Groups

139 Pages·1993·8.465 MB·English

by  Mark Cartwright (auth.)

GROUPS
DIMENSIONS OF
MATHEMATICS
Groups
Mark Cartwright
M
MACMILLAN
© Mark Cartwright 1993
All rights reserved. No reproduction, copy or transmission of
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issued by the Copyright Licensing Agency, 90 Tottenham Court
Road, London WlP 9HE.
Any person who does any unauthorised act in relation to this
publication may be liable to criminal prosecution and civil
claims for damages.
First published 1993 by
THE MACMILLAN PRESS LTD
Houndmills, Basingstoke, Hampshire RG21 2XS
and London
Companies and representatives
throughout the world
ISBN 978-1-349-12125-0 ISBN 978-1-349-12123-6 (eBook)
DOI 10.1007/978-1-349-12123-6
A catalogue record for this book is available
from the British Library.
To Leslie,
for understanding
CONTENTS
Preface ix
1 NUMBERS, VECTORS AND MATRICES 1
1.1 The natural numbers 1
1.2 The integers 5
1.3 The rationals 7
1.4 The reals 9
1.5 Vectors 11
1.6 Matrices 14
1.7 The integers modulo n 19
1.8 The integers modulo p 22
2 PERMUTATIONS 25
2.1 Shuffling shells 25
2.2 Permutations 26
2.3 Composition of permutations 28
2.4 Systems of permutations 32
3 SYMMETRY 37
3.1 Symmetry of plane figures 37
3.2 Types of symmetry 39
3.3 Symmetries and matrices 42
3.4 Arithmetic on symmetries 47
3.5 Systems of symmetries 50
3.6 Solids 52
3.7 The Platonic solids 55
3.8 Rotational symmetries 56
3.9 Tilings 60
vii
CONTENTS
4 GROUPS 64
4.1 Axioms 64
4.2 Examples of groups (and non-groups) 67
4.3 Cayley tables 70
4.4 Isomorphism 75
5 INSIDE GROUPS 79
5.1 Powers 79
5.2 Generators 82
5.3 Cyclic groups 85
5.4 Subgroups 87
5.5 Finding subgroups using Cayley tables 90
6 THE CLASSIFICATION PROGRAMME 95
6.1 Why classify groups? 95
6.2 Using the Latin square property 98
6.3 Using groups we know about 101
6.4 Isomorphic tables 103
6.5 Failed groups 106
7 COUNTING COLOURED SHAPES 111
7.1 Black and white tiles 111
7.2 The group approach 113
7.3 Justifying the method (i) 115
7.4 Justifying the method (ii) 119
7.5 A three-dimensional example 121
Glossary 124
Index 129
Vlll
PREFACE
It is not easy to decide what to put in a volume like this; or, having decided
what to include, how to put it in. What follows is, then, a very personal
approach to introducing groups and group theory.
To justify it, I would say that the real stumbling block that students find
in approaching the subject is its abstract nature. I have therefore included a
fairly extensive discussion of examples before identifying them as groups,
the hope being that this stock of examples might provide a 'map of the
land'. I have also presented a lot of the 'formal' theory from the point of
view of (comparative concrete) Cayley tables.
There will be people who will wish that I had introduced (say) the cycle
notation for permutations, or proved Lagrange's theorem. To these I say:
there is plenty of time for those things in a university course. This book is
all about what groups are, and the frame of mind in which to approach
them.
The final chapter is, I admit, a bit courageous: a development of (only a
slight bowdlerisation of) the orbit-stabiliser theorem. This is here because
of its great practical relevance, very rare in elementary group theory. It is
possible to do this without introducing cosets or a formal idea of group
actions.
Finally, I must commit to print my immense debt of gratitude to my wife
Leslie, for reading the manuscript and (most encouragingly) liking it.
Thanks too go to the series editors and to the staff at Macmillan for their
support and guiding hands.
ix
1 NUMBERS, VECTORS AND
MATRICES
1.1 THE NATURAL NUMBERS
It is a fair bet that your first conscious introduction to mathematics was
when you learned to count: one apple, two apples, and so on. Well,
professional mathematicians too start their activities by setting up a count
ing system. There is one slight difference between the system they adopt
and the system you became familiar with: they include the number zero at
the beginning. (Zero apples is a perfectly sensible counting number; it is, in
fact, exactly the number of apples I have eaten so far today.) This system,
the sequence
0, 1, 2, ... ' (1.1)
is called the natural numbers. The natural numbers form such a useful
system that they are given a special name: N.
Within the system of natural numbers, it is possible to do arithmetic. If I
have three apples, and you give me two more, I end up with five apples. In
the abstract, we speak of addition being an operation on the natural
numbers: given any two natural numbers, there is a third natural number
called their sum. The behaviour of the apples is an application of the
operation of addition, since it happens that 3 + 2 = 5.
Addition is a very natural idea (which is why you were taught it so
young). In view of this, it is perhaps not surprising that it should be a
'well-behaved' operation. For instance, it makes no difference to me
whether you give me three apples and then a further two, or whether you
give me two apples to start with and then three more; I end up with five
apples either way. In the natural numbers, this becomes the rule that:
m+n=n+m for any m, n (1.2)
1
GROUPS
We say that addition is a commutative operation.
Again, suppose you give me five apples; later on you give me two apples,
then another three apples. That is the same sequence of operations as
giving me five apples, then another two apples, followed some time later by
three apples. This example shows that 5 + (2 + 3) is the same as
(5 + 2) + 3. In general, for any three natural numbers m, nand p:
(m + n) + p = m + (n + p) for any m, n, p (1.3)
This property is called associativity, and we say that addition is associative.
The argument of counting apples proves the commutativity and the as
sociativity of natural number addition.
A second abstract operation is the operation of natural number multi
plication: for any two natural numbers, there is a third natural number
called their product. For example, 4 x 7 = 28. As an application of this, if
you give me four apples every day for seven consecutive days, then you
have given me a total of 28 apples. Like addition, multiplication is com
mutative and associative:
mXn=nXm for any m, n (1.2a)
(m x n) x p = m x (n x p) for any m, n, p (1.3a)
Unlike in the case of addition, these facts are not proved simply by the
apple example. There is no obvious reason why four apples a day for seven
days should result in the same total as seven apples a day for four days.
Instead, we can use some neat geometrical ideas to make these proper
ties of multiplication clear. For commutativity, consider a rectangle with
sides of length m metres and n metres, for any natural numbers m and n
(see figure 1.1). Divide it into metre squares. Looked at one way, there are
····~
1-----+--+--+
.... U
~----&...I
J
L---..L.----'--1 ....
Figure 1.1 One way of seeing the commutativity of natural number
multiplication is by dividing an m x n rectangle into unit squares.
2