0 c + c). (iii) ar = ra => r- 1 ix = ar- 1 =>

(ar 1)2 = ar - 1ar - 1 = a2r - 2 = I.

(iv) Substitute a = QT in ar = ra. (v) A(! = A => AQr = Ar => Ar(! = Ar. (vi)/ = {Qr) 2 = (!'C(!T = (!(!H = /, so r is uniquely defined, and Q a-r

-1

.

31 (i) translations and glide-reflections. (ii) rotations. (iii) reflections and the

identity. 32

=

3 Groups of permutations of 1R: and C

38

33

w-z z-w arg - - = arg - t- z z-t

arg t-z Z(1.

-

W(1.

z-w

Z(J.

-

((1.

z -

34---

=

35 Arg z 36

Z(1. Z(1.

-

2n - arg

z.

z- w = --_, z- t

W(1. -

t

((1.

37 Arguing as in qn 1, Dis a subgroup. From qns 34 and 36 D consists of

rotations and translations. It is a rotation through the same angle. 38 The reflections and glide-reflections are the only opposite isometries of the

plane. 39 e'0 = I when 0 = 0. e' 0 = - I when 0 40 z -

z and z -

41 z -

z, z -

42

W

2

= -

I

- z.

- z, z - z and z -

'! -

= n.

/3 ,

1·

'!ly

W

J

=

- z.

I. 0 = 2n/3, 4n/3, 0.

z, z - wz, z - w2z.

43 z -

45 0 = ½n, n,

½n, 0.

46 z -

z, iz, - z and - 1z.

47

± Z, ± iz, ± z, ± iz.

Z -

48 It has been rotated through ½n and enlarged by a factor 2. 50 S is a subgroup, argue as in qn 3. 51 a

= rew so z - az is the enlargement z - rz followed by the rotation z - e' 0z.

52 1/(2i -

w

=

I) = -

iz - ,t -

+ti.

ti.

If

(1.

is a quarter-turn about this point

Answers to chapter 3

39

If /3 is an enlargement by a factor of 2 from this centre

::/3 = 2z + ! + fi, and w./3 = 2iz - 1. so z 1---> 2iz -

1 is a spiral similarity.

=

53 p = ap + c p = c/(1 - c:). ex is a direct similarity being the product of a rotation and an enlargement with centres at p. a = 1 gives a translation. a = e' 0 gives a rotation. lal cf. I gives a spiral similarity. 54 z1--->z

+ b - a. ,- 1a.

55 ex: z

(b - a)z

1--->

+

a; }': z

1--->

(d - c)z

+

c; a

= ex· \

56 A translation is uniquely defined by the image of the origin.

57 (i) Transitive because it contains the translation group. (ii) Cannot map O 1---> 0 and I

1--->

2 by the same isometry.

58 (i) yes, (ii) yes. 59 p =- c/(1 - r). bis an enlargement by a scale factor r. centre p. (i) Transitive because it contains the translation group. (ii) Cann0t map O 1---> 0 and I 1---> i by the same dilatation.

4 The 1\Johius group

To extend our investigation of transformations of the plane from the similarity group of chapter 3 (with transformations of the form z 1--+ az + b) to the Mobius group with transformations of the form

m+b . z 1--+ ---d, we need to count z 1--+ 1/z as a transformat10n of the CZ+

plane, and we can do this if we adjoin a single point 'oo' to the plane. This 'completed plane' is then like the surface of a sphere in a sense which we will establish with stereographic projection. As a preparation for the idea of a 'completed plane' we will start this chapter by considering a real line with a single 'point at infinity' adjoined. This 'completed line' then has a certain similarity to the circumference or a circle, and we can relate some algebraic transformations of the completed line to geometric transformations of the circle. The symbol 'oo' will take on those properties which we give it, and no others are to be presumed. It does not denote a real or complex number, however large. Concurrent reading: Knopp, chapters 3, 4, and 5; Pedoe, chapter 6; Ford, chapters 1 and 2.

The -completed line, IR u {oo} 1 Let ON be a line segment of unit length and let I: be the circle on ON as diameter. We define a function n: of I: - {N} to the tangent at 0 by n: A 1--+ A', where NAA' are collinear. Is n: an injection, a surjection, a bijection? The function n: is called projection from N.

2 If A and B are points of I: such that AB is parallel to ON but distinct from it, show that angle ONA + angle ONB = one right angle. Use the Euclidean theorem about the angle in a semicircle.

The completed line, IR u {oo}

41

N

A'

= 0, what is the distance OA'? What is OA' · OB', where B' = B1t?

3 If angie ONA 4

5 If the points on the tangent at O are labelled with their directed distances from 0, and the points of l: - {N} are given the labels of their images under n, what is the label for B if the label for A is x (where x is not equal to O)? Use these labels to describe a reflection of l: about the diameter perpendicular to ON. Ignore the points O and N at first, and then describe what happens to them separately. We now label N with the symbol oo and use the algebraic description x 1--+ 1/x to describe the transformation of the whole circumference. 6 Under an anticlockwise rotation through an angle ex about the centre of l:, show that the image of the point of l: which was labelled tan 0 in qn 5 is the point which would have been labelled tan (0 + ;«). Use the Euclidean theorem that the angle at the centre of the circle is twice the angle at the circumference standing on the same arc. Deduce that for all real points of the circle, except for the point x = tan (½n - ½a) = I/tan ½a, this rotation may be expressed in the form

+a ..

X XI-+---.

1 - xa'

where a = tan ½ex- What is the image of oo under this rotation? What point has oo as its image under this rotation? Can you identify these points from the algebraic expression X

+ a?

I - xa·

42

4 The M obius grJup

(It is intuitively useful to imagine what happens to this expression as x gets large, and to decide what values of x make this expression large.) 7 If the diameter ON is rotated through an angle a about the centre of the circle and we call the image of ON under this rotation d, show that under the reflection in d, the transformation of points on the circumference of the circle is given by tan 0 f-+ - tan (0 - :x) using the labelling of qn 5. Give an algebraic expression for this transformation analogous to that at the end of qn 6. What is the image and the pre-image of a::, for this transformation? 8 Verify that there is no real number x such that

ax+ b ex+ d

a

= ~•

assuming that e i= 0, and acJ. - be i= 0. Prove that if the real number r is not equal to a/c, then there is a unique real number x such that

ax+ b = r. ex+ d What real number x does not have a real image under the transformation ax+ b? ex+ d State the image and the pre-image of ax+ b X f-+--ex + d is in s~u{:o} and c i= 0. X f-+---.

9 If e

=

:JJ

;f the function

0 and ad - be i= 0, verify that

ax+ b

X f-+---

CX

+

d

is an affine transformation of 1H (see qn 3.3). If this is to be a function in S1 O} may be expressed in the form

+

b

CZ+

d

az

ZI--->---.

with a, h, c and d real numbers, and ad - be > 0. Assume that the fixing of the half-plane implies the fixing of its boundary. Subgroup fixing the unit circle

50 Use qn 32 to show that every circle through the two points wand 1/it· cuts /zl = 1 orthogonally. If a is a Mobius transformation which maps the circle /zl = 1 onto itself, what can you say about a circle passing through wa and (1/w)a? Deduce that the inversion z 1---> 1/z interchanges; wa and (1/w)a. 51 If a Mobius transformation a maps the circle /zl = 1 onto itself and fixes the point 0, explain why it must also fix oo and so have the form z 1---> e10z for some 8.

52 If the Mobius transformation az

+ +

b d

Zl--->---

CZ

maps the unit circle Jzl

I to itself. prove from qn 50 that

cz +

bz + a --

J az+E

dz+c

for all complex numbers z. By putting z = 0, - a/b, oo, show that a/J = b/c = c/b = d/a, and d.:duce that d = ei 8a and c = ei 8 fj for some 8. By a judicious choice of multiple, show that every transformation fixing the unit circle /zl = 1 may be expressed in the form z

az

+

I--> /"'

DZ+

b a-

'

and that every transformation of this form fixes the unit circle.

53 Show that a transformation fixing the unit circle /zl fixes the interior of the circle and if and only if aa

= 1 as in qn 52,

- bb

> 0.

Summary

Definition The set C u {oo} is called the Mobius plane. qn 13 Definition A Mobius transformation is a transformation of the qn 13 Mobius plane of the form az

+

b

Zl--->--CZ d

+

51

Summary

where ad - be I:- 0, and CfJ 1----+ oo when c = 0, and - d/c 1----+ w 1----+ a/c otherwise. Definition The cross-ratio R(z 1, z 2; z3, Z4) of four distinct points of qns 11, 15 the M6bius plane is Z3 -- Z1

Z3 -

IZ4

-

Z1

z2/ Z4 -

Z2

with appropriate adjustments when one point is w. Theorem The Mobius group is the full group of cross-ratioqns 15, 16 preserving transformations of the Mobius plane. Theorem A Mobius transformation is uniquely determined by the

qns 17, 45 images of three points. Theorem Four points of the Mobius plane are concyclic or colqns 21-24 linear if and only if their cross-ratio is real. Theorem The set of circles and lines is transformed to itself by a qns 25, 26 M6bius transformation if we adopt the convention of adjoining CfJ to each line. Definition If ON is a diameter of the sphere l: and II is the tanqn 35 gent plane at 0, then the projection from N of l: - {N} onto II is called stereographic projection. Definition The transformation of the Mobius plane z 1----+ 1/z which qn 36 transposes O and CfJ is called the inversion in the unit circle lzl = I. Theorem If two points are transposed by the inversion in the unit qns 34, 37 circle, then any circle through those two points is mapped onto itself by this inversion. Theorem The (absolute value of the) angle between two circles is qns 39--41 equal to the angle between their images under z 1----+ 1/z. Theorem The angle between two circles is equal to the angle qn 42 between their images under any M6bius transformation. Theorem The subgroup of the Mobius group fixing the upper half · qn 47 plane contains all transformations of the form

az

+b

ZI----+---,

CZ+

d

where a, b, c, d are real and ad - be > 0, and no others. Theorem The subgroup of the M6bius group fixing the interior of qn 53 the unit disc contains all transformations of the form

z

I----+

az Ez

where

+b + a'

aa - bE

> 0, and no others.

4 The Mobius group

52

Historical note

The first use of the equation z'

= az + b CZ+

d

was by L. Euler (1777). It was B. Riemann (1854) who first adjoined 'oo' to a plane to make z ....... I/ z a continuous transformation. The notion of cross-ratio-preserving transformations is due to M. Chasles (1837), who was considering the cross-ratio of lengths of real line segments. The first use of complex numbers to describe projective transformations was by von Staudt (1856). The systematic study of circle-preserving transformations was pursued by A.F. Mobius (1852~6) using synthetic methods. The interpretation of Mobius' results using complex numbers and the transformation

az

+ +

b d

Zf-->---

CZ

is due to F. Klein (1875) and others. Stereographic projection was used by Ptolemy (A.D. 140): its use for labelling the points of the sphere with C u {oo} and its use in associating transformations of the sphere with transformations of the plane is due to F. Klein and is described in his Lectures on the Icosahedron (1884). The subgroup of the Mobius group fixing the upper half plane is the group of direct isometries of the non-Euclidean plane with lines taken as semicircles centred on the real line. The subgroup of the Mobius group fixing the interior of the unit circle is the group of direct isometries of the non-Euclidean plane with lines taken as circular arcs orthogonal to the unit circle. Both these models of the hyperbolic plane are due to H. Poincare (1881). The first major text book covering the whole of this area is by R. Fricke and F. Klein (1897).

53

Answers to Chapter 4

Answers to chapter 4 1 n is a bijection.

=

2 ONA

NAB

=

NOB.

3 tan 0.

4 ONB = ½n - ONA, so OB' = cot 0 and OA' · OB' = 1. 5 B is labelled 1/x. x >--+ I /x.

6 tan 0 >--+ tan (A

+ b)

is

X + a I - xa

X f--+ - - - .

I !a ,-. x 7 tan

1 a.

f--+ -

tan (0 - '.() is

(J f--+ -

a -

X

X f--+ - - - .

I+ ax

-

I/a

+

8 e(ax X

x;

f--+

b)

f-+

-

Ila.

= a(cx + d)

= ad -

=

- d/c has no real image. - die 9

be

=

0.

dr - b - er+ a f--+ er.; f-+

a/e.

X) f--+ X)_

10 Provided ad - be # 0, all four are permutations of lR u {a:;}.

1l Tis a subgroup by the argument of qn 3.1. Thus a transformation of iR u { x;} preserves cross-ratio if and only if it has the form

ax+ b

X f--+ - - - .

ex+ d

13 (i) x;

f-+

x. (ii) - die

f--+

x

f-+

a/e.

14 All but

be - ad e

z f--+---z

which requires be - ad # 0. First and third are direct similarities, the fourth is a translation. 15 Yes.

4 The Mobius group

54

17 Use the algebra of qn l l. 18 The surface of the sphere with oo at N.

19 The circle centre ½, radius ½Transformation z f-+ 1/z transposes points so circle is mapped to line. 20 z and c lie on a circle through w and t.

21 0 or n so the number is real. 22 0 or n.

23 0 or n. 24 (i) The circle through z, t and w. (ii) The line through z, t, and w, and the point 00.

25 Since Mobius transformations preserve cross-ratio, they preserve real crossratios, so the image of a circle is either a circle or a line with oo 26 As in qn 25 the image is either a circle or a line with oo.

27 If they are parallel, a translation is sufficient. If they intersect, a rotation is sufficient. Every direct isometry is a Mobius transformation. 28 Choose three points a, b, con the circle and three points a', b', c' on the line, then by qns 12 and 17 there is a Mobius transformation mapping a f-+ a', bf-+ b', cf-+ c' and from qn 25, the circle maps onto the line with oo.

29 ei 0 f-+ e -iii_ 2 f-+ ½- jzj = 2 maps to izl = ½- 1R - {O} ➔ 1R - {O}. The line is reflected about the real axis, but not pointwise. 30 {r. 1/r, e' 0} ..... {1/r, r, e-ie}. If the line joining Oto ei 8 meets the circle again at z, 1 ·lzl = r·(l/r) so jzj = 1, and z = ew. C' is the reflection of C in the

real axis. C' cuts

izl = 1 orthogonally.

33 rs = 1. rand s are transponsed. 34 /J = ¢. rs

=

~

is mapped to its reflection in tJ-e real axis.

I. rew • se' 8 = I.

35 z f-+ 1/z. In each plane through ON the reflection acts like the real transformation x f-+ I /x. 36 z

37

~

f-+

I/.:' maps circles and lines to circle and lines.

is mapped to itself. but not pointwise.

38 Any circle through P and P' is orthogonal to lzl = I, so any such circle is mapped to itself under z f-+ 1/z. So S -➔ S. Thus Sn C ➔ S n C'. Since Sn C is a single point, so is S n C. 39 C 1 and S 1 have a common tangent at P, C 2 and S2 have a co, .1mon tangent at P so angle between C1 and C2 = angle between S1 and

s~.

55

Answers to chapter 4

40 (i) Argument of qns 38 and 39 holds so long as P # P' and P # 0. (ii) Circle image is parallel to tangent at 0.

41 Apply argument of qns 38 and 39 to the point of intersection off lzl 42 z

H

cz + d, z

H

z, z H

=

1.

1/z,

be - ad

ZH---z, C

z

z + a/c each preserve angles of intersection and their product is

H

+

b

CZ+

d

az

ZH---.

(i) z

43

az

H

+ b,

(v) z

.. az (u ) z H - -1 CZ+

.. ') (Ill

Z H

(iv) z

H

a;;:

(VI.)

a

H

az

+

az a -

ZH----.

CZ+

+c+

1 - a, C

d - a

------, CZ+ d

az,

The set of all transformations of the given form form the stabiliser in each case. 44 If z is a fixed point,

az + b = ---

z

CZ+

d

so cz2 + (d - a) z - b = 0. This quadratic equation has either 1 or 2 roots. If c b/(d - a) are fixed.

=

0, then oo and

45 Only the identity fixes three points. If two transformations a, fl map three points in the same way, then (/.p- 1 fixes the three points and aP- 1 = 1. a Ha', b t--· b', c H c'.

+ 4bc =

0.

b fixes oo, and fixes no other point if a

=

46 From qn 44, equal roots when (d - a)2

47 z

H

az

+

48 If real line is fixed qns 11 and 17. 49 z

H

az

OH

a1, 1 Hai, oc

Hai,

where

+b

CZ+

d

preserves upper half plane if Im (z) > 0 implies Im (az + CZ+

b) > 0.

d

I.

ai, a2,

ai are real from

4 The Mobius group

56

50 0, w, I ;i;:, are collinear. Product of distances from O = I. Every circle through w and I /w cuts l=I = 1 orthogonally so by angle preservation every circle through wa and (1/w) a cuts lzl = I orthogonally. The line through wa and U 111>) a cuts lzl = I orthogonally and so passes through 0. ·r: z 1---+ I /z, way = wya. 51 With 1' as in qn 50, 0a and I 1---+ ei 0•

=

0 so wa

=

Oycr:

= 0ay =

0y

=

CX),

0

1---+

0.

y as in qn 50, zct1' = zyx. dJ = aa implies ldl = la! so la/JI d/a = ei 0 (say).

52 With az

+

CZ+

b

d

az

+

cD 1---+

I and

b

Eei 0z + ae' 0.

Now multiply top and bottom by e-+,e.

=

=

1---+ b/a. lb/al < I lbl < 1a1 aa - bo > 0. If w is interior, there is a circle through O and w not intersecting lzl This circle's image cannot intersect lzl = I.

53 Cl

oo

I.

5 The regular solids

In chapter 3, we found the group of isometries of the real plane and some of its subgroups. In this chapter we will identify three finite groups of isometrics of real 3-dimensional space. These will be the groups of rotational symmetries of the regular tetrahedron, the cube and the icosahedron respectively. As regular solids, these may be inscribed in a sphere, and theu any symmetry n

= 3.

4 n · 120" < 360° => n < 3, contradiction. 5 (i) tetrahedron, (ii) octahedron, (iii) cube, (iv) icosahedron, (v) dodecahedron. 6 Three rotations stabilise one vertex. Each vertex has four images. At most 3 · 4

=

12 rotations in all.

8 The six transpositions give the reflections. The 4-cycles give neither reflections nor rotations. 9 Three rotations stabilise one vertex. Each vertex has eight images. At most

3 · 8 = 24 rotations in all. Four rotations stabilise one face. Each face has six images. 4 · 6 = 24. 10 Yes.

11 4-cycles, 90'. 3-cycles, 120°. Transpositions or pairs, 180°.

12 Every rotational symmetry of a cube is a rotational symmetry of it8 inscribed octahedron and vice-versa.

13 Five rotations stabilise one vertex. Each vertex has 12 images. Three rotations stabilise one face. Each face has 20 images. 14

± 72°: (12345), (15432), (12534), (14352), (13425), (15243), (13254), (14523), (14235), (15324), (12453), (13542). ± 144°: (13524), (14253), (15423), (13245), (14532), (12354), (12435), (15342), (12543), (13452), (14325), (15234). ± 120°: all 3-cycles. ± 180°: all pairs of transpositions. Ai.

15 Can be identified through qn 5 with the rotational symmetries of an icosahedron.

16 Argue as in qn 3.24. yix is a rotation about OB; ix/3 is a rotation about OC; yp is a rotation about OA. Locate A so that the angle between the planes OAB and OBC is half the angle of the rotation about OB, and so that the angle between the planes OBC and OAC is half the angle of the rotation about OC.

6 Abstract groups

We start this chapter by abstracting the properties of S.4 to which we drew attention in qn 1.34 and use these properties to define a group in a way which does not depend on permutations or geometry and in which the elements of the group need not eve_n be functions. For two functions, composition has heen presumed to be the method of combination. For two numbers, addition or multiplication are possible methods of combination. In general the phrase 'binary operation' is used to describe the combination of two elements to make one element. Some immediate consequences of the group axioms are identified in qns 5-9. · · Concurrent reading: Green, chapters 4 and 5, Fraleigh, sections 2, 3, 6 and?. Axioms

So far every group we have considered has been a subgroup of a symmetric group. We now widen the definition of a group to include those sets, G, with a binary operation ( •), not just composition of . functions, which satisfy the four conditions which we insisted on for symmetric groups, namely closure If a, b E G, then a·b E G. associativity If a, b, c E G, then (a·b)·c = a·(b·c). identity There exists e E G such that a·e = e·a = a for all a E G. inverses For each a E G, there exists an a - 1 such that -1 -1 a·a = a ·a = e. i Name some sets of numbers which under the binary operation of addition form groups according to the definition above.

Subgroups

63

2 Name some sets of numbers which under the binary operation of

multiplication form groups according to the definition above.

= (a - b) - c? Do any of the sets you have chosen for qn 1 form groups under subtraction?

3 For what numbers does a - (b - c)

= (a/b)/c? Do any of the sets you have chosen for qn 2 form groups under division?

4 For what numbers does a/(b/c)

5 If e andf are elements of a group (G, ·) and for every element a in G, a · e = e · a = a = a · f = f • a, prove that e = f [The identity is unique.]

6 If (a1) • a = e, a1 is called a left inverse for a. If a· (a') = e, a' is called a right inverse for a. By considering the product a1·a·a' prove that

every left inverse is equal to every right inverse and hence that the inverse of a is unique. 7 If (G, ·) is a group with given elements a and b, prove that there is a unique x in G such that a • x = b, and a unique y in G such that }" a = b. What is the inverse of a- 1? 8 Show that every group (G, ·) contains a unique element a such that , a- = a·a = a. 9 If a and bare elements ofa group (G, ·),find (ab)(b- 1a- 1), and deduce that (ab)- 1 = b- 1a- 1• Give an example to show that

(ab)- 1 = a- 1b- 1 may be false. Subgroups

When a subset Hof a group (G, ·) has the property that (H, ·) is itself a group, then H is called a subgroup of G. When G #- H, H is called a proper subgroup. The smallest subgroup containing a given element a is denoted by and is said to be generated by a. Such subgroups, generated by a single element, are useful in analysing the structure of groups and are said to be cyclic. Finite cyclic groups lead to the notion of the order of an element. For the element a, the order is the least positive integer n such that d' = e. 10 Name some subgroups of (C, +) and some subgroups of (C - {O}, x).

11 If His a subgroup of G, use qn 8 to show that these two groups have the same identity. Deduce that the inverse of an element in His the same as its inverse in G.

12 If Hand Kare both subgroups of G, prove that H n K is also a subgroup of G.

6 Abstract groups

64

Give examples from the groups D2 and S3 to show that H u K need not be a subgroup of G.

13 If the group (IR, +) contains a subgroup (H, +) and l is in H, what other elements of IR can we be sure lie in H? Must every integer lie in H? (In the light of this question (Z, +) is said to be the subgroup of (IR, +) generated by I, and this is expressed symbolically by writing ( 1) = (£'., + ).) 14 If(£'., +) contains a subgroup H and 2 is in H, what other elements of £'. can we be sure must lie in H? (The subgroup generated by 2, (2), in (£'., +) consists of the even integers.) 15 If ix = (1234) and {J = (12)(34) denote symmetries of a square descrited by permutations of rts vertices, find a2, cl, pa, {Ja 2 and {JaJ, as permutations of the vertices, and desctibe them geometrically as symmetries of the square. This is an example of the dihedral group D4 • What is the smallest subgroup containing a:, that is, (ix), the subgroup generated by a:? Find the subgroups (a: 2 ), (aJ), (/3), ({Jix), (f3a: 2 ), ({JixJ) and (I) respectively.

1□4 2

3

Order 16 If a is a reflection of the Euclidean plane, say, a: z H z, then a: 2 = I, the identity function, and ix is said to have order 2. The reflection a generates the subgroup {I, a:} of Sc. If {J is a rotation of the Euclidean plane through 120°, say, {J:z H e2xi/Jz, find the least positive. power n of {J such that fJ" = I (n is then called the order of {J) and identify the skbgroup of Sc which is generated by {J.

17 Make a list of elements of SJ and state the order of each element. What is the subgroup of SJ generated by (123)?

18 State the orders of the elements of D4 as given in qn 15. Cyclic groups: groups generated by one element 19 For any element gin. a group (G, · ), we define g° = e, g 1 = g, and then gn+I = (gn)g for positive integers n as in qn 2.7. Also we define g-n = (g- 1 f as in qn 2.7 for positive integers n. Assuming that the results of qn 2.7 apply in this more general context, show that the set {g'lr = 0, ± 1, ±2, ... } is a subgroup of G. Must

Groups generated by two elements

65

any subgroup of G which contains the clement g contain the whole of this set? As before, we denote this subgroup by g

< >.

.

20 A subgroup of the form (g) in any group, that is to say, a subgroi•p generated by one element is known as a cyclic subgroup. Find a cyclic subgroup of Ss containing exactly five elements. Find a cyclic subgroup of S6 containing exactly six elements .

21 If a is a group element of order 5, how many distinct elements are there in (a)? What are the orders of each of these elements?

22 If a is a group element of order 6, how many distinct elements are there in (a)? What are the orders of each of these elements? 23 If a is a group element of order 6, find all the subgroups of (a). Are all these subgroups cyclic?

24 If a and b are elements of a cyclic group, must it follow that ab

= ba?

25 If a is an element of a group and a has order n, what can be said about the order of a - I? 26 The group (Z:'., +1) is a cyclic group generated by l. If (H, +) is a subgroup of (Z:'., +) and the least positive integer in His a, prove that H cari only contain the elements of (a), and deduce that every subgroup of (Z, +) is cyclic. 27 If G is an infinite cyclic group, prove that every subgroup of G is cyclic.

28 If G is a cyclic group generated by an element g of order n, we know from qn 19 that G = (g) = {g'!r e Z}. Prove that G = (g) = {g'IO ~ r < n}, and that then elements in this set are distinct. Let H be a subgroup of G and of all the elements of H let g" be that with least positive exponent. Prove that if k is a positive integer less than a, then no element of the form g1a+k may belong to H. Deduce that H = (g"). ~

Groups generated by two elements If a group contains the elements a and b, then it clearly contains both (a) and (b). What other elements it must contain depends on the relation between the elements a and b. The smallest subgroup containing a and b is called the subgroup generated by a and b and is denoted by (a, b). The simplest relation between a and bis the equation ab = ba, as, for example, when a and b are disjoint cycles of a permutation group. When such an equation holds, we say that a and b commute, or satisfy the coriimutative law.

6 Abstract groups

66

= ba, prove by induction on n that a"b = ba". If, in addition, the element.bis given to have order 2, by filling in the multiplication table shown,

29 If a and b are elements of a group and ab

prove that the set {a", ba"ln E Z} is closed, and then complete the proof that it forms a subgroup, the subgroup (a, b). 30 If a and bare elements of a group and ab = ba, prove that a"b 2 = b2a". (It is only necessary to use qn 29, no induction is required.) If, in addition, b has order 3, by filling in the multiplication table shown, prove that the set

{a", ba", b2 a"I n E Z} is closed, and then complete the proof that it forms a subgroup, the subgroup (a, b). 31 If a and b are elements of a group and ab = ba, prove that amb" = b"am for all integers m and n. (Use qn 29 and an induction on n.) Deduce that {amb"I m, n E Z} forms a subgroup, the subgroup (a, b). 32 If a and pare elements of a group and (ab) 2

ab

= a2 b2, prove that

= ba.

33 If a and b are elements of a group and ab

= ba, prove that

(ab)" = a"b" for all integers n. 34 If a = (12) a!l,d b = (345), what is the order of ab?

35 If a = (12) and b = (34567), what is the order of ab? 36 If a

= (1234) and b = (567890), what is the order of ab?

37 If a and b are disjoint cycles of length m and n respectively, and the highest common factor of m and n is k, what is the order of ab?

Dihedral groups

67 \

38 Give an example to show that if a and b are elements of a group and a has order 6 and b has order 10, then ab need not have order 30 even when ab = ha. When ab = ha for any two elements a, b of a group (G, •), then (G, ·) is said to be abe/ian. Dihedral groups Another relationship that may hold between two elements a and b of a group is ab = ha- 1 and in this case too it is possible to give a complete account of. the elements of the ftl'OUp (a, b). We will concentrate only on such groups for which b has order 2. 39 If a

= (123), b = (23), so that a 3 = b2 = e, verify that ab = ba- 1

and, by listing the permutations a 2 , ha and ba2 , ve~ that (a, b) = S3. 40 If a = (1234) and b = (12)(34), so that a 4 = b2 = e, verify that ab = ba- 1 and, by listing the permutations a 2, a3, ha, ba1 and ba3, verify that (a, b) describes the group of symmetries (J{ a ~uare as in qn 15. 41 If a = (12345) and b = (25)(34), so that a5 = b 2 = e, verify that ab = ba - 1. L"1st th e permu tat"10n a2, a 3, a4, ba, b, a 2, ba 3 an-.1 ha4

and, by labelling vertices of a regular pentagon appropria,.:ly with the digits 1, 2, 3, 4 and 5, name geometric symmetries which correspond to these permutations. 42 If a and b are elements of a group, b has order 2 and ab = ba - ,, prove that bah = a- 1• Evaluate the product (bab)(bab) in two ways to establish that ba2b = a- 2 and hence that a2b = ba- 2 • Extend this method to establish that d' b = ba-" for all integers n. By filling in

the multiplication table shown,

prove that the set {a", ba"I n e Z} is closed, and then complete the proof that it forms a subgroup, the subgroup (a, b). 43 A group G is generated by an e!ement a of order n and an element b of order 2 with the relation ab = ha_,. Prove that G contains 2n elements and that every element of the form ha' has erder 2. Tffi.s

6 Abstract groups

68

is the abstract definition of the dihedral group D•. The name dihedral (two-faced) stems from the fact that in the plane, the group of symmetries of a regular n-gon is of this type. See qns 3.41-47. Groups generated by larger sets of elements 44 Express each element of S3 as a product of transpositions. This possibility may be expressed by writing ((12), (13), (23)) = S3.

45 Use qn 2.20 to show that the set of transpositions ins. generates S•. 46 Evaluate'(la)(lb)(la) and deduce that the set of transpositions {(12), (13), ... , (In)} generates s •. 47 Evaluate (12)(13), and also evalu,ate (134)(132) as a product of disjoint cycles, and deduce that every even permutation may be expressed as a product of 3-cycles. That is, the 3-cycles generate A •.

48 Explain how to express a translation as a product of two reflections

and how to express a rotation as a product of two reflections. This establishes that every transformation of the form z H e;8z + c is the product of two reflections. Deduce that every transformation of the form z H e;8 z + c is a product of at most three reflections, and that the set of reflections generates the Euclidean group. 49 Since every element in the group of direct isometries is a product of

two reflections, deduce that every product of four reflections or of any product of an even number of reflections is equal to a product of two reflections. Prove also that the product of any odd number of reflections is either equal to a single reflection or to a product of three reflections. Isomorphism When can we say that two groups are the same? In some cases this may be obvious. For example, if the elements and the relation between those elements are written down using roman letters, we have the same group as if they had been written down using Greek letters. In qns 2.49 and 3.47 and qn 40, we have found different ways of representing the same group of symmetries. We now examine this kind of possibility. 50 The group of symmetries of a rectangle, D2, may be exhibited as a subgroup of S4 by labelling the vertices of the rectangle with the

numbers l, 2, 3 and 4. See qn 2.53. • What is geometrically the same group may be exhibited as a

69

Isomorphism

different subgroup of S4 by labelling the vertices of a rhombus and exhibiting-its group of symmetries. See qn 2.46. Again, each of the elements of this geometrical group may be expressed as an isometry in Sc, or a mapping of coordinate pairs (x, y). See qn 3.41. Exhibit tht;se four presentations of this geometric group consisting of the identity, two reflections and a half-turn.

51 Exhibit the group of symmetries of an equilateral triangle, DJ (i) as a subgroup of S3 by labelling the vertices 1, 2, '3; (ii) as a subgroup of S6 by labelling the points one-third of the way along each of the sides. Give the two lists of elements in such a way that it is plain which elements of the two groups match. 52 When there is a one-one matching of the elements of a group G with the elements of a group G' with gin G matching g' in G ', then the function G -+ G' given by g f-+ g' is called an isomorphism and the groups G and G' are said to be isomorphic, provded that (ab)' = a'b', for every pair of elements a, b in G. We then write G ~ G '. If a = (123) and ri:z -+ e¼"iz exhibit multiplication tables for each of the groups {e, a, a2} and {/, ri, ri 2 } and say how the elements should be matched for an isomorphism between these two groups. 53 By using the convention that the product ab appears in the row with a at the left and b at the top, a binary operation on a finite set may be exhibited by means of a multiplication table.

For the table given here, say how you would find the identity, check for closure, and then find an inverse fqr each element. d

a

b

C

a b-

a b

C

d

d

C

C

C

b a d

d

d

C

a b

b a

Use qn 7 to explain why the elements in a row(or a column) ofa group table are all different. ·

6 Abstract groups

70

S4 Because the multiplication table or Cayley table for a group exhibits

products, the matching of two Cayley tables is a check on the structure-preserving property (ab)' = a'b'. Which of the following Cayley tables for a group may be rearranged so as to establish an isomorphism with the group exhibited in qn 53?

p I q r

p

I

q

r

I p r

p I q r

r q I p

q r

q

I

I I m

I

I

p

I m

I m

I

n

n

n I

m

n

m

n I

n I

I

I n

55 The Cayley table

2 3

2 3

2

3

2 3 I

3 I 2

is that of a group, G. Make a Cayley table for the permutation group {(1), (123), (132)} and check that G is isomorphic to A 3 under the matching I H (I), 2 H (123), 3 H (132). Exhibit the functions x H x • 1, x H x • 2 and x H x · 3 of G. What can you say about the matching a H [x H x · a]? 56 For each of the groups with Cayley tables

I 2 3 4

1 2 3 4

2

3

4

2 3 4 I

3

4

4 1 2

1 2 3

2

construct the four permutations x H x • 4. Is the matching a H [x case?

x

2

1 2 3 4

2 3 4 H H

3

4

3

4

4

3 2

4

3

2

x · I, x H x · 2, x H x • 3 and x · a] an isomorphism in each

57 For any group (G, ·) show that the set of functions of the form x H x · a forms a subgroup of SG. By matching the function

Summary

71

•x · a with the element a show that this subgroup of SG is

x H isomorphic to (G, · ). It is this theorem (Cayley, 1878) which shows that our starting point in chapter I was sufficiently general to give an isomorphic image of every possible group.

58 Establish that the group of translations of IR is isomorphic to (IR, 59 Establish that the group of translations of C is isomorphic to (C,

+ ). + ).

60 In the group of ratio-preserving transformations of IR, prove that the stabiliser of O is isomorphic to (IR - {O}, x ). 61 In the Mobius group, use the fact that the stabiliser of oo is the group of direct similarities to prove that the stabiliser of O and oo is isomorphic to (IC - {O}, x ). 62 Use the logarithmic function to establish that the multiplicative group of positive real numbers is isomorphic to (IR, + ). 63 If a cyclic group contains n distinct elements, what can be said about the order of one of its generators? 64 If two cyclic groups each contain n distinct elements, establish that they are isomorphic. The symbol C is used to describe such groups up to isomorphism. 65 Show that two infinite cyclic groups are isomorphic. 66 If ct.: G --+ G' is an isomorphism and e is the identity in G and e' is the identity in G ', prove that er:t. = e'.

67 If ct. : G --+ G' is an isomorphism and g E G, prove that (g- 1)ct. is the inverse of gr:t. in G '. In other words, (g- 1)r:t. = (gc1.)- 1• 68 If c1.: G --+ G' is an ismorphism and g is an element of order n in G, prove that gr:t. is an element of order n in G '. 69 Determine whether either the full group of symmetries of a cuboid with three unequal sides or the full group of symmetries of a solid swastika is isomorphic to D4 •

6 Abstract groups

72

Summary

Definition A set G on which a binary operation · has been defined is called a group when (i)a·b E G for all a, b E G (closure), (ii) a· (b · c) = (a· b) • c for all a, b, c E G (associativity), (iii) there exists e E G such that a· e = e · a = a for all a E G (identity), (iv) for each a E G , there exists a- 1 E G such that a·a- 1 = a- 1 ·a = e. (inverses). Theorem The identity in a group is unique. The inverse of each qns 5, 6, element is unique. Solutions for x of a· x = b and

7

x-a

=

bare each unique.

Definition If His a subset of a group (G, · ), and (H, ·) is a group, qn 10 then H is called a subgroup of G. Theorem The intersection of two subgroups is a subgroup.

qn 12 Definition If h

E G, the smallest subgroup of G which contains h, qn 13 or, the intersection of all subgroups of G which contain h, forms a subgroup which is denoted by a- 1(a · a)

=

a- 1a

=>

a

=

e.

9 (ab)(b- 1a- 1) = [(ab)b- 1 ]a- 1 = aa- 1 = e. b- 1 a- 1 is the unique solution to (ab)x = e. If a = (12) and b = (13), then (ab)- 1 =f a- 1 b- 1 •

+ ), (0, +) and (IR, +) are subgroups of (C. + ).

10 (Z,

11 The unique solution of a2 = a in His in fact a solution in G. If e is the common identity and a E H, then ax = e has a unique solution in H and in G.

12 Check identity, inverses, closure. 13 l

+ I, l + 1 + 1, etr. and their inverses.

14 2

+

15 a 2

2, 2

=

+

2 + 2, etc. and their inverses.

(13)(24), half-tum.

16 3. (/3)

/h. = (13), reflection. (::.:) = {I,

ci:, / ,

::.:3}.

= {I, /3, /3 2}.

17 order I: (l); order 2: (12), (13), (23); order 3: (123), (132). 2

18 order 1: /; order 2: IX ,_ /3, /31X, 19 Obviously closed.

20 a

=

/30: 2, /31X J; order 4:

g° is th~ identity.

g-n

=

J

a.

(g'')- 1• Yes.

(12345). (:.i) has five elements.

21 Five elements. Order l: /; order 5: a,

a2, a3,

a4 •

22 Six elements. Order 1: /; order 2: a3; order 3:

· 23 {I}, {I, a 3 }, 24 If a

CJ.,

=

{I,

g" and b

25 Since a'(a ~

1r =

a2,

a4; order 6: a,

a 2 , a 4 }, {I, a, a2 , a 3 , a4, a 5 }. All cyclic.

=

g"', then ab

/,

(a -

Ir =

=

g"+m

=

g"'+"

=

I and no lower power.

ba.

a5.

Answers to chapter 6

75

26 If b EH, b # ka, then for some k, ka < b < (k giving a smaller positive element of H.

+

l)a and 0 < b - ka < a,

= (g) then each subgroup contains an element g' of least positive exponent. Then argue as in qn 26.

27 If G

28 If g"

I then g'

=

=

g,-kn, so every element of (g) is in the given set.

=

If O ,:; r, s < n and g'

=

g', then gr-s

g5-r

=

I so either r

=

s or the

"+\

order of g is less than n. H contains (g") and so g 10 • If H contains g 1

H contains /. Contradiction. 29 Given for n = I. Assume anb = ban. Then

an-,- 1b

=

(an·a)b

=

an(ab)

=

a\ba)

E.g. d" bd = am+/ b, and (ban)(ba 1) bd" is ba-n.

=

=

(anb)a

=

(ban)a

=

ban+I_

an+/. Closed. Identity a0 , inverse of

33 n = I. trivial. Assume (ab)" = anbn, then

(abr 1

=

(abY(ab)

35 10.

34 6.

=

anbn(ab)

36 12.

38 a = (12)(345), b

=

d'abnb by qn 29.

37 mn;k.

= (12)( 67890).

43 G = {a', ba110,;; i,j < n}. 44 (I) = (12)(12), (123) = (12)(13), (132) = (13)(12). 47 (12)(13) = (123). (134)(132) = (12)(34). In a string of transpositions every adjacent pair may be written either as a 3-cycle (if they overlap) or as a product of two 3-cycles (if they are disjoint). 48 A translation is the product of two reflections in parallel axes. The length of the translation is twice the distance between the axes. See qn 3.21.

49 See qn 3.37. 4

50

2

3

(!) (12)(34) (14)(23) (] 3)(24)

(1) (13) (24) (13)(24)

z z

f-+

z

i---+

z

I---►

f-+

z i -Z -z

(x, y)

f-+

(x, y) (x, _v) (x, y)

f-+

(x, y) (x, - y)

f-+ ( f-+ ( -

X, y) x, - y).

6 Abstract groups 51

76

l

ti

2

c

d

3

(123)....., (ace)(bdf) and (23) 52 e

+-+

I, a

+-+

ex, a

2

+-+

+-+

(af)(be)(cd), for example.

'

ex·.

53 Two elements in the same row are ax and ay. ax = ay

=>

x = y.

54 {I,p, q, r}. 55 x

1-+

x·2 is (123). a 1-+ [x

t-+

x·a] is an isomorphism.

57 If ex:x t-+ x·a, and /J:x,....... x·b, then ex/3:x establishes the isomorphism. 62 xy

t-+

log xy

=

log x

t-+

(x·a) ·b

=

x·(ab) which

+ logy.

63 Each has order n.

66 a· e = e • a = a, so (a· e)oc eex is the identity.

=

(e • a)ex

67 g·.g- 1 = g- 1 ·g = e, so (g·g- 1):x g -lex· gex = eex = identity.

68 g;ex

=

(gexf. g"

=

e

=>

(gex)"

=

=

=

all., so aex • eex

(g- 1 ·g)oc

=

=

eex • aex

=

ecx, sogex·g- 1ex

aex and

=

eex so gex has order n or less. But

(gexr = eex => gmex = eex and ex is one-one so gm = e. That is, elements of the same order must be matched under an isomorphism. 69 Number of elements of order

Cuboid Solid swastika D4

2

4

7 3 5

4

2

So no isomorphism possible, since elements of the same order must be matched by qn 68.

7 Inversions of the Mobius plane and stereographic projection \

'

In this chapter we elaborate one example of a group generated by a set of elements, the group generated by all inversions, which includes the Mobius group as a subgroup, and we examine one particular isomorphism. The group generated by the inversions is in fact the full group of circle-preserving transformations of the plane, though we will not complete the proof of this since it depends on the fact that the identity function is the only function IR --+ IR which is both an isomorphism of the additive group of real numbers and also an isomorphism of the multiplicative group of real numbers. The isomorphism which we will study is between the full group of circle-preserving transformations of the Mobius plane, and a group of transformations of the surface of a sphere. The isomorphism is set up using stereographic projection and the interest in this isomorphism depends upon the preservation of geometric structure under this kind of projection. Concurrent reading: Forder, pp. 19-29; Hilbert and Cohn-Vossen, pp. 248-68; Yaglom, pp. 55-7; Knopp chapters 3, 4 and 5; Hille, pp. 38-4~, 46-58; Klein, chapter 2.

Inversion 1 If L is a circle with centre O and radius a and Sis a circle which cuts L orthogonally and, moreover, a diameter of L meets Sat the points A and B, what is the value of OA · OB? (See qn 4.31.)

2 If L is a circle with centre O and radius a and A and Bare two points on a radius of L with OA · OB = a2, then A and Bare said to be ~nverse points with respect to L. Must any circle through A and B cut L orthogonally provided A #- B?

7 Inversions of the Mobius plane and stereographic pr-ojection

78

3 If I: is a circle and A and B are two points in the plane of I: such that

every circle through A and B meets I: orthogonally, must A and B' be inverse points with respect to I:? If o,; is a Mobius transformation, what can you say about Ao,; and Bo,; in relation to I:o,;? What happens if I;o,; is a straight line? 4 For any circle I: of the plane, we define an inversion in I: as a

transformation of the Mobius plane which transposes every pair of inverse points and also transposes oo with the centre of I:. In the case where I: is the circle l=I = 1, does this definition coincide with that in qn 4.36? What are the fixed points of an inversion in I:?

5 If S is a circle orthogonal to I:, what is the image of S under the inversion in I:? 6 If L is a diameter of I:, what is the image of L under the inversion in

I:? 7 If z and z' are inverse points with respect to the circle lzl = a, use the working of qn 4.34 to show that if z = re;o then z' = (a 2/r) e; 0• 8 If two circles have the same centre and distinct radii, a and b, by taking

the centre as the origin compute the product of successive inversions in the two circ1es. Give a geometric description of the composite transformation. 9 Prove that every direct similarity can be expressed as the product of

two reflections and two inversions.

10 Use the fact that z f-+ 1/z is the product ofa reflection and an inversion (as in qn 4.36) to prove that every Mobius transformation is a product of reflections and inversions, an even number in total. 11 Describe as ·many similarities or geometric analogies as you can between a reflection and an inversion in a circle. Jf we use the

phrase 'inversion in a straight line' to mean a reflection, how does the group of transformations of the Mobius plane generated by the inversions relate to SM and to the Mobius group? 12 If z and z' are inverse points with respect to a circle with centre s and radius R, show that if z - s = rei0 then z' - s = (R2 /r)ei8 and deduce that the inversion in this circle has the form z f-+ R 2/(z - s) + s. Show that for a suitable choice of a, b, c and d, this may be expressed in the form z

az + b cz + d'

f-+---

with ad - be ¥- 0.

Stereographic projection

79

13 Show that the-product of two inversions is a Mobius transformation. 14 What can be said about a transformation which is a product of an even number of inversions?

15 Give a general algebraic form for any transformation which can be expressed as a product of an odd number of inversions. 16 Must any transformation in the group generated by inversions be either a Mobius transformation or a product of a Mobius transformation and a reflection and so preserve the set of lines and circles in the Mobius plane? Any such transformation is called a circular transformation, and in fact the full group of circular transformations is generated by the inversions, though we will not prove this.

17 In the diagram of qn 4.1, prove that NA· NA' = N0 2 • Deduce the image of the circle on NO as diameter under an inversion in a circle with centre N and radius NO. 18 If A and B are points on the circumference of a circle :E, what is the image of the line AB under the inversion in :r? 19 If :r is a circle with centre O and / is a line which does not cut :r, let A be the foot of the perpendicular from O to /, and let B be any other point of/. Let A' be the inverse of A and B' be the inverse of B with respect to L. Show that B' lies on the circle on OA' as diameter. What is the inverse of/ with respect to :E?

Stereographic projection Questions 20--28 establish that the image of a circle under stereographic projection is either a circle or a straight line. The method is to ~onsider stereographic projection as part of a 3-dimensional inversion.

20 If :E is a sphere with centre O and radius a, and we define inversion in L as a mapping of 3-dimensional real space with the point oo added such that two points on a radius of :r, A and B are transposed when OA · OB = a2 and ·O is transposed with oo, use qn 17 to determine the image of a tangent plane to :r under an inversion in :r. 21 Using the diagram of qn 4.1 and the definition of qn 4.35, use the notion of inversion in a sphere to describe stereographic projection. 22 With the help of qns 18 and 19 determine the image of a plane (other than a tangent plane) under inversion in a sphere. 23 What is the image of a sphere under inversion in a sphere?

7 Inversions of the Mobius plane and stereographic projection

80

24 In 3-dimensional real space, we let I: denote the surface of a sphere with centre O and radius a, and we let P denote the foot of the perpendicular from O to a plane IT. Determine the nature of I: n IT (i) when OP > a, (ii) when OP = a, (iii) when OP < a. 25 Is every circle on a sphere the intersection of a plane with that sphere?

26 By considering stereographic projection as the restriction of a 3-dimensional inversion, determine the image of a circle under stereographic projection.

27 Using the notation of qn 4.35, determine the image of a circle through N under stereographic projection. 28 What is the image of the set of all circles on a sphere under stereographic projection? Having established that under stereographic projection circles are projected onto circles or straight lines, we must establish one more fundamental property of stereographic projection, namely that it preserves angles of intersection. Access to a transparent or wire sphere will be helpful. 29 If two straight lines in the plane intersect in the point P', and P' is the stereographic projection of the point P, what can you say about the curves on the sphere which are projected onto the two given straight lines?

If two smooth curves intersect at a point P, then the angle between the curves is defined to be the angle between the tangents to the curves at P.

Stereographic projection

81

30 The first step in showing that sttereographic projection preserves angles is to equate the angle between two lines in the image plane with the angle between two lines through the point of projection. Let NS be the diameter of a sphere L and let TTN and TT 5 be the tangent planes to L at N and S respectively, so that L is like a ball between the two boards II.v and TT 5 • Are the planes TTN and ·ns parallel? Let / be a line in the plane TT 5 , and let the plane containing the point N and the line l cut the plane TTN in n. Why must the lines l and n be parallel? If / is the stereographic projection of the circle C on L, why must n be a tangent to C?

N

n

C

31 If /1 and 12 are lines in the plane TT 5 and we define two corresponding lines n 1 and n 2 in the plane TTN as in qn 30, why is the angle between /1 and /2 equal to the angle between n1 and n2? If the lines / 1 and 12 are the stereographic projections of the circles C 1 and C 2, why is the angle between /1 and /2 equal to the angle of intersection of C 1 and C2 ? What point on the sphere is projected to the point of intersection of /1 and lz?

From qn 31, the preservation of angles under stereographic projection is now immediate. The next step is to apply these two results, the preservation of circles and the preservation of angles under stereographic projection to the image of orthogonal circles under stereographic projection and hence to determine the transformation of the sphere which corresponds to the transformation of inversion jn the plane. 32 If two circles on a sphere cut orthogonally, what can be said about

their images under stereographic projection? 33 If a circle S and two points A and B on the surface of a sphere have images S', A' and B' respectively under stereographic projection, and A' and B' are inverse points with respect to S', what may be said about circles in the plane through A' and B' (see qn 2)? What may be deduced about circles on the sphere through A and B? 34 It is not easy to visualise orthogonal circles on the surface of a sphere

7 Inversions of the Mobius plane and stereographic projection

82

until we have identified, for each circle on the sphere, a unique tangent cone. If Pis a point on a circle S which lies on the surface of a sphere L, how many lines are there, through P, which are tangents to the sphere L and perpendicular to S at P? 35 If Lis a sphere with centre O and radius a, and Sis a circle lying on the surface of L with centre B and radius b, identify a point Von the line OB such that every tangent from V to L intersects S. The point V is called the vertex of the tangent cone to S. 36 If C and S are circles on the surface of the sphere L which cut orthogonally at the two points P and Q, what can be said about the tangents to the circle Cat the points P and Q? What other

point must lie in the plane of the circle C? 37 If every circle on the surface of a sphere through the two points A and B of the sphere meets the circle Son the sphere orthogonally, how must A and B relate to the vertex of the tangent cone to S? ' 38 If the circle S' in the plane is the stereographic projection of the circle S on the sphere, what transformation of the surface of the sphere would correspond to inversion in S' on the plane? 39 How must the answer to qn 38 be adapted if the circle S is a great

circle, that is, an equator of the sphere? 40 Use the 3-dimensional analogue of qn 3.21 to say what you can about

the transformation in the plane which corresponds to a rotation of the sphere about a diameter. Riemann sphere We now suppose that the sphere L has unit diameter, ON, and we let the tangent plane at O be a Gauss plane with O at 0, and we label the points of L - { N} with the set C, using their images under stereographic projection from N as labels. We further label N with oo, and refer to the sphere so labelled as the Riemann sphere. 41 Verify that a reflection of the Riemann sphere in the diametral plane perpendicular to ON is z ....-. I/z. 42 Why must every rotation of the Riemann sphere be a Mobius trans-

formation? 43 How are the points z,

z,

-z,

-z,

1/z, 1/z, -1/z, -1/z located on

the Riemann sphere? 44 What is the point diametrically opposite z on the Riemann sphere?

45 If a is a rotation of the Riemann sphere, why must ( -1/z)a

= -

I/(za):

Riemann sphere ~

If

ex:z

az 1--+

+

CZ+

83

b d

is a rotation of the Riemann sphere, by considering the images of 0 and oo, prove that ab + cd = 0, and deduce that

~ a~ =

cc ( be

aa

+

cc

bb

+

dJ.

and that bb ( be

~ a~ =

By considering the images of 1 and - 1, prove that aa + cc = bb + dJ, and deduce that bb = cc and aa = dJ. Since !al = ldl, we may put d = aei 8 • Now deduce that - b = cei8 so that

a :z

az

1--+ _

+b + aeie •

[jeiB z

By multiplying top and bottom by a suitable number show that every rotation of the Riemann sphere has the form

az ZI--+

-

+

b

5z + a. .

i7 Let a denote the transformation of the Riemann sphere az + b ZI--+

-

5z + a.

(i) Prove that the image of a pair of diametrically opposed points on the Riemann sphere under ex is a pair of diametrically opposed points, that is, ex preserves diameters. (ii) Use qn 4.25 and qn 28 (the preservation of circles) to deduce that the image of a great circle under ex is a great circle. (iii) Use qn 4.44 to prove that a fixes a pair of diametrically opposed points, C and D, say. (iv) Must the image of a circle through C and D, under ex, be a circle through C and D? (v) Identify a unique great circle orthogonal to all the circles through C and D. Use qns 4.42 and 31 to show that this circle is fixed by ex. (vi) What would be the image of the circle of (v) under a rotation of the sphere about the diameter CD? (vii) Use the fact that the angle between two great circles through C and D is equal to the angle between their images under ex to show that ex coincides with a rotation at C, D and on the circle of (v). (viii) Let P and Q be any two points on the Riemann sphere which are

7 Inversions of the Mobius plane and stereographic projection

84

not diametrically opposed and let S be the centre of the sphere. If AB is the diameter perpendicular to the plane SPQ, by considering the preservation of great circles and angles under et, compare the spherical triangles APQ and ArxPaQa, and deduce that the great circle arc PQ is equal in length to the great circle arc PetQet so that et acts as an iso-metry on the Riemann sphere. 48 If we let a denote stereographic projection from .the sphere to the plane, exhibit a function which establishes an isomorphism between the circular transformations of the Mobius plane and the circle-preserving transformations of the sphere. Summary

Definition Two points A and B are inverse with respect to a circle qn 2 with centre O and radius a, if they lie on the same radius through O and OA · 0 B = a2 • Also O and oo are inverse points. · Theorem Two distinct points A and B are inverse with respect to qn 3 a circle 1: if and only if every circle through A and B cuts 1: orthogonally. Definition An inversion of the Mobius plane with respect to a qn 4 given circle transposes pairs of inverse points with respect to the given circle and also transposes the centre of the circle and oo. Theorem Every Mobius transformation is a product of an even qn 14 number of inversions. Theorem The group generated by the inversions maps the set of qn 16 circles and lines onto the set of circles and lines. Theorem Under stereographic projection the set of circles on the qn 28 sphere is mapped onto the set of circles and lines of the Mobius plane. Theorem The angle between two curves is preserved under stereoqn 31 graphic projection. Theorem A conicar projection on the sphere corresponds to an qn 38 inversion of the plane under -stereographic projection. Historical note

A. F. Mobius' study of the transformations generated by inversions (1852-56) included the complete enumeration of the circle-preserving transformations of the plane. The connection between transformations ·of the plane and transformations of the surface of the sphere was identified by F. Klein in 1875 and fully explored in his Lectures on the· Icosahedron (1884).

85

Answers to chapter 7

Answers to chapter 7 1 a2• 2 Yes, see qn 4.32. 3 Mobius transformations preserve circles (qn 4.25) and orthogonality (qn 4.42) so A!X and B!X are inverse points with respect to :E!X. If :E!X is a straight line, it is the perpendicular bisector of A!XB!X. 4 The points on the circumference of :E.

5 S is mapped onto itself. 6 L is mapped onto itself with centre and oo interchanged. 8 z

(b 2/a 2)z, an enlargement.

H

b is the composite of z H rz and z H e;8 z + b. z H rz is the product of two inversions from qn 8. z H ei8 z + b is the product of two reflections from qn 6.48.

9 If a

= re;8 , z H az +

10 Use qn 4.14. 11 Both have order 2. Both map circles and lines to circles and lines and preserve absolute values of angles of intersection. Inversions generate a subgroup of SM. Even products of inversions give the Mobius group. 12 a

= s, b = R2

-

ss, c =

I, d

= - s.

13 Let

az + b !X:ZH--- and CZ + d Calculate !Xp.

P:

z

H

Az Cz

+ B. +D

14 It is in the Mobius group.

15 z

az+b H

---

cz + d

with ad - be '# 0.

16 Yes. 17 NA

=

ON cos 8, NA'

=

ON sec 8. A

a, I: n TI= 0- (ii) When OP= a, I: n TI (iii) When OP < a, I: n TI = a circle centre P.

=

{one point}.

25 Yes, because a circle is a plane figure. 26 A circle on I: is the intersection of I: and a plane TI. The image of I: under inversion is the tangent plane; ihe image of TI from qn 22 is a sphere. Sphere and tangent plane intersect in a circle. 27 Plane section through N meets the tangent plane in a line.

28 The circles and lines of the plane. 29 They are both circles through N. 30 Yes, / is in TI 5 , n is in TIN but TI 5 II TIN so I and n do not meet. But they are, c0planar, so they must be parallel. C and n are coplanar and n meets C at N. If n met C at an additional point, this point would project onto I and n would intersect /. 31 /1 lln 1, /2

/ 2 lln 2• n1 is a tangent to C1, n2 is a tangent to C2• Angle between /1 and = angle between n 1 and r. 2 = angle between C 1 and C2• The point of

C 1 r, C2 distinct from N. 32 Images cut orthogonally. 33 Circles through A and B orthogonal to S.

34 Just one. 35

S

0

~ B

V

36 They lie in the plane of C and are perpendicular to S, so they pass through the vertex of the tangent cone to S. 37 Every plane through A and B cuts S orthogonally, and so contains the vertex of the tangent cone of S. So the vertex lies on AB.

38 Projection from the vertex of the tangent cone of S. 39 Tangent cone becomes tangent cylinder. Parallel projection. This amounts to

reflection in the plane of the great circle. 40 Rotation of sphere

= product of two reflections in diametral planes. Each of

1

Answers to chapter 7 these reflections corresponds to an inversion in the plane. A rotation of the sphere corresponds to a product of two inversions. 42 Use qns 13 and 40.

43 At the vertices of a cuboid. 44 - I/z.

45 Because diameters map to diameters. 48

r1.f---+OWT-I.

87

8 Equivalence relations

This chapter puts in formal terms every child's instinctive activity of sorting a collection of things into different kinds. We will say that things count as being of the same kind when they are related by an equivalence relation, or they are equivalent. When we have gathered together all things of the same kind we call the set an equivalence class. Concurrent reading: Green, chapter 2.

1 Two lines in a plane may be parallel, perpendicular or neither. If/ II m, does it follow that m II I? If l l. m, does it follow that m l. l? These results make both parallelism and perpendicularity symmetric relations. Is orthogonality of circles a symmetric relation? Is the divisibility of natural numbers a symmetric relation?

I m and m II n, does it follow that l II n? (We allow a line to be parallel to itself.) If l l. m and m l. n, does it follow that l l. n? These results make parallelism, but not perpendicularity, a transitive relation. Is orthogonality of circles a transitive relation? Is divisibility of natural numbers a transitive relation?

2 If I

3 If a set Pis partitioned into disjoint subsets A 1, A 2, A 3, ••• , where Aj n A1 = 0 unless i = j, we call the Aj, parts of P and the set of all

Ai a partition of P. Check the three following results. (i) If x E P, then x lies in the same part of P as x. (ii) If x, y E P and x lies in the same part of P as y, then y lies in the same part of P as x. (iii) If x, y, z E P, and x lies in the same part of P as y, and y lies in the same part of P as z, then x lies in the same part of P as z.

.,,..-,

8 Equivalence relations

89

4 If R is a relation on P which is reflexive (i.e. x R x for all x e P), symmetric (i.e. x R y implies y R x) and transitive (i.e. x R y and y R z imply x R z), and for each a e P we define Ra = {xix Ra, x E P}, prove that (i) R. is not empty, (ii) if Ra('\ Rb is not empty, then Ra s;; Rb at1d Rb s;; R,., so Ra (iii) the sets Ra for all a e P form a partition of P.

= R,,,

These sets are called the equivalence classes of P under the equivalence relation R. S We can exhibit a relation on a finite set by something that looks like

a multiplication table, by listing the elements of the set across the top and down one side, and then the entries in the table(✓ for yes, x for no) indicate whether the elements are related or not. Thus indicates that a R b. For each of the following relations on the set {a, b, c} determine whether it is reflexive, symmetric or transitive. If it is an equivalence relation, state the equivalence classes.

a b c

a b c

a

X X X

a

✓

✓

b

X

✓ ✓

b

✓

C

X

✓

C

✓

X X X X

✓

✓

a b c

a b c a

✓

✓

✓

a

b

X

✓ ✓

b

X X X ✓ X

C

X X

✓

C

X

✓

X ✓

Exhibit two other tables for the same set, each of which defines an equivalence relation, and state the equivalence classes in each case. 6 The most familiar equivalence relation is that of equality ( =) on a set.

What are the equivalence classes in this case?

-

7 A relation R. is defined on the set of integers Z by a R b when a - b is an even number. (i) Check that R is reflexive, symmetric and transitive (that is, R is an equivalence relation). (ii) Find the sets R 1, R 2, R 3 and R 4 according to the notation of qn 4. The two classes here are called residue classes modulo 2 and a R b · is usually written a = b (mod 2) in this case.

•

8 Equivalence relations

90

8 A relation R is defined on the set of complex numbers, IC, by z R w when lzl = lwl. · (i) Check that R is an equivalence relation. (ii) Illustrate the sets R 1 and $ 2 in a diagram of the Gauss plane. 9 In the group of isometries of the plane, we denote the subgroup fixing 0 (consisting of the rotations about O and the reflections with axes

thrQUgh 0) by M 0 , and we use this group to define a relation R on the points of the plane by A R B when Aµ = B for some ft e M0• (i) Prove that R is an equivalence relation. (ii) Find the equivalence class containing a point distant 1 unit from

0. (iii) Identify the equivalence classes, which, in this case, are called orbits of the group M0 •

10 If G is the group of symmetries of a regular pentagon, draw a regular

pentagon and mark the possible images of a vertex under the ten elements of the group. These form the orbit of a vertex. Mark another point P of the pentagon and the images of this point under the ten elements of G. These points form the orbit of P under G. 11 If G is any group of permutations on a set A, and we define a relation

RonAby a R b when alX = b for some IX e G, prove that R is an equivalence relation on A. The equivalence classes of A under Rare called the orbits of G. When G has only one orbit, G is said to be transitive (qn 3.54). 12 If IX e Sn, what is the ordinary name for the orbits av' for all a E IR?

2 A function Vm(F)

-> V,,(F) which preserves the structure of vector addition and the structure of scalar multiplication, in the sense that if v !---> v' and u !---> u' then v + u !---> v' + u' and av !---> av'

for all a E F, is called a linear transformation of Vm(F). What is the image of the zero vector under any linear transformation? 3 Show that tpe set of images under a linear transformation form a

subspace of the codomain. This subspace is called the image space of the transformation. 4 Show that the full subset of the domain with image the zero vector

under a linear transformation is a subspace of the domain. This subspace is called the kernel of the transformation.

12 Linear transformations

115

5 Under a linear' transformation show that the image of a I-dimensional

subspace of the domain is either a I -dimensional subspace or the zero vector. How would you describe this result geometrically if the domain and coc.omain was either Vi(R) or V3(1R)? 6 If a: Vi(F)

-+ Vi(F) is a linear transformation and a: (l, 0) 1-+ (a, b), what is the image of (x, 0) under a? If a: (0, I) 1-+ (c, d), what is the image of (0, y)? Use (x, 0) + (0, y) = (x, y) to deduce the image- of (x, y) under a. The conventional expression for this is

(X:(x, y)

1-+

(x, y) (;

!).

Check that every transformation of this form is a linear transformation. 7 If a is a linear transformation of Vi(IR) such that a:(x, y)

1-+

(x, y) ( _ 1 1 . ~)

use some squared paper to illustrate the images of points in 7L 2 • 8 If OABC is a parallelogram in 1112 what are the possible geometrical

configurations of the images of these four points under a 11 -it:ar transformation Vi(R) -+ Vi(IR)? With the help of qn 5 descibe the image of a parallelogram lattice under such a transformation. 9 If a: Vi(F) -+ Vi(F) is a linear transformation such that a:(l, 0) and a: (8, l) 1-+ b, prove that a:(x, y) xa

+ yb.

xa

1-+

+ yb is also written (x, y) (:)

.

Check that every transformation of this form is a line_ar transformation. 10 If a: Vj(F)

-+

Vi(F) is a linear transformation such that

a:(l, 0, 0)

1-+' (a,

a:(O, l, 0)

1-+

b),

(c, d), and

a:(0, 0, 1) H (p, q), find the image of (x, y, z) under a. We conventionally write .:(x, y, z) >-+ (x, y, z) 11 If a: Vi(F)

a:(l, 0)

1-+

-+

G:)-

Vj(F) is a linear transformation such that

(a, b, c), and

1-+

a

116

12 Linear transformations

cc:(0, 1)

r+

(p, q, r),

find the image of (x, y) under a:(x, y) r+ (x, y) (;

12 If cc: Vm(F)

--+

We conventionally write this

! ~)-

V.,(F) is a linear transformation such that under a

(1, 0, 0, ... , 0)

r+ (a 11 , a 12 ,

(0, 0, 0, ...

r+

1

ry,_

l)

a 13 , . .. , a1.),

(am 1, am2 , am 3 ,

••• ,

am.),

then we write al I

G12

a13

aln

G21

a22

G23

a2.

The rectangular array of a;,s is called a matrix, strictly an m x n matrix. and the definition above defines a product of a row vector (an m-tuple) and an m x n matrix. How could you describe the result of the product d' an m-tuple and an m x n matrix? 13 Must every transformation of the kind exhibited in qn 12 be a linear transformation? Summary

Definition A function a: v;.,(F) --+ V.,(F) is said to be a linear transqn 2 formation when (u + v)cc = ua + vcc and ),(ua) = (Au)o:, for all vectors u, v of the domain, and all scalars ). in F. Theor.:m The image space of a linear transformation is a subqn 3 space of the codomain. Theorem The kernel of a linear transformation (the subset mapqn 4 ped to the zero vector) is a subspace of the domain. Theorem Every linear transformation Vm(F) ~ V.,(F) is uniquely qn 12 defined by the images of the basis vectors ( 1, 0, 0, ... , 0), (0, I, 0, ... , 0), etc. of the domain, and can be expressed as v r+ vA, where the images of the basis vectors are the rows of the matrix A.

Historical note

117

Historical note

Linear transformations were first studied as 'substitutions' and applied to quadratic forms in number theory during the eighteenth century. The substitution of (ax + by, ex + dy) for (x, y) transforms .x2 + y2 to (ax + by) 2 + (ex + dy) 2 and interest focussed on those substitutions for which the two forms took on precisely the same set of integer values. The matrix notation was developed by A. Cayley (1858) as he applied substitutions to homogeneous polynomial expressions in n variables (which he called quantics) of which the quadratic forms of Lagrange and Gauss were a special case.

12 Linear transformations

118

Answers to chapter 12 1 (i), (ii), (v), (vi), (vii), (viii). 2 If O

f-+

O', then v

=

v

+

0

f-+

v'

+

O'

=

v', so O' is the zero vector.

f-+ u', then v + u f-+ v' + u' so the set of images is closed under vector addition. Also av f-+ av', so the set of images is closed under scalar multiplication and the images form a subspace.

3 If v f-+ v' and u

4 v f-+ 0 and u f-+ 0

=>

v

+

u f-+ 0

+ 0 = 0 and av f-+ aO = 0.

f-+ u' then Sp(u)-> Sp(u'). Ifu' = 0 then Sp(u') = {O}. A line through the origin is either mapped to a line through the origin or to the origin itself.

5 Ifu

6 cx:(x, 0)

f-+

(xa, xb). cx:(O, y)

f-+ (

ye, yd), so cx:(x, y)

f-+

(xa

+ ye,

8 Either a parallelogram or four collinear points.

Either another parallelogram lattice or a set of collinear points. 12 An n-tuple.

xb

+

yd).

13 The general linear group GL(2, F)

This chapter is about those linear transformations which are permutations of a 2-dimensional vector space, sometimes Vi(F) and sometimes VilR). The work with Vi(F) develops the general theory. When VllR) is under discussion, diagrams will represent our ideas faithfully and the full range of geometrical language is at our disposal. Concurrent reading: Birkhoff and MacLane, chapter 8, sections 3, 4 and 6; Burn (1977). Singular and nonsingular transformations

The first five questions of this chapter determine when the linear transformation

belongs to the group S,,2(FJ· When this happens, such transformations are said to be nonsingular. The rest of the chapter is only concerned with the set of nonsingular linear transformations which forms the general linear group GL(2, F). 1 If (0, 0), (a, b) and (c, d) are noncollinear points of IR 2 , find the area of the parallelogram with vertices at these three points and (a + c, b + d). 2 If (x,

y)

H

(x,

y)(: !) is a linear transformation of Vi{IR), what is

the area of the image of the unit square (0, 0), (1, 0), (0, l) and (1, l) under this transformation? 3 If o:: (x, y)

H

(x,

y)(; !) is a linear transformation of Vi(F),

13 The general linear group GL(2, F)

120

show that the image space of r:,. is Sp((a, b), (e, d)). Deduce from qn 11.25 that if ad - be = 0 then r:,. is not a surjection onto Vz(F). By considering the images of the vectors (0, 0), (d, -b) and (- e, a) show that if ad - be = 0, then r:,. is not one-one. The number ad - be is called the determinant of the matrix. When the determinant of a matrix is 0, the matrix and its related linear transformation are said to be singular. 4 If ad - be =I 0, so that the linear transformation r:,. of qn 3 is nonsingular, use qn 11.24 to show that r:,. is a surjection Vi_(F) --+ Vi_(F). If it was the case that (x 1 , y 1 )r:,. = (x 2 , y 2 )r:t., show that (x 1 - x 2 )(a, b) = (y 2 - Y1)(e, d), and deduce that r:,. is an injection. 5 Use qns 3 and 4 to formulate a theorem about surjections and injections which are linear transformations Vi_(F) --+ Vi_(F).

The group of nonsingular transformations

In qns 6--14, matrix multiplication is defined so as to be compatible with the composition of linear transformations and the group of nonsingular linear transformations is identified. The isomorphism between matrices and linear transformations is so well used in ordinary mathematical Ian:. uage that the two are often, and quite helpfully, confused. 6 If we

v

.. "e

!) by A and the matrix(~

the matrix(:

;) by P

and we L · rt. denote the linear transformation (x, y) 1-+ (x, y)A and let f3 denote ;,,e linear transformation (x, y) 1-+ (x, y)P, then the composite linear transformation r:1./3 is given by (x, y) 1-+ [(x, y)A]P. Find the matrix corresponding to the linear transformation r:1./3. This matrix is defined to be the matrix product AP. 7 If/ is the matrix (~

formation (x, y) 8. If A

= (;

H

~), how could you describe the linear trans(x, y)fl

!) is a nonsingular matrix with determinant A, find

! -!;~}

the product of A with ( _ ;~ transformation (x, y) transformation.

H

Deduce that the linear

(x, y)A has an inverse which is a linear

9 If A and B are 2 x 2 matrices over the same field prove that the determinant of AB = (det of A)· (det of B).

10 If A is a 2 x 2 matrix, and if A has an inverse with respect to the

identity matrix(~

~) , prove that the determinant of A =I 0.

Shears

121

11 Use the definition of matrix multiplication and the associativity of

composible functions (qn 1.32) to prove that products of 2 x 2 matrices are associative.

12 Prove that the set of 2 x 2 nonsingular matrices over a given field forms a group under matrix multiplication. 13 If Vis a vector space, and a: V -+ Vis a linear transformation which is also a bijection, then a is called a nonsingular linear transformation of V. Must the nonsingular linear transformations of V form · a s9bgroup of Sv?

14 Prove that the group oflinear bijections J--;(F) -+ J--;(F) is isomorphic to the group of 2 x 2 nonsingular matrices over F. Either one of these groups is denoted by GL(2, F) and called the general linear group of Vi(F). The centre of the general linear group

In questions 15-19, the elements of GL(2, F) which commute with all the rest of the group are identified and are shown to form a subgroup.

15 A matrix of the form ( ~ ~) in GL(2, F) is sometimes denoted by al and called a scalar matrix. Under the isomorphism of qn 14 what linear transformations of Vi(IR) correspond to scalar matrices? 16 Let

A=(; !),1= (~

!)andK=

G ~}If Al= JA,provethat

= 0. If AK = KA, prove that a = d and b = 0. Deduce that if A is in GL(2, F) and A commutes with every matrix in GL(2, F) then A is a scalar matrix. Check that scalar matrices always commute with all other matrices of GL(2, F).

a = d and c

17 Does the subset of scalar matrices form a subgroup of GL(2, F)? 18 In any group G, prove that the subset C = {c Icg forms a subgroup of G. This subgroup is called the centre of G.

=

gc for all g E G}

19 Identify the centres of D 2 , D 3 and D 4 • Shears

In questions 20-24, some shears of Vi(IR) are identified, and a general definition of a shear in v.(F) is given. One example is given of the isomorphism of the additive group of the field with the group of shears with a given axis.

13 The general linear group GL(2, F)

122

20 Under the isomorphism of qn 14, illustrate the linear transformation

G ~). This transforma-

of Vi(!R) which corresponds to the matrix

tion is called a shear on the x-axis. A shear of Vi(IR) has a line of fixed points and every point off this line is moved parallel to this line.

21 Find the line of fixed points of the shear (x, y)

~ (x,

y)a 7).

22 Find the line of fixed points of each of the shears (i) (x, y) (ii) (x, y) (iii) (x, y)

~ (x, y) ~

(x, y)

~ (x, y)

+ y(a, 0), where a + x(0, b), where b + (ex + dy)(a, b),

-:/=-

0,

-:/=-

0,

where (a, b) is not the zero vector and ca

+

db

= 0.

23 If v ~ v + f(v)a, where /(v) is a scalar and a a constant vector, is a linear transformation of V,,(F), prove that the function f is a linear transformation V,,(F) ...... F. When /(a) = 0, the form given here is the general definition of a shear (or transvection) of a general vector space. Show that each of the shears in qn 22 may be expressed in the form given here by finding the function f and the constant vector a in each case.

24 Show that the matrices of the type GL(2, F) isomorphic to (F,

(! ~) form a subgroup of

+ ).

More subgroups of GL(2, F) 25 Show that the nonsingular matrices of the type(~

~) form a

subgroup of GL(2, F) isomorphic to the multiplicative group F-{0}. What are the fixed points and fixed lines of the corresponding group of linear transformations of Vi(IR)?

26 How would your description of the fixed points and fixed lines in qn 25 be affected if the real numbers were replaced by an arbitrary field and you were confined to the language of vector spaces and groups (e.g. if lines through the origin always had to be referred to . as I-dimensional subspaces)? 27 Show that ~he nonsingular matrices of the type ( ~

~) form a

group. What are the fixed points and fixed lines of the corresponding group of linear transformations of Vi(IR)'? Matrices of this type are called diagonal matrices.

Summary 28

123

If G = (IR-{0}, ·) show that the group of diagonal matrices in GL(2, IR) is isomorphic to the direct product G x G.

29 Modify your description of the fixed points and fixed lines in qn 27 so that it would be valid over any field. 30 Show that the matrices of the type(:

~), where a, d "I= 0, form

a group under matrix multiplication. What are the fixed lines and fixed families of lines of the corresponding group of linear transformations of Vi(IR)? Matrices of this type are called (lower) triangular matrices.

31 How would your description of the fixed lines in qn 30 be affected if the real numbers were replaced by an arbitrary field? 32 In G L(2, Z 3 ) find the orders of the group of scalar matrices, the group of shears on the x-axis, the group of diagonal matrices and the group of triangular matrices. Determine the order of the full group by counting the possible pairs of images of (I, 0) and (0, I). There are nine vectors in Vi(Z 3 ). How many of these may be images for (l, 0)? After determining the image for ( l, 0), how many may be images for (0, I)?

The orthogonal group In chapter 21 we will link the orthogonal group with isometries in GL(2, IR) and GL(3, IR), but here we simply offer a method of defining a subgroup of GL(2, F) which is quite different from any of those we have used so far. 33

If A = (: !), we define the transpose of A, AT = (:

~). Prove

that det A = det AT and for any two 2 x 2 matrices A and B that (ABf = BTAT. 34 Prove that the 2 x 2 matrices A such that A· AT= /form a subgroup of GL(2, F). This subgroup is known as the orthogonal group.

Prove that if A ae 35

+

bd

=

ab

!) then a

= (: + ed =

2

+

c2 + d 2 = ± 1.

b2 =

0 and ad - be

= 1,

If(; !) is in the orthogonal subgroup of GL(2, IR) and if = cos 0, d = cos ¢, prove that 0 ±

either x 1 = x 2 and y 1 = y 2 or Sp((a, b), (e, d)) is I-dimensional. 5 If ad - be

If ad - be 6 (ap + br ep + dr

i= 0 then a: (x,

!) is a bijection.

y)..,.. (x, y) (:

= 0 then a is neither a surjection nor an injection. aq eq

+

+

bs) . ds

7 The identity :nap on Vi(F).

8 Product

= /.

9 From qn 6, (ap

= = 10 A· A- 1

+

br)(eq

apds

+

ds) - (ep

+

dr)(aq

+

bs)

+ brcq - cpbs - draq

(ad - be)(ps - rq).

= I implies det

A· det A- 1

= 1 so det

A i= 0.

12 Closure follows from qn 9. Identity from qn 7. Inverses from qn 8. 13 Composite of two bijections is a bijection. Composite of two linear trans-

formations is linear. Inverse of linear bijection is linear. 14 The function mapping [(x, y)..,.. (x, y)A] to A is an isomorphism.

15 Enlargement~ with centre (0, 0). 17 Yes. 19 D 2 is all centre. Centre of D3 is {e}. Centre of D 4

= {e, a2} see qn 9.31.

20 Each point of the x-axis is fixed. Each line parallel to the x-axis is mapped

onto itself. 21 The y-axis. 22 (i) y = 0. (ii) x = 0. (iii) Sp((a, b)).

23 u + /(u)a + v + /(v)a = u + v + /(u + v)a => /(u + v) = /(u) + /(v), and k(v + /(v)a) = kv + /(kv)a => k/(v) = /(kv). (i) a = (a, 0). f(x, y) = y. (ii) a = (0, b), f(x, y) = x. (iii) a = (a, b), f(x, y)

=

ex

+

dy . .

Answers to chapter 13 24

(! ~)

H

127

a gives the isomorphism.

25 The y-axis is fixed pointwise. Lines parallel to the x-axis are fixed but not pointwise. 26 Sp(0, l) is fixed pointwise. Cosets of Sp(l, 0) are also fixed but not pointwise.

27 Only the x- and y-axis. 28 (~

~)

H

(a, b).

29 Sp(l, 0) and Sp(0, l) are fixed but not pointwise.

30 The x-axis is fixed but not pointwise. Every line parallel to the x-axis is mapped to a parallel line. 31 Refer to Sp(I, 0) and its cosets.

32 !Scalar matrices! = 2, !Shears on x-axisl = 3, !Diagonal matrices! = 4, !Triangular matrices! = 12. Eight possible images for (1, 0), then six possible for (0, I) so IGL(2, Z 3)1 = 48.

= B · BT = I then AB· (AB) T = ABBT AT = I, so subset closed. AT = A - 1 and (AT) T = A, so inverse is orthogonal. det (A • AT) = det I and det A = det AT=> det A = ± 1. Other equations follow from A • AT = I or AT • A = l

34 If A • AT

35 b2 = sin 2 0, e2 = sin 2 q>, ae + bd be

(

=

cos 2 0 - l

=

-sin 2 0. 0 =

cos 0 sin 0) d (cos 0 - sin 0 cos 0 an sin 0

= 0 gives sin (4> ± 0) = 0. 0 = ± 4> => ±4> + n => be = 1 - cos 2 0 = sin2 0. sin 0) - cos 0 ·

14 The vector space V3 (F)

Angles, distances, areas and volumes in VllR) can be calculated with the help of two operations on vectors, the scalar and vector products. In combination, these enable us to define the determinant of a 3 x 3 matrix and to establish that 3 x 3 determinants have similar properties to 2 x 2 determinants. Throughout this chapter, as in preceding chapters, we will think of vectors as points. Concurrent reading: Birkhoff and MacLane, chapter 7, section 9; Maxwell chapter 13.

Scalar products 1 If a = (a 1 , a 2, a 3) and b formula to prove that

= (b 1, b2, b3) are points of iR 3 use the cosine

+ a3b3 J(a~ + a~ a~) (bf + b~ + bi) Assume that the distance Oa = ✓

b x v and v f---> a · v

=

=

F.

v • a.

10 Use qn 5. If three leading edges are coplanar, the volume of the parallelepiped is 0.

11 All 0. 12 a·(b x c) = a 1(b 2 c 3 - b 3 c2 ) + a 2(b 3 c 1 - b 1c 3 ) + a 3 (b 1c 2 - b2 c 1). Even permutations have a positive coefficient, odd permutations have a negative coefficient.

13 a·b x c = b·c x a= c·a x b -c · b x a.

=

-a·c x b

14 (a· b x c) I. -I

~),

-1

-¼ ½

-¼

=

-b·c x a=

Answers to chapter 14

I ( 6 3 (iii) -4 6 16 2 I

-1) -2 5

135

.

16 If a · b x c #- 0 then A(l/(a · b x c))B

= /.

17 Compare details with expansion in qn 12. If Bis the right inverse of AT, then AT B = 1 so (AT Bf = IT and BT A = I. 22 If AA - i = I then det A • det A - i = I so det A #- 0. 23 Use qns 1.24 and 1.28.

24 Argue as in qns 13.13and 13.14. 25 (I, 0, 0)a = a, (0, I, 0)a = b, (0, 0, l)a = c, so (x, y, z)a = xa + yb + zc. det A = 0 => a in kernel. From qn 7, band c are in the kernel. The kernel is the whole space only if b x c = 0, but this only occurs if b E Sp(c) or c E Sp(b) from qn 6, and then we have Sp(a, b, c) = Sp(a, b) or Sp(a, c). 26 A singular => Ar singular. If AT has rows a, b, c then B constructed as in qn

14 makes AT B = zero matrix. BT A = (A 7 Bl = zero matrix. so the rows of Br, i.e. b x c, c x a and a x c are mapped to 0. At least one of these is nonzero unless a, b, c are in a I-dimensional subspace. If thest: lie in a I-dimensional subspace then any vector perpendicular to it is in the kernel.

27 Enough to guarantee commutativity with

and their transposes. 28 pa

+

qb

+

29 I, 4. 8, 168

re = 0. Every point on px

=

7 · 6 · 4.

+

qy

+

rz = 0 is fixed.

15 Eigenvectors and eigenvalues

The actual matrix for a linear transformation often hides the geometrical character of the transformation. For many linear transformations a geometrical view may emerge from finding those vectors which are transformed as if by an enlargement with centre the origin. Thes~ vectors are called the eigenvectors of the transformation and their related scale factors are called their eigenvalues.

1 Under the linear transformation of Vi(IR) given by a: (x, y) r-+ (x,

.v)(~ _~),

which represents a reflection on the x-axis, determine whether 0, v and va are collinear points when (i) v = (1, 0), (ii) v = (0, l) and (iii)v = (l, 1). 2 Under the linear transformation of Vi(lli) given by a: (x, y) r-+ (x,

y)G ~),

which represents a shear on the x-axis, determine whether 0, v and va are collinear points when (i) v = (I, 0), (ii) v = (0, 1) and (iii} V = (I, I). 3 Under the linear transformation of Vi(IR) given by a: (x, y) r-+ (x,

y)(~ ~),

which is known as a two-way stretch, determine whether 0, v and va are collinear points when (i) v = (I, 0), (ii) v = ( 5, 0), (iii) v = (0, l ), (iv) v = (0, 4). (v) v = ( l, l ). 4 If 0, v and va are collinear, how may va be expressed in terms of v?

Characteristic equation

137

5 If a: Vn(F) ➔ V.(F} is a ltnear transformation, then a vector v i= 0 such that va = },v for some scalar ). is called an eigenvector of a with eigenvalue }.. What are the possible eigenvalues when a is a rotation of Vi(IR) about the origin? 6 If a linear transformation is an isometry, what real eigenvalues may

it have? 7 Illustrate the image of a square lattice in Vz(IR) under the transformation (x, y) .-. (x,

y)G D

and then identify the images of the points (2, 1) and (I, - 1). Are these eigenvectors of the transformation and if so, what are their eigenvalues?

8 If o: is a linear transformation of Vn(F) to itself, explain why every nonzero vector in the kernel of a. is an eigenvector of ex. · 9 If ( 1, 0) is an e1genvector of

a.: (x,

y) .-. (x, y)(: !) what can be said about. b?

If (0, 1) is an eigenvector of :x, what can be said about c? If (1, 0), (0, 1) and ( 1, l) are eigenvectors of o: what can be said about a, b, c, d? How would you then describe the transformation r1. geometrically? If a = d and b = c = 0, what are the eigenvectors of a.?

a, does it follow that 2v, 3v and in fact kv are eigenvectors of o: for every scalar k?

10 If v is an eigenvector of a linear transformation

11 If u and v are eigenvectors of a linear transformation oc and both u and v have the same eigenvalue )., what conclusion may be drawn about the vectors of Sp(u, ,·)?

Characteristic equation 12 If u is an eigenvector of the linear transformation V ;-.

v(; ;),

and u has eigenvalue ),, what can be said about u in relation to the linear transformation

v.-. v(a : ).

d

~

;_)?

15 Eigenvectors and 'eigenvalues

138

Deduce from qns 13.3, 13.4 and 13.5 that the matrix •

(a - 2 C

b

is singular. 13 The equation det (a.~ 2

d

~

equation of the transformation (a

C

.h).

2) = 0 is called the ch~racteristic

v v(: !) and of the matrix H

d

By solving the characteristic equations of the transformations

y)(; ;), (x, y)G 1), (x, y)( _ 1 i}

(i) (x, y)

H

(ii) (x, y)

H

(iii) (x, y)

H

(iv) (x, y)

H

(x, y)( _

(v) (x, y)

H

II (x, y)\ 2

(x,

i ~), 2)

4 ,

. (x, y) H (x, y) (cos0 (v1) . 0 sm

0)

sin -cos 0 '

determine the possible eigenvalues of these transformations. 14 From the equation (x, y) (;

D=

(4x, 4y), determine the

eigenvectors of qn 13(i) with eigenvalue 4. Similarly, determine the eigenvectors of this transformation with eigenvalue 1. 15 Find the eigenvectors of the transformation qn 13(iii).

16 Find the eigenvectors of the transformation (x, y)

H

(x,

y)(~ ~),

assuming a ¥- b. 17 Let A denote the 3 x 3 matrix

Use an argument similar to that of qn 12 to show that if the transformation v H vA has an eigenvector with eigenvalue 2, then the m~trlv

)

d - A

Similar matrices

139

is singular. This matrix is often denoted by A - ).[ and the equation det (A - )J) = 0 is called the characteristic equation both of the transformation v f-+ vA and of the matrix A. 18 By first solving its characteristic equation, fir;id the eigenvalues and eigenvectors of the transformation v f-+ vA, where A

2

1

3

I

= ( -2 1

Similar matrices 19 Let A be an n x n matrix, and let u be an eigenvector of the linear transformation v f-+ vA, with eigenvalue A., let Ube a nonsingular n x n matrix and let w = uu- 1• Find the images of w under the transformations V f-+ vU, V f-+ vUA and V f-+ vUAu- 1 and deduce that w is an eigenvector ofv f-+ vUAu- 1 with eigenvalue A.. Illustrate this result on a drawing of Sp(w) and Sp(u). 20 Let v f-+ vA be a linear transformation of Vi(F) with eigenvectors u 1 and u2 with eigenvalues A. 1 and A.2 respectively. Suppose further that the matrix U with rows u1 and u2 is nonsingular. Find the images of (1, 0) and (0, 1) under the transformation V f-+

vUAu- 1,

and deduce that UAU- 1

J}

= (~1

21 Use qn 15 to find a matrix U such that

u( _1 ;)u-

1

= (~ ~).

Check your result.

22 Find a matrix U and a diagonal matrix D such that

u( _--;~ 1~)u-

1

=

D.

Show that there are two distinct possible diagonal matrices D for appropriate choices of U.

23 Let v f-+ vA be a linear transformation of VJ(F) for which ui, u2 and u3 are eigenvectors with eigenvalues A- 1, A. 2 and A. 3 respectively.

15 Eigenvectors and eigenvalues

140

Suppose further that the matrix U with rows u1 , u2 and u3 is nonsingular. Determine the matrix UAu- 1• 24 If v H vA is a linear transformation of Vi(F) for which u 1, u2 and u3 are eigenvectors with eigenvalues ), 1, ). 2 and ). 3 respectively, find a matrix U such that ).I

UA =

(

Q Q)

). 2 0 U 0 ).3 whether or not-u is nonsingular. 0 0

25 Js there more than one matrix U which satisfies the conditions of qn 24?

26 The matrices A and B are said to be similar if for some matrix M, B

= M' 1AM. Show that similarity is an equivalence relation on the set of

n x n matrices over a given field. Use qn 19 to show that similar matrices have the same eigenvalues.

27

If(~: !;) and (;;

;;) arc similar matrices, by considering their

characteristic equations, prove that a 1 + d 1 = a 2 + di and a 1 d1 - b1 c 1 = a2 d2 - b2 c2 • Also give a direct proof that similar matrices have the same determinant. Change of basis

28 For which pairs of vectors u, v docs Sp(u, v) (i) u ;:: (I, 0), ,. (ii)

U

(iii)u (iv)

U

= (), )),

V

= =

V

= =

=

Vi(F)?

(0, I),

(0, 0),

= (-2,4), = (), 3).

(1,-2),v (I, 2),

When Sp(u, v) = Vi(F). the vectors u and v arc said to form a basis for V2(F).

29 If a is a linear transformation of V.(F) with eigenvectors u and v with eigenvalues ). and µ. respectively, use qn 10 to show that if v E Sp(u) or u e Sp(v), then ,1, = µ.

30 If a is a linear transformation of Vi(F) with eigenvectors u and v with eigenvalues ). and µ, respectively, use qn 11.25 to show that if

i. # µ, then the matrix(:) is nonsingu1ar and deduce that u and v form a basis for V2(F).

Shears

141

31 If the linear transformation v - vA of ViF) has two distinct eigenvalues, deduce from qn 20 that A is similar to a diagonal matrix.

32 ff Y. is a linear transformation of V,,(F) with eigenvectors u, v and w with eigenvalues i., µ and I', respectively, and w E Sp(u, f), use qn 11.28 to show that either v = i., 1· = ;1 or ;_ = µ. 33 If a 2 x 2 matrix A has just one eigenvalue i.. is it possible that there exists a basis of V2 (F) consisting of eigenvectors of \' - v A? Is there necessarily a basis of ViF) consisting of eigenvectors of

v - vA? 34 If Sp(u, w) = V2 (F), then every vector v in V2(F) has the form xu + yw, and for a given v, the scalars x and y are unique from qn 11.28. When v = (2, 3), find x and y when (i) u

= =

(I, 0) and w

(0, !),

(I, 2) ar,d w

=

(I, 3),

(iii) u = (I, I) and w

=

(I. - l).

(ii) u

The scalars x and y are called the coordinates of v relative to the basis u, w. 35 If a: v - vA is a linear transformation of V2(F) = Sp(u, w), and uA = pu + qw and wA = ru + sw, deduce that

so that

If, furt4er, we denote the vector xu :.

=

I and det A

det AB- 1

=

I

=>

=

det S · det B

=

det B.

AB- 1 ~ S where det S

=

3 Number of cosets = number of distinct values of determinant !SL(2, Z1 )1 = IGL(2, Z3 )l/2 = 24.

4 4. 24 • 20/4

=

5 det A= det B

1.

= 2.

120. =>

det A- 1 B =I.Yes.

6 Left and right cosets are subsets with same determinant. Yes.

1 F - {O}. 8 Multiplication. 9 If QC: z .- az + b, y: z >-- cz + d, QC)': z .- caz QC>-- !al takes QC)'>-- jcal = !al· 1£·j. Yes.

+ cb +

d. Now the function

11 Definition of homomorphism gives closure. Identity and inverses from qn I 0.

12 Yes, of the group of vectors under vector addition. 13 SL(2, F). 14 The subgroup of direct isometries.

15 g 1QC = eQC, g 2QC = eQC 16 (k 1K)QC

=

=>

K1QC • g 2 QC = eQC ·ea=> (g 1g 2 )QC

k 1rt•gQC = KQC.(gk 2 )a: = gQC·k 2 rt

=

17 Kirt = Ki:X => (g,rt)(g 2 a:)- 1 =ea=> (g 1gi 1 )rt (K1 1K2) a: = ea: => Ki 'Ki e kernel.

= eQC, etc.

gQC.

=

ea:=> g 1gi 1 E kernel. Also

18 Ki, Ki in same right coset - g 1gi I e kernel - g 1a: = Kia: - g1 1g 2 e kernel - Ka, Ki in same left coset. Every left coset of kernel is a right coset. 19 Each coset is a singleton and the homomorphism is an injection (one---0ne). The homomorphism is then called an isomorphism.

20 Definition gives closure. Identity is N. Na- 1 is inverse of Na. 21 {e, a 2 }

{a, {b, ba 2 } {ba, ba 3 } a3 }

{a, a3 } {e, a2} {ba, ba3 } {b, ba2 }

{b, ba2 } {ba, ba3 }

{ba, ba3 }

{e, a2}

{a,

{a,

{e, a 2 }

a3 }

{ b, ba 2 } a3 }

22 Follows directly from qn 9.32. 23 (i) and (ii) g 1rt = g 2:x - (g 1g 2 1):x = e:x - g,g:; 1 EK - Kg 1 Obvious. (iv) g 1:x • g 2:x = (g 1g 2)QC >-- Kg 1g 2 = Kg,· KK2•

= Kg2• (iii)

154

16 Homomorphisms

24 The cosets of N are N and G - N which must be both left and right. Quotient

group~ C2 • 25 (i) An, (ii) Cn, (iii) direct similarities, (iv) direct isometries. 26 Yes. 27 g"+n

= g" • g".

Yes.

28 {O, ±3, ±6, ... }. 29 g 0

=

30 If N

gb.

=

(n) we need an element g

G of order n. Then map I

E

31 The elements of Zn are then cosets of (n) in Z. (n)

f--->

g.

+ I generates Z,..

= pq then (kn + p)(/n + q) is a multiple of n and there are nonzero elements of Zn whose product is zero.

32 If n

= 0 (mod p). But a prime number has no pr~per divisors. ax ay a(x - y) 0 x y (mod p). p - I different numbers for p - I different entries so all are there including I. Argument establishes closure and existence of inverses in (ZP - {O}, · ).

33 Not closed only if ax

= =

= = =

34 Group has order p - I. So for any element x, we have xk = I with k a factor of p - I. Thus xp-I = I (mod p). When xis a multiple of p obviously

both x and xP

=

0 (mod p) so xP

=

x

(mod p).

35 Z + p/q has order q. 36 If er:: z

23.

f--->

ewz

+

c, the mapping er:

f--->

ei 0 establishes the isomorphism using qn

17 Conjugacy

Every equivalence relation is a way of isolating a particular kind of sameness. The sameness which the equivalence relation of conjugacy identifies in the Euclidean group is the sameness between two reflections or the sameness between two half-turns. In the symmetric group, th.e sameness of conjugacy is the sameness of permutations with the same cycle structu~e. Much of the significance of this relation derives from the fact that the subgroups which are formed by the union of whole conjugacy classes are precisely the normal subgroups.

1 Express the permutations (1432) (12) (1234) and (132) (12) (123) as single cycles.

2 If ;1. is an element of S 4 , find the images of loc,

2;1., 3ix and 4ix under the permutation ix- 1(12)ix. Write the permutation ix- 1(12)ix in cycle form. The important point to remember is that .;1.- 1(12)ix is always a transposition.

3 Exhibit the image of the set

{(l), (123), (132), (23), (13), (12)}

under the mapping given by y

f-->

(l 432)y(l 234).

Describe the original set and the image set as subsets of S 4 • Fixed points of conjugate elements 4 If ;1. and /3 are permutations of a set containing the element m, prove that m;1. = m implies (m/3) 13- 1:1.{J = m/3 and conversely. 5 If cJ. is a rotation of the plane and /3 is any isometry of the plane, use . qn 4 to show that 13- 1:1./3 has one and only one fixed point. What sort of isometry is 13· 1 af]?

17 Conjugacy

156

6 lfo is a reflection of the plane and pis any isometry, what can you say about the fixed points of

p- 1(/.p?

What kind of isometry is

p- 1(/.p?

7 If r is a translation of the plane and f3 is any isometry, how many fixed points does p-- 1r/3 have? By expressing r as a product of two reflections, show that p- 1r/3 may also be expressed as a· product of two reflections, and deduce that p- 1r{3 ii; a translation.

8 If R is the group of rotations of the plane with centre O and r is the translation mapping O to I, show that every member of the set r- 1Rr.fixes I and is in fact a rotation. Illustrate. 9 Two elements x, y of a group G are said to be conjugate when there exists a g e G such that x = g- 1yg. · In the Euclidean group of the plane, what can you say about the isometries which are conjugate to (i) a reflection, (ii) a rotation, (iii) a translation. (iv) a glide-reflection? 10 Exhibit the mapping of the elements of the group D4 qn 6.40 given by.

= yx- 1 E c.. Thus C.x = C.y «> x·'ax = y- 1ay. SonumberofcosetsofC. = number of elements conjugate to a. Index of c. divides IGI.

Answers to chapter 17

165

38 A 3-cycle :J. has eight conjugates in S 4 so C, has order 3. Thus' C, = {I, :J., :J. 2}. These are all in A. so :J. has four conjugates in A 4 . {I}, {(12)(34), (13)(24), (14)(23)}. {(123), (214). (341), (432)}, {(132). (241), (314), (423)}.

39 The conjugacy class of one 3-cycle contains four 3-cycles. The inverses of these four belong to another conjugacy class.

40 If A 4 had a subgroup of order 6. it would have index 2 and thus be normal, contradicting qn 39. 1 2345' . . 41 ( I 2 3 4 5) (de) = ( ! c e so one 1s even and one is a b c d e ,a odd. So there is an even permutation in S5 such that I f--> a. 2 f-> b and 3 f--> c. This makes A 5 triply transitive. If :t. is the even permutatwn of these cr:- 1 (123):t. = (abc) so all 3-cycles are conjugates in A 5 •

b

42.

d)'

20. The order of C, in Ss is 6. C0 rn = {(!), (123), (132), (45). (123) (45), (132) (45)} in S 5 • C11 ~!, n A,. {(I), (I 23), (132)}. The conjugacy class of ( 123) in As contains 60 3 dements, i.e. all 3-cycles.

=

43 Even: (a) (h) (c) (d) (e); (a) (be) (de); (a) (h) (ale); (abcde). Odd: (a) (b) (c) (de); (a) (hcde); (ab) (ccle). 45 Let fJ = (ab) (cd). then :J.- 1 fJ:J. is closed. per:· 1 {3:J. = (cde).

E

:V since ,Vis nonnal and fJ:J.- 1 /h EN since !\/

46 Argue as in qn 45. (bed). 47 There are elements of three· cycle types in As distinct from the identity. The last three questions have dealt with the three types, and show that any one in a normal subgroup N of A 5 implies N = As, Cyclic groups of prime order are simple.

48 fJ:1.- 1{3:1. = (13) (24). (:J.. fi) contains elements of order 2, 3 and 5 so order of (:1., fJ) = 2 · 3 · 5 · k = 30k. But a subgroup of index 2 would be normal, so k = 2. 49 There are 24 5-cycles in S5 • so centraliser of er: has index 24 and so order 5. Thus C, = {I, :1., :J. 2, :1.3, :J. 4 }. These are all even permutations. So in A 5, C, has order 5 and so index 12. Conjugacy class of :1. in As contains 12 elements.

50 The line /er: is also perpendicular to a and is inclined at an angle 0 to /. The lines lfJ and /:1.fJ are both perpendicular to afJ and inclined to each other at an angle 0. Now A E a AfJ E afJ and AfJ(fJ- 1CJ.{3) = AfJ so afJ is 1 the line of fixed points o.f p- CJ.{3. Moreover, lfJ(P- 1CJ.{3) lcr:fJ, so fJ 1xfJ is a rotation with axis afJ through an angle 0.

=

51 The conjugate of a rotation is a rotation through the same angle. Thus

17 Conjugacy

166

rotations through ± 72° may be conjugate to each other, but not conjugate to rotations through ± 144°.

53 g- 1xg = g- 1yg

=> x = y so mapping is one-one. g- 1xyg so mapping is structure-preserving.

=

(g- 1xg) (g- 1yg)

54 Yes, by qn 16.11.

55 b = x- 1ax

=> Ix,.

=

=

(x- 1)ix · aix • xix

(xix)- 1 aix (xix) for any

automorphism ix.

56 A normal subgroup is a union of conjugacy classes. Yes. 57 Yes.

58 Yes. 59 If ix is an even permutation, (12)ix(l2) is even. (12)ix(l2) = (12)P(l2) => ix = {3. (l2)ixP(l2) = [(12)ix(l2)] [(12)P(l2)]. If ix = (123), (12)ix(l2) = (213) which is not in the same conjugacy class as (123) in A 4, so by qn 57 this is not an inner automorphism. 60 For an abelian group, every inner automorphism is the identity map. 61 Composite of x x

1---+

1---+

g:;1(g1 1xg1)K2

g 1 1xg 1 and x

=

1---+

g 2 1xg2 is

(g1g2)- 1x(g1g2)

so the set of inner automorphisms is closed. g 1 = e gives identity map. g2 = g 1 1 gives 'inverse map. 62 Automorphisms of G are elements of S6 by definition. If (xy)ix = xix · yix and (xy)p = xp • yp then (xy)IXP = [xix· yix]P = xixP • yixp so the set of automorphisms is closed. The identity map is an automorphism. (xix- 1 • yix- 1)1X = xy so xix- 1 • yix- 1 = (xy)ix- 1•

18 Linear fractional groups

If we examine the effect of the elements of the general linear group on I -dimensional subspaces we obtain a transitive permutation group which is a homomorphic image of the general linear group. This provides a setting in which the Mobius group and the projective group on a line may be seen as examples of an infinite class of triply transitive groups. In the development of this chapter, the I-dimensional subspaces of V2 (F) become the set of objects being permuted in a permutation group. This is the step by which we move from a vector space to a projective space, in this case from a 2-dimensional vector space to a projective line. The I-dimensional subspaces are the points of the projective line. More generally the I-dimensional subspaces of VnCF) form an (n - I )-dimensional projective space. Because different vectors, e.g. (x, y) and (kx, ky), lie in the same I-dimensional subspace, either pair of coordinates uniquely determines a projective point provided k #- 0. In such circumstances we sometimes talk of using homogeneous coordinates for the projective points. Permutations of I-dimensional subspaces 1 Let

:r:: (x, y) -

(x, y) (;

;) be a linear transformation of Vi(IR). Find

the images of the points (2, I), (4, 2), (6, 3) and generally (2k, k) under this transformation. Do the images of these points all lie in the same I-dimensional subspace? If the subspace {x(2, l)lx E IR} is mapped to the subspace {x(s, l)lx E IR} under a, what is the numbers? Is the subspace {x(3, l)lx E IR} mapped to a subspace {x(t, !)Ix E IR} under :r:? If so, find t. For a given real number m, must the subspace {x(m, !)Ix E !RI} be

18 Linear fractional groups

168

mapped to a subspace of V2 (1R) under 'Y.? Supposing that + 5 i:- 0. can you find a number m' in terms of m such that {x(m. !)Ix E IR} is mapped to {x(m'. l)lx E IR} under a? 2 Let

3m

/J: (x.

y)

1-+

2k (x, y) ( 4k

3k) Sk

be a linear transformation of VilR), and let k i:- 0. Find the image of the subspace {x(m. I )Ix E IR} under the transformation /J, and again. presuming that 3m + 5 i:- 0, find a number m' in terms of m such that {x(m. l)jx E IR} is mapped to {x(m', l)jx E IR} under /J. 3 Is every I-dimensional subspace of V2 (1R) either of the form {x(m, l)jx E IR} for some suitable choice of m. or of the form {(x. 0)jx E IR}? If the I -dimensional subspaces of Vi IR) are la belled with the elements of the set IR u {::c} as follows {x(m, l)jx E IR}

((x. O)jx

E

IR}

1-+

m,

1-+ OCJ,

exhibit the permutation of IR u { oc;} which is induGed by 'Y. (the a of qn l ). State explicitly which mis mapped to oc. and what the image of YJ is. 4 Is every I-dimensional subspace of V2(l\) either of the form {x(m, l )jx E &:1' 3 } for some suitable choice of m, or of the form {(x. 0)jx E Z. 3}? Find the image of each of the subspaces {x(0, !)Ix E Z. 3}. {x(l, l)jx E J\}. {x(2. l)jx E Z.J} and {(x, 0)lx E Z. 3} under the linear transformation a:(x,y)1-+(x,y)(1

:).

• Does every nonsingular linear transformation of V2 (Z. 3 ) necessarily permute the I-dimensional subspaces? If the I-dimensional subspaces of Vi(ZJ are labelled with. the elements of the set Z 3 u {oo} as follows {x(m, l)lx E Z. 3}

{(x, 0)lx E Z 3}

1-+

1-+

m,

oo,

exhibit the permutation of Z 3 u {oo} which is induced by ~- Give a general algebraic expression for the image of m, provided m + I # 0? Write down another linear transformation of Vi(Z 3) which permutes the I-dimensional subspaces in exactly the same way as a. 5 If ex and fJ are nonsingular linear transformations of Vi(F) which permute the I-dimensional subspaces in exactly the same way, what is the effect of ap- 1 on the I-dimensional subspaces of Vi(F)?

Permutations of I-dimensional subspaces

169

If (I, 0)!J./3- 1 = (s, 0) and (0, 1)!J./3- 1 = (0, t). prove that (1, l)!J./3- 1 = (s, t) and deduce thats = t. If -:t./3- 1 :(x, y) r-+ (x, y)A, what kind of matrix is A? 6 Treating GL(2, 1\) as a group of matrices, find the centre of this group using qn 13.16. Write down three distinct cosets of this centre, and find the permutations induced on the !-dimensional subspaces of V2 (l 3) by the six transformations derived from these matrices. 7 If under the nonsingular linear transformation of Vi(F) y) r--+ (x, y) (~

!J.: (x,

~)

the I-dimensional subspace {x(m, I)lx E F} is mapped to the I-dimensional subspace {x(m', I)lx E F}, a~suming that cm + d -:/= 0, express m' in terms of m. What is the image of the subspace {x(- d, c)lx E F} under a? What is the image of the subspace {(x, O)lx E F} under a? 8 If we attempt to define a mapping F -+ F by X r--+

ax+ b ex+ d '

where a, b, c, d E F and c -:/= 0, what element of F must be deleted from the domain? 9 Use the equation

+ +

b b ad - be = +----x d d d( ex + d) to show that when ad - be = 0, the mapping

ax

ex

ax+ b

XH--CX

+

d

is not an injection of the domain F - { -d/e}. 10 If ad - be -:/= 0 and c -:/= 0, find the field elements if the mapping given by ax+ b CX + d

XH---

is a bijection of F - { - d/c} to F - {s}. 1.1 If for each m E F, a nonsingular linear transformation a of Vi(F) maps the I-dimensional subspace {x(m, l)lx E F} onto the I-dimensional subspace {x(am + b, l)lx E F} where a -:/= 0, what is the image and the pre-image of the subspace {(x, 0) Ix E F} under a? 12 When ad - be -:/= 0 we define a linear fractional transformation of Fu {oo} by ax+ b !J.:XH d • ex+ when c

=

0, ooa

= oo. -and when c

-:/= 0, ( -djc)a

= oo and

18 Linear fractional groups

170

oor:t. = a/c. The set Fu { x} is called the projectii•e line over the field F. Write down the six linear fractional transformations of Z 2 u {x} and determine the corresponding permutations of {0,1,co}. The homomorphism GL(2, F)

->

LF(F)

13 Prove that the mapping of the group GL(2, F), of matrices under matrix multiplication, onto the set of linear fractional transformations under composition, given by (~

;) H

[ X H

::: :

!]

1s a homomorphism. Deduce that the linear fractional transformations form a group under composition. This group is denoted by LF(F). 14 If in qn 13 we take F = Z. 3 , can you distinguish between the images

of the matrices (~

~) and (~

D

under the homomorphism?

15 What is the kernel of the homomorphism of qn 13? 16 From qn 13.32 determine the order of GL(2, Z 3) and of its centre C. Deduce the order of the group LF(Z 3 ). 17 ExhibitthetransformationsxH 1/(x + l)andxH 1/xinLF(Z 3)as permutations of the set {O, I, 2, co} and deduce from qn I 7.23 that LF(-:f.. 3 ) is isomorphic to S 4 • 18 With the notation of qn 11.8, determine the number of vectors in V2 (F4 ) and the number of vectors in a I-dimensional subspace. Deduce the order of GL(2, F4 ). How many matrices are there in th~centre of this group? _Use these results to determine the order of LF(F4 ), by applying the fundamental theorem on homomorphisms. 19 With the notation of qn 11.8, exhibit the transformations x H b/(x + b) and x H bx + I in LF(F4 ) as permutations of the set {0, I, a, b, x} and deduce from qn 17.48 that LF(F4 ) is isomorphic to A 5 • The quotien~ group PGL(2, F) 20 Apply the fundamental theorem on homomorphisms to qn 13 to establish that if C is the centre of the group GL(2, F), then the quotient group

Projective special linear group PSL(2, F)

171

GL(2, F) C is isomorphic to the group LF(F).

C

. GL(2 F) The quotient group

is known as the projectfre general linear group PGL(2, F). Since PGL(2, F) ~ LF(F), we will use the two names interchangeably.

21 If a, b and care distinct elements of F, what are their images under the linear fractional tram.formation x-a c-b XI-+--•--? x-b c-a Deduce that the linear fractional group is triply transitive on the projective line F u {oo}. 22 Prove that the only linear fractional transformation to fix oo, 0 and 1 is the identity. Deduce from qn 21 that a linear fractional transformation is uniquely determined J:?y three points and thei ... images. 23 If Fis a field containing q elements, show that LF(F) contains (q + l)q(q - !) elements. Deduce from qn 20 that GL(2, F) contains (q + l)q(q - 1) 2 elements, and verify this directly.

24 What can you say about a linear transformation of Vi(F) with three eigenvectors, none of which is a scalar multiple of the others?

25 (Optional) By labelling the faces of a cube with the elements of Zs u {oo} in such a way that the linear fractional transformations x i-+ 2x and xi-+ 3/x both correspond to rotational symmetries of the cube, show that there is a subgroup of PGL(2, Zs) which is isomorphic to S4 • Projective special linear group PSL(2, F) 26 If C denotes the centre of the group GL(2, F), use the working of qn 13.16 to show that the centre of the group SL(2, F) is SL(2, F) n C = C5 • 27 If Fis a finite field containing q elements and q is even, prove that the multiplicative group of F contains no elements of even order, so that x = I is the only solution of the equation x 2 = 1. Deduce that for such a field, the centre of the group SL(2, F) consists of the identity alone. 28 If Fis a finite field containing q elements and q is odd, prove that the

i72

18 Linear fractional groups

additive group of F contains no elements of even order, so that = 0 is the only solution of the equation x + x = 0. Deduce that + 1 and - 1 are distinct, so that x2 = 1 has exactly two solutions. Show that for such a field, the centre of the group SL(2, F) contains just two distinct elements. x

29 For F = Z 2 , "ll. 3 , F4 and "ll. 5 find the orders of GL(2, F), SL(2, F), the centres of each of these two groups, and the quotient groups GL(2, F)/C and SL(2, F)/C5 • 30 Let S denote SL(2, F), Cs the centre of Sand let n denote the homomorphism of qn 13. Use qn 16.11 to show that Sn is a subgroup of LF(F). What is the kernel of the homomorphism TC: S -+ Sn? Use the fundamental theorem on homomorphisms (qn I 6.23) to establish that the quotient group Sf Cs is isomorphic to Sn. The quotient group S/Cs is known as the projective special linear group P SL(2, F). 31 With the notation of qn 30, suppose that A is an element of GL(2, F) such that An E Sn. Prove that for some element BE SL(2, F), AB-' lies in the centre of GL(2, F), and deduce that the determinant of A is a square.

32 If ad - be = r2 =I 0, prove that ax+ b Xf-+---

CX

+

d

lies in Sn (using the notation of qn 30) by constructing a matrix in SL(2, F) with the same image under

TC.

33 From qns 30, 31 and 32 deduce that PSL(2, F) is isomorphic to the ·subgroup of LF(F) consisting of those elements with determinant a

square. 34 By examining the elements x f-+ x + l and x that PSL(2, J\) is isomorphic to A 4 •

h+

2/x of LF("ll. 3), prove

35 (Optional). Label the twelve faces of a regular dodecahedron with the six elements of "ll. 5 u {oo} in such a way that x f-+ x + l and x f-+ 4/(x + 4), as elements of LF("ll. 5 ) correspond to rotational symmetries of the dodecahedron. Verify, either from qn 29 or from qn 5.15 that the group generated by these two linear fractional transformations has order 60 or less. From qn 5.5(iv) deduce that these rotations of the dodecahedron correspond to rotational symmetries of a regular icosahedron and from qn 5.14 to elements of A 5 • From qn 17.48, prove that PSL(2, "ll. 5) is isomorphic to A 5 • 36 Use qn 20 to prove that PGL(2, C) is isomorphic to the Mobius group,

Summary

173

and use qn 33 to prove that PSL(2, C) is also isomorphic to the Mobius group.

37 Use qns ,7.46 and 7.47 to show that there is a homomorphism of the group of nonsingular complex matrices of the form (:

-!) onto

the group of rotations of a sphere. Summary

Theorem

qns 7, 12

The elements of GL(2, F) permute the I-dimensional subspaces of Vi(F). If th~ I-dimensional subspaces are labelled {x(m, l)lx E F} H m, {(x, O)lx E F} H oo, then the permutation induced by (: ~) in GL(2, F) is precisely the linear fractional transformation XH

Theorem

ax+ b ex+ d'

where oo is fixed if c = 0 and - d/c is mapped to oo and oo is mapped to a/c if c # 0. The mapping of GL(2, F) given by

qn 13 (:

Theorem

qn 20

Theorem

qns 21, 22

Definition

qn 30

~) H

[ X H

;; :

!]

is a homomorphism with kernel the centre of GL(2, F) onto the linear fractional group LF(F). If C denotes the centre of GL(2, F) then the quotient group GL(2, F)/C, known as the projective general linear group PGL(2, F), is isomorphic to the linear fractional group LF(F). The linear fractional group is triply transitive on the projective line F u {oo} and each transformation is uniquely determined by its action on three distinct elements. If Cs denotes the centre of SL(2, F) then the quotient group SL(2, F)/Cs is known as the projective special

linear group PSL(2, F). PSL(2, F) is isomorphic to the subgroup of linear qn 33 fractional transformations for which ad - be is a

Theorem

square. Further reading: Rotman, chapter 8; Carmichael, sections 68 and 71.

{

18 Linear fractional groups

174

Historical note The linear fractional groups for different fields arose independently. We have already seen, how, for the field of complex numbers, it was studied synthetically by A. F. Mobius (1852-56). For the field of real numbers, it appeared in the work of von Staudt (1847) as the projective group on a line, with elements formed by a sequence of projections from one line to another in the real projective plane. For the field lp, the linear fractional group and its subgroup with square determinants was studied by E. Galois (1832) who introduced the symbol oo with the meaning we attach to it in this chapter. For arbitrary finite fields, the linear fractional group was studied by E. H. Moore (1893) who established the simplicity of PSL(2, F) for fields of order greater than 3. The homomorphism of GL(2, F) to the linear fractional group is implied in the work of E. Galois (1832) and J. A. Serret (1866), and was used by A. Cayley (1880) to determine properties of linear fractional transformations.

Answers to chapter 18

175

Answers to chapter 18 1 s= m'

fr.

=

t = ;.

2m 3m

+ + + +

4

5

2m 4 3m 5 2m + 4 3 m 1-> - - - , -i 1-> oo, oo 1-> t, 3m + 5 4 {x(0, l)} ➔ {.x(l, l)} ➔ {.x(2, l)} ➔ {.x(l, 0)} ➔ {.x(0, l)}. Yes from qns 12.5 2 m'

=

and 13.5. (0 12 oo). m

1->

1/(m

+

1). (~

~).

5 cx.p- 1 fixes each I-dimensional subspace. (1, I) = (1, 0) + (0, l) so (1, l)cx./J- 1 = (s, 0) + (0, t). If the subspace {x(l, l)lx E F} is fixed by ap- 1, s = t. A is a scalar matrix.

for example. Matrices in the same coset permute the I-dimensional subspaces in the same way. 7 m'

=

am+ b. cm+ d

{x(-d, c)} ➔ {(x, O)}

-->

{.x(a, c)}.

8 -d/c. 10 s

=

a/c.

11 {(x, 0)} is fixed by qn 13.5. 12 (0): X I-> X, (0loo):x 1-> 1/(x + I), (0ool):x 1-> (x + 1)/x, (01):x 1-> x + I, (0oo): x 1-> 1/x, (loo):x 1-> x/(x + 1).

13 The set of linear fractional transformations is a subset of SFu{oo}. Since this set is the image of a group under a homomorphism it is a group by qn 16.11. 14 No.

15 The scalar matrices. 16 IGL(2, l'. 3)1 = 48, !centre[ = 2, so [LF(l'. 3)1 = 24.

18 Linear fractional groups

176

+ l). ( ::x:;0): x ,_. 1/x.

17 (:x:,012): x ,_. 1/(x

18 16 vectors in all. Four in a I-dimensional subspace. /GL(2. F 4 )/ 180. Three matrices in centre. ILF(F4 )/ = 60. 19 (0labx:):x,_.b.(x

+

b). (0la):xf-->bx

+

=

15 · 12

=

1.

20 C is the kernel of the homomorphism. 21 a,_. 0, b ,_. x, cf--> l. Argue as in qn 4.12. 22 To fix XJ, c = 0. To fix XJ and 0, b = c = 0. To fix x, 0 and 1, b = c = 0 and a = d. If /3 and y have the same effect on a, b and c and C1. is the transformation of qn 21 then C1. 1(py- 1 )C1. fixes 0, 1 and x. From qn 22, C1. - I /J;, - l (1 = 1, so /3 = y. 23 LF(F) is triply transitive on Fu { x }, i.e. q + 1 elements. So .:tJ may map to q + 1 elements. 0 to q elements and 1 to q - 1 elements. Thus /GL(2, F)/CI = (q + l)q(q - 1) and since C contains q - 1 elements, IGL(2, F)I = (q + 1)q(q - 1)2. Since V;(F) contains q2 vectors, GL(2, F) has order (q2 - I )(q2 - q).

24 It is an enlargement since three I-dimensional subspaces are fixed. Compare qns 5 and 15.9.

25 (1243): x ,_. 2x. (Ox)(l3)(24): x ,_. 3/x. 4 ::x:;

2

0

3

Now use qns 5.10 and 17.23.

26 Qn 13.16 shows that each element in the centre of SL(2, F) is in the centre of GL(2, F).

27 The multiplicative group of Fcontains q - I elements, an odd number, so by Lagrange's theorem every element of this group has odd order. ( a 0) is in the centre of SL(2, F) ..., a 2 = 1. O a

29 F l2

li F~ ls

/GL(2, F)/

ISL(2, F)I

6 48 180 480

6 24

/Cs/ 1 2

60

ICI 1 2 3

120

4

2

/GL(2, F)/CI

/SL(2, F)/Csl

6 24 60 120

6 12 60 60

Answers to chapter 18 31 If An E Sn then for some BE S. An_

=

Bn. so AB- 11r = In and AB- 1 is a

~)say.Thus det AB- 1 = a1 , so det A = d-.

scalar matrix,(~

32

177

G;: 1-:)-

33 From qns 31 and 32.

ax+ b CX + d

Xf-->---

is in Sn if and only if ad - be is a square. So n is a homomorphism of SL(2. F) onto the subgroup of LF(F) with square determinants. The

kernel of this homomorphism is C5 , and the fundamental theorem gives the result.

34 Both (012): x f--> x + I and (0oo)(l2): x f--> 2/x have determinant 1. An element of order 2 and an element of order 3 generate A 4 by qn 17.40. Result follows from qns 29 and 33 .

35 (01234): x

f-->

x

+

.

I and (01 cc )(243): x

f-->

4/(x

+

4) each have determinant

1.

36 In C every element is a square, so PGL(2, C)

~ PSL(2,

C).

19 Quaternions and rotations

In this chapter we first extend the algebra of matrices by defining matrix addition and establishing the distributive laws. We then use matrices to define the quaternions and develop some of their special algebraic ·properties. Finally we show how the grpup of inner automorphisms of the quaternions is isomorphic to the group of rotations of 3-dimensional space with a given fixed point. Concurrent reading: Birkoff and MacLane, chapter 8, section 10; Rees, pp. 42~4; Curtis, chapter 5; Coxeter (1974), chapter 6. Addition of matrices

In the first two questions we develop the algebra of matrices in a quite general context. We define addition on matrices of the same shape in a way which coincides with componentwise addition, and we show that this addition is compatible with our definition of matrix multiplication in the sense that both distributive laws hold. 1 Ifv .-. vA and v .-. vB are both linear transformations V,,(F) - Vm(F), prove that v .-. vA + vB is a linear transformation. This establishes the existence of a matrix C such that vA + vB = vC for all vectors v. We now define A + B = C and this operation is called addition of matrices. By considering v = (1, 0, ... , 0), (0, 1, ... , 0), etc., show that C is formed from A and B by componentwise addition. 2 If A, Band Maren x n matrices over ihe same field, use the definition of A + B to prove that

+ B) = MA + MB. In addition, by making an appeal to the linearity of v .-. vM,

M(A

179

Algebra of quaternions

prove that (A

+

B)M

=

+

AM

BM.

Algebra of quaternions

We now study the set of matrices which we considered in the last question of the last chapter, and in qns 3-8 obtain the basic algebraic properties of this set. 3 We will call any complex 2 x 2 matrix of the form ( : quaternion, and we write every quaternion - w

(_::!: :~!0

=

aG +

;) a

~)+b(~ -~)+c(_~ ~)

d (~ ~)

· where a, b, c and d are real numbers, using the notation of scalar

(t !!) = k ~ !) . ~). k = (~ ci}

multiplication of matrices, namely Let i

= (~ -~} j = ( _ ~

Prove that i2 = j2 = k 2 = - l, and that ij = k, ji = - k, jk = i, kj = - i, ki = j, ik = - j, to establish that the set of eight matrices { ± l, ± i, ± j, ± k} forms a group under matrix multiplication. We use the notation of this question throughout the rest of the chapter. 0

4 Let

oi

denote the function with domain the quaternions and codomain

Vi (IR) given by oi: (

a+ ib

c

+

0

i

-c+1'd a-1'b

H

(a, b, c, ri).

Prove that a is a bijection, and that under a the structure of matrix addition corresponds with vector addition, and the structure of the scalar multiplication of matrices corresponds to scalar multiplication of vectors. It is the isomorphism a which justifies the description of the quaternions as a 4-dimensional vector space over the real numbers.

= al + bi + cj + dk, where a, b, c, and dare real numbers, we define the conjugate quaternion A = al - bi - cj - dk.

5 If A

19 Quaternions and rotations

180

Use qns 2 and 3 to show that AA

=

(a 2

+

b2

+

c2

+

d 2 )1

(det A)/. The quaternion A given above is said to be a real quaternion if b = c = d = 0. Check that the real quaternions commute, under multiplication, with all quaternions. 6 Use qns 2 and 3 to show that the set of quaternions is closed under matrix multiplication. Use qn 5 to show that every nonzero quaternion has a multiplicative inverse which is a quaternion. Which of the axioms for a field can we now claim are valid for the quaternions under the operations of matrix addition and matrix multiplication? 7 Justify each of the equalities (AB)(AB) = ( uA · aA => uA ·a = 0. So the plane through O perpendicular to 0a is fixed by v

1-►

=

0

vA.

14 A has one of the first two fonns of qn 10. If >>> '\

'

\.

\.

' ' ' ' ' '

7

V

7

/

V

V

'

7

/

' ' ' xxxxxx

22 Discrete groups fixing a line

206

+ c and z 1--+ e; 0z + c each map the real line onto itself, determine a necessary restriction on the value of c by considering the image of 0. By further considering the image of 1, determine a necessary restriction on the value of e; 0•

2 If the isometries z 1--+ e; 0z

3 Give geometric descriptions of the transformations Zt--+

z + c, -z +

C,

ZI--+

-z +

C,

Z I--+

Z

z

I--+

+

C,

for real values of c. Deduce that the necessary conditions on c and e; 0 in qn 2 for isometries to fix the real line are in fact sufficient conditions. Does the set of all isometries of these types form a group? 4 Prove that the reflection z

1--+

z lies in the centre of the group of

isometries fixing the real line.

5 Let G be a group of isometries fixing the real line, but not stabilising any point. (i) If G contains a half-turn (X with centre A and p is an isometry in G which moves A, describe p- 1(Xp geometrically, and hence describe (X(p- 1,y_p) geometrically. (ii) If G contains a reflection with axis perpendicular to the real line, devise an argument similar to that of (i) to prove that G must contain a translation. (iii) If G contains a glide-reflection, why must G contain a translation? Deduce that G must contain a translation. A homomorphic image: the point group 6 Establish that the mapping n of the group of similarities defined by

n: [z 1--+ az + b] 1--+ [z 1--+ az] and n: [z 1--+ az + b] 1--+ [z 1--+ az] is a group homomorphism. What is the kernel of n? If G 1s a group of similarities, then the group Gn is called the point-group of G. (Beware! Gn is not in general a subgroup of G.) 7 What is the image of the full group of isometries fixing the real line under the homomorphism n of qn 6?

Discrete groups of transformations

207

8 If G is a subgroup of the full group of isometries fixing the real line and Gn (as in qn 6) consists of the identity alone, what can be said about the elements of G? 9 If G is a subgroup of the group of isometries fixing the real line and Gn = {[z H z], [z H - z]}, what can be said about the elements of G?

10 If G is a subgroup of the group of isometries fixing the real line and Gn = {[z r-➔ z], [z H - z]}, what can be said about the elements of G?

When discussing symmetries of the plane, we distinguish between the isomorphic groups C2 and D, by presuming that C2 contains a halfturn and D 1 contains a reflection. Thus the point group in qn 9 is C2 , and the point group in qn 10 is D 1• 11 If G is a subgroup of the full group of isometries fixing the real line and Gn = {[z 1--+ z], [z H z]}, give an example to show that

although the point group is D 1, the group G need not contain a reflection. 12 If G is a subgroup of the full group of isometries fixing the real line and Gn = D 2 , give an example to show that G need not contain a reflection in the real line. First choose a suitable pattern and then describe its symmetries. Discrete groups of transformations

13 A group of transformations of the plane to itself is said to be discrete when, for each point P of the plane, it is possible to draw a circle centre P within which there are no other points of the orbit of P. Which of the following groups are discrete: (a) the full group of rotations with centre 0, (b) the full group of translations fixing the real line, (c) D6 , being the full symmetry group of a regular hexagon in the plane? 14 If a discrete group qfisometries is given which stabilises a point 0, by

considering the orbit of a point other than Q, establish that if the group contains any rotations distinct from the identity then it contains a rotation through a minimum angle. Use qns 21.24 and 21.26 to show that the group is finite, and either cyclic or dihedral. 1--+ z + c is defined to be lei. If a discrete group of isometries of the plane is given which contains a translation, prove that the group contains a translation of minimal length.

15 The length of the translation z

22 Discrete groups fixing a line

208

16 Deduce from qns 5 and 15 that a discrete group which fixes a line and does not stabilize a point contains a translation of minimal length. Classification of frieze groups

17 If a discrete group of isometries fixes a line but does not stabilise a point, it is called a frieze group. A pattern whose group of symmetries is a frieze group is called a frieze pattern, or simply a frieze. Exhibit a frieze pattern for which the frieze group contains only translations. Must this frieze group be cyclic? The symbol C 00 is sometimes used to denote an infinite cyclic , group. 18 Exhibit a frieze pattern for which the frieze group contains half-turns, but neither reflections nor glide-reflections. If a minimal translation in this group is r : z H z + 1 and a is a half-turn in the group, verify that ara = r- 1 • Because a2 is the identity, this relation justifies referring to ( r, a) as D By choosing the origin at the centre of ct, exhibit all the elements of this frieze group as mappings of complex numbers. 00 •

19 Exhibit a frieze pattern for which the frieze group contains reflections

with axes perpendicular to the fixed line, but neither half-turns nor glide-reflections. If a minimal translation in this group is r : z H z + 1 and Q is a reflection in the group, verify that Q!Q = r- 1• Because Q1 is the identity, this relation justifies referring to ( r, Q) as D By choosing the imaginary axis as the axis of e, exhibit all the elements of this frieze- group as mappings of complex numbers. If the imaginary axis is not the axis of e and e : z H - z + r for some real number r, determine all the elements of< r, e) in this case as mappings of complex numbers. 00 •

20 If a frieze group contains a minimal translation z H z + 1, what glide-reflections may be in the group? If a frieze group contains only translations and glide-reflections, prove that it is a cyclic group, and exhibit a frieze with this group of symmetries. 21 If a frieze group contains any two of the three types of isometry,

a half-turn, a glide-reflection, a reflection with axis perpendicular to the fixed line, prove that it contains all three types of isometry. Exhibit a frieze pattern for which the frieze group includes

Summary

209

isometries of these three types, but not a reflection in the fixed line. If z H z + l is a minimal translation in this group and z H - z is a half-turn in the group, exhibit all the elements of the group as mappings of complex numbers. By a suitable choice of generators, show that this group is abstractly isomorphic to D'XJ. 22 If a frieze group contains a minimal translation z H z + l, the reflection z H z and no other reflections, prove that it may not contain any half-turns, and exhibit all the elements of the group as mappings of complex numbers. Show that this grbup is abstractly isomorphic to C2 x C 00 • Use qns 4 and IO. I I. Draw a frieze pattern with such a frieze group. 23 If a frieze group contains a minimal translation r: z H z + l and the reflection(!: z r-+ z, and contains some other isometry not in