Discovering Group Theory

218 Pages·2017·1.848 MB·english

by  Tony Barnard, Hugh Neill

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CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2017 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
Version Date: 20160725
International Standard Book Number-13: 978-1-138-03016-9 (Paperback)
Library of Congress Cataloging‑in‑Publication Data
Names: Barnard, Tony (Mathematics professor) | Neill, Hugh. | Barnard, Tony
(Mathematics professor). Mathematical groups
Title: Discovering group theory / Tony Barnard and Hugh Neill.
Other titles: Mathematical groups
Description: Boca Raton : CRC Press, 2017. | Previous edition: Mathematical
groups / Tony Barnard and Hugh Neill (London : Teach Yourself Books,
1996). | Includes index.
Identifiers: LCCN 2016029694 | ISBN 9781138030169
Subjects: LCSH: Group theory--Textbooks. | Algebra--Textbooks |
Mathematics--Study and teaching.
Classification: LCC QA174.2 .B37 2017 | DDC 512/.2--dc23
LC record available at https://lccn.loc.gov/2016029694
Visit the Taylor & Francis Web site at
http://www.taylorandfrancis.com
and the CRC Press Web site at
http://www.crcpress.com
Contents
Preface ......................................................................................................................xi
1. Proof ..................................................................................................................1
1.1 The Need for Proof ...............................................................................1
1.2 Proving by Contradiction ....................................................................3
1.3 If, and Only If ........................................................................................4
1.4 Definitions ..............................................................................................6
1.5 Proving That Something Is False ........................................................6
1.6 Conclusion ..............................................................................................7
What You Should Know..................................................................................7
Exercise 1 ...........................................................................................................7
2. Sets .....................................................................................................................9
2.1 What Is a Set? .........................................................................................9
2.2 Examples of Sets: Notation ..................................................................9
2.3 Describing a Set ...................................................................................10
2.4 Subsets ..................................................................................................11
2.5 Venn Diagrams ....................................................................................12
2.6 Intersection and Union .......................................................................13
2.7 Proving That Two Sets Are Equal .....................................................14
What You Should Know................................................................................16
Exercise 2 .........................................................................................................16
3. Binary Operations ........................................................................................19
3.1 Introduction .........................................................................................19
3.2 Binary Operations ...............................................................................19
3.3 Examples of Binary Operations ........................................................20
3.4 Tables ....................................................................................................21
3.5 Testing for Binary Operations ...........................................................22
What You Should Know................................................................................23
Exercise 3 .........................................................................................................23
4. Integers ...........................................................................................................25
4.1 Introduction .........................................................................................25
4.2 The Division Algorithm .....................................................................25
4.3 Relatively Prime Pairs of Numbers ..................................................26
4.4 Prime Numbers ...................................................................................27
4.5 Residue Classes of Integers ................................................................28
4.6 Some Remarks .....................................................................................32
What You Should Know................................................................................32
Exercise 4 .........................................................................................................33
5. Groups .............................................................................................................35
5.1 Introduction .........................................................................................35
5.2 Two Examples of Groups ...................................................................35
5.3 Definition of a Group ..........................................................................37
5.4 A Diversion on Notation ....................................................................39
5.5 Some Examples of Groups .................................................................40
5.6 Some Useful Properties of Groups ...................................................43
5.7 The Powers of an Element ..................................................................44
5.8 The Order of an Element ....................................................................46
What You Should Know................................................................................49
Exercise 5 .........................................................................................................49
6. Subgroups ......................................................................................................51
6.1 Subgroups ............................................................................................51
6.2 Examples of Subgroups ......................................................................52
6.3 Testing for a Subgroup .......................................................................53
6.4 The Subgroup Generated by an Element .........................................54
What You Should Know................................................................................56
Exercise 6 .........................................................................................................56
7. Cyclic Groups ................................................................................................59
7.1 Introduction .........................................................................................59
7.2 Cyclic Groups .......................................................................................60
7.3 Some Definitions and Theorems about Cyclic Groups .................61
What You Should Know................................................................................63
Exercise 7 .........................................................................................................63
8. Products of Groups .......................................................................................65
8.1 Introduction .........................................................................................65
8.2 The Cartesian Product ........................................................................65
8.3 Direct Product Groups .......................................................................66
What You Should Know................................................................................67
Exercise 8 .........................................................................................................67
9. Functions ........................................................................................................69
9.1 Introduction .........................................................................................69
9.2 Functions: A Discussion.....................................................................69
9.3 Functions: Formalizing the Discussion ...........................................70
9.4 Notation and Language .....................................................................71
9.5 Examples ..............................................................................................71
9.6 Injections and Surjections ..................................................................72
9.7 Injections and Surjections of Finite Sets ..........................................75
What You Should Know................................................................................77
Exercise 9 .........................................................................................................77
10. Composition of Functions ...........................................................................81
10.1 Introduction .........................................................................................81
10.2 Composite Functions ..........................................................................81
10.3 Some Properties of Composite Functions .......................................82
10.4 Inverse Functions ................................................................................83
10.5 Associativity of Functions .................................................................86
10.6 Inverse of a Composite Function ......................................................86
10.7 The Bijections from a Set to Itself .....................................................88
What You Should Know................................................................................89
Exercise 10 .......................................................................................................89
11. Isomorphisms ................................................................................................91
11.1 Introduction .........................................................................................91
11.2 Isomorphism ........................................................................................93
11.3 Proving Two Groups Are Isomorphic ..............................................95
11.4 Proving Two Groups Are Not Isomorphic ......................................96
11.5 Finite Abelian Groups ........................................................................97
What You Should Know..............................................................................102
Exercise 11 .....................................................................................................102
12. Permutations ................................................................................................105
12.1 Introduction .......................................................................................105
12.2 Another Look at Permutations ........................................................107
12.3 Practice at Working with Permutations .........................................108
12.4 Even and Odd Permutations ...........................................................113
12.5 Cycles ..................................................................................................118
12.6 Transpositions ...................................................................................121
12.7 The Alternating Group .....................................................................123
What You Should Know..............................................................................124
Exercise 12 .....................................................................................................125
13. Dihedral Groups .........................................................................................127
13.1 Introduction .......................................................................................127
13.2 Towards a General Notation ............................................................129
13.3 The General Dihedral Group D .....................................................131
n
13.4 Subgroups of Dihedral Groups .......................................................132
What You Should Know..............................................................................134
Exercise 13 .....................................................................................................134
14. Cosets ............................................................................................................137
14.1 Introduction .......................................................................................137
14.2 Cosets ..................................................................................................137
14.3 Lagrange’s Theorem .........................................................................140
14.4 Deductions from Lagrange’s Theorem ..........................................141
14.5 Two Number Theory Applications .................................................142
14.6 More Examples of Cosets .................................................................143
What You Should Know..............................................................................144
Exercise 14 .....................................................................................................145
15. Groups of Orders Up To 8 .........................................................................147
15.1 Introduction .......................................................................................147
15.2 Groups of Prime Order.....................................................................147
15.3 Groups of Order 4 .............................................................................147
15.4 Groups of Order 6 .............................................................................148
15.5 Groups of Order 8 .............................................................................149
15.6 Summary ............................................................................................151
Exercise 15 .....................................................................................................152
16. Equivalence Relations ...............................................................................153
16.1 Introduction .......................................................................................153
16.2 Equivalence Relations.......................................................................153
16.3 Partitions ............................................................................................155
16.4 An Important Equivalence Relation ...............................................157
What You Should Know..............................................................................159
Exercise 16 .....................................................................................................159
17. Quotient Groups .........................................................................................161
17.1 Introduction .......................................................................................161
17.2 Sets as Elements of Sets ....................................................................163
17.3 Cosets as Elements of a Group ........................................................164
17.4 Normal Subgroups ............................................................................165
17.5 The Quotient Group ..........................................................................167
What You Should Know..............................................................................169
Exercise 17 .....................................................................................................169
18. Homomorphisms ........................................................................................171
18.1 Homomorphisms ..............................................................................171
18.2 The Kernel of a Homomorphism ....................................................174
What You Should Know..............................................................................175
Exercise 18 .....................................................................................................176
19. The First Isomorphism Theorem.............................................................177
19.1 More about the Kernel ......................................................................177
19.2 The Quotient Group of the Kernel ..................................................178
19.3 The First Isomorphism Theorem ....................................................179
What You Should Know..............................................................................182
Exercise 19 .....................................................................................................182
Answers ...............................................................................................................183
Index .....................................................................................................................217
Preface
This book was originally called Teach Yourself Mathematical Groups, and pub-
lished by Hodder Headline plc in 1996. In this new edition, there is some new
material and some revised explanations where users have suggested these
would be helpful.
This book discusses the usual material that is found in a first course on
groups. The first three chapters are preliminary. Chapter 4 establishes a
number of results about integers which will be used freely in the remain-
der of this book. The book gives a number of examples of groups and sub-
groups, including permutation groups, dihedral groups, and groups of
residue classes. The book goes on to study cosets and finishes with the First
Isomorphism Theorem.
Very little is assumed as background knowledge on the part of the reader.
Some facility in algebraic manipulation is required, and a working knowl-
edge of some of the properties of integers, such as knowing how to factorize
integers into prime factors.
The book is intended for those who are working on their own, or with
limited access to other kinds of help, and also to college students who find
the kind of reasoning in abstract mathematics courses unfamiliar and need
extra support in this transition to advanced mathematics. The authors have
therefore included a number of features which are designed to help these
readers.
Throughout the book, there are “asides” written in shaded boxes, which
are designed to help the reader by giving an overview or by clarifying detail.
For example, sometimes the reader is told where a piece of work will be used
and if it can be skipped until later in the book, and sometimes a connection
is made which otherwise might interrupt the flow of the text.
The book includes very full proofs and complete answers to all the ques-
tions. Moreover, the proofs are laid out so that at each stage the reader is
made aware of the purpose of that part of the proof. This approach to proofs
is in line with one of our aims which is to help students with the transition
from concrete to abstract mathematical thinking. Much of the student’s pre-
vious work in mathematics is likely to have been computational in character:
differentiate this, solve that, integrate the other, with very little deductive
work being involved. However, pure mathematics is about proving things,
and care has been taken to give the student as much support as possible in
learning how to prove things.
New terminology is written in bold type whenever it appears.
At the end of each chapter, a set of key points contained in the chapter are
summarized in a section entitled What You Should Know. These sections are
included to help readers to recognize the significant features for revision
purposes.
The authors thank the publishers for their help and support in the produc-
tion of this book. In particular, they thank Karthick Parthasarathy at Nova
Techset and his team for the excellent work that they did in creating the print
version from the manuscript.
Tony Barnard
Hugh Neill
May 2016
1
Proof
1.1 The Need for Proof
Proof is the essence of mathematics. It is a subject in which you build secure
foundations, and from these foundations, by reasoning, deduction, and
proof, you deduce other facts and results that you then know are true, not
just for a few special cases, but always.
For example, suppose you notice that when you multiply three consecutive
whole numbers such as 1 × 2 × 3 = 6, 2 × 3 × 4 = 24, and 20 × 21 × 22 = 9240,
the result is always a multiple of 6. You may make a conjecture that the prod-
uct of three consecutive whole numbers is always a multiple of 6, and you
can check it for a large number of cases. However, you cannot assert correctly
that the product of three consecutive whole numbers is always a multiple of
6 until you have provided a convincing argument that it is true no matter
which three consecutive numbers you take.
For this example, a proof may consist of noting that if you have three con-
secutive whole numbers, one (at least) must be a multiple of 2 and one must
be a multiple of 3, so the product is always a multiple of 6. This statement is
now proved true whatever whole number you start with.
Arguing from particular cases does not constitute a proof. The only way
that you can prove a statement by arguing from particular cases is by ensur-
ing that you have examined every possible case. Clearly, when there are infi-
nitely many possibilities, this cannot be done by examining each one in turn.
Similarly, young children will “prove” that the angles of a triangle add
up to 180° by cutting the corners of a triangle and showing that if they are
placed together as in Figure 1.1 they make a straight line, or they might mea-
sure the angles of a triangle and add them up. However, even allowing for
inaccuracies of measuring, neither of these methods constitutes a proof; by
their very nature, they cannot show that the angle sum of a triangle is 180°
for all possible triangles.
So a proof must demonstrate that a statement is true in all cases. The onus
is on the prover to demonstrate that the statement is true. The argument that
“I cannot find any examples for which it doesn’t work, therefore it must be
true” simply isn’t good enough.
2 Discovering Group Theory
b
b
a c c a
FIGURE 1.1
“Proof” that the angles of a triangle add to 180°.
Here are two examples of statements and proofs.
EXAMPLE 1.1.1
Prove that the sum of two consecutive whole numbers is odd.
Proof
Suppose that n is the smaller whole number. Then (n + 1) is the larger
number, and their sum is n + (n + 1) = 2n + 1. Since this is one more than
a multiple of 2, it is odd. ■
The symbol ■ is there to show that the proof is complete.
Sometimes, in the absence of such a symbol, it may not be clear
where a proof finishes and subsequent text takes over.
EXAMPLE 1.1.2
Prove that if a and b are even numbers, then a + b is even.
Proof
If a is even, then it can be written in the form a = 2m where m is a
whole number. Similarly b = 2n where n is a whole number. Then
a + b = 2m + 2n = 2(m + n). Since m and n are whole numbers, so is m + n;
therefore a + b is an even number. ■
Notice in Example 1.1.2 that the statement says nothing about the result
a + b when a and b are not both even. It simply makes no comment on any
of the three cases: (1) a is even and b is odd; (2) a is odd and b is even; and
(3) a and b are both odd.
In fact, a + b is even in case (3) but the statement of Example 1.1.2 says
nothing about case (3).