Essential Group Theory

97 Pages·2.499 MB·English

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Essential Group Theory
Michael Batty
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Michael Batty
Essential Group Theory
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Essential Group Theory
© 2012 Michael Batty & bookboon.com
ISBN 978-87-403-0301-8
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Essential Group Theory Contents
Contents
Introduction 9
1 Sets and Maps 10
1.1 Sets 10
1.2 Maps 11
1.3 Equivalence Relations and Partitions 12
1.4 Modular Arithmetic 13
2 Groups 14
2.1 Binary Operations 14
2.2 Groups: Basic Definitions 15
2.3 Examples of Groups 17
3 Subgroups 21
3.1 Definition of a Subgroup 21
3.2 Cosets 22
3.3 Lagrange’s Theorem 23
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Essential Group Theory Contents
4 Generators and Cyclic Groups 24
4.1 Orders of Group Elements 24
4.2 Generating Sets 25
4.3 Cyclic Groups 26
4.4 Fermat’s Little Theorem 27
5 Mappings of Groups 28
5.1 Homomorphisms 28
5.2 Isomorphisms 29
6 Normal Subgroups 31
6.1 Conjugates and Normal Subgroups 31
6.2 Cosets of Normal Subgroups 32
6.3 Kernels of Homomorphisms 32
7 Quotient Groups 35
7.1 Products of Cosets 35
7.2 Quotient Groups 35
8 The First Isomorphism Theorem 37
8.1 The First Isomorphism Theorem 37
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Essential Group Theory Contents
8.2 Centres and Inner Automorphisms 39
9 Group Actions 40
9.1 Actions of Groups 40
9.2 The Orbit-Stabilizer Theorem 41
10 Direct Products 43
10.1 Direct Products 43
10.2 Direct Products of Finite Cyclic Groups 43
10.3 Properties of Direct Products 44
11 Sylow Theory 47
11.1 Primes and p-Groups 47
11.2 Sylow’s Theorem 48
12 Presentations of Groups 51
12.1 Introduction to Presentations 51
12.2 Alphabets and Words 53
12.3 Von Dyck’s Theorem 56
12.4 Finitely Generated and Finitely Presented Groups 57
12.5 Dehn’s Fundamental Algorithmic Problems 58
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Essential Group Theory Contents
13 Free Groups 60
13.1 Reduced Words and Free Groups 60
13.2 Normal Closure 61
13.3 Torsion Free Groups 62
14 Abelian Groups 64
14.1 Commutator Subgroups and Abelianisations 64
14.2 Free Abelian Groups 65
14.3 Finitely Generated Abelian Groups 67
14.4 Generalisations of Abelian Groups 67
15 Transforming Presentations 69
15.1 Tietze Transformations 69
15.2 Properties of Tietze Transformations 71
16 Free Products 74
16.1 Free Products 74
16.2 A Normal Form for Free Products 76
16.3 The Universal Property of Free Products 77
16.4 Independence of Presentation 78
16.5 Decomposability 78
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Essential Group Theory Contents
17 F ree Products With Amalgamation 80
17.1 Free Products with Amalgamation 80
17.2 Pushouts 81
17.3 Independence of Presentation 85
18 HNN Extensions 86
18.1 HNN Extensions 86
18.2 Relation to Free Products with Amalgamation 87
18.3 The Higman-Neumann-Neumann Embedding Theorem 90
19 Further Reading 92
20 Bibliography 94
21 Index 95
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Essential Group Theory Introduction
Introduction
This short book on group theory is partly based on notes from lectures I gave at Trinity College Dublin
in 1999 and at Edinburgh University in 2001. The first part of the book is an introduction to elementary
group theory. It doesn’t aim to cover everything that might be in your introductory course in abstract
algebra, but just to give a summary of the important points. Perhaps you could read it before you begin
your course, if you want to read something gentle in advance, or it could be a starting point for revision.
The second part of the book is about free groups and presentations of groups. This would typically appear
in a second course on group theory.
Group theory can seem very abstract and strange when you first encounter it. It involves a different
mindset and most likely you will not have done this type of mathematics before. But it has a set of
techniques and beauty of its own and is worth perservering with, and you will find that the Gs and Hs
will soon come to life.
As with a lot of university mathematics, it depends very much on the language and logic of sets and
their elements, containment, equality and maps and unless you understand these fully, it is unlikely that
you will be able to apply them in the context of group theory, where there is even more to think about.
Most of the elementary proofs in group theory involve these simple but important techniques. I can’t
stress this enough and have included a preliminary section on sets and maps before the main theme of
group theory starts.
I hope that there are as few mistakes as possible, but if you find any, have suggestions to improve the
book or feel that I have not covered something which should be included please send an email to me at
[email protected]
Michael Batty, Durham, 2012.
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Essential Group Theory Sets and Maps
1 Sets and Maps
This section is primarily for reference, as you will probably have seen most of these definitions before.
But please at least skim lightly through them as a reminder and refer to them later when necessary.
1.1 Sets
Sets are at the very foundation of mathematics. They are difficult to define formally, in order to avoid
things going wrong. This can be done, with various systems of axioms. But it’s a subject in its own right.
Better for now if we just naively think of them as collections of elements, and take that as a starting
point. We will liberally use set notation throughout the book, summarized as follows:
• x X means x is an element of the set X.
∈
• x / X means x is not an element of the set X.
∈
• x,y means the set consisting of x and y.
{ }
• X Y or Y X means all the elements of X are in Y. We say X is a subset of Y or Y
⊂ ⊃
contains X.
• X =Y means X Y and Y X. This is how sets are proved to be equal, so such proofs
⊂ ⊂
have two parts.
• X Y will also include the possibility that X =Y. We will not use notation like X Y.
⊂ ⊆
• X Y means the union of X and Y, the set of elements that are in X or Y (or both).
∪
• X Y means the intersection of X and Y, the set of elements that are in X and Y.
∩
• X Y means the set of elements that are in X but not in Y.
−
• denotes the set containing no elements.
∅ {}
A special type of set is a pair which has two elements. This can be either unordered, in which case it is
of the form
x,y
{ }
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