William a adkins, steven h weintraub algebra (1992) 978 1 4612 0923 2

Graduate Texts in Mathematics

136

Editorial Board
S. Axler F.W. Gehring K.A. Ribet

Springer Science+Business Media, LLC

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Graduate Texts in Mathematics

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4

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TAKEUTIlZARING. Introduction to
Axiomatic Set Theory. 2nd ed.
OXTSCHAEFER. Topological Vector Spaces.
HILTON/STAMMBACH. A Course in
Homological Algebra. 2nd ed.
MAC LANE. Categories for the Working
Mathematician. 2nd ed.
HUGHES/PIPER. Projective Planes.
SERRE. A Course in Arithmetic.
T AKEuTIlZARING. Axiomatic Set Theory.
HUMPHREYS. Introduction to Lie Algebras

and Representation Theory.
COHEN. A Course in Simple Homotopy
Theory.
CONWAY. Functions of One Complex
Variable l. 2nd ed.
BEALS. Advanced Mathematical Analysis.
ANDERSON/FuLLER. Rings and Categories
of Modules. 2nd ed.
GOLUBITSKy/GUILLEMIN. Stable Mappings
and Their Singularities.
BERBERIAN. Lectures in Functional
Analysis and Operator Theory.
WINTER. The Structure of Fields.
ROSENBLATT. Random Processes. 2nd ed.
HALMOS. Measure Theory.
HALMOS. A Hilbert Space Problem Book.
2nd ed.
HUSEMOLLER. Fibre Bundles. 3rd ed.
HUMPHREYS. Linear Algebraic Groups.
BARNESiMACK. An Algebraic Introduction
to Mathematical Logic.
GREUB. Linear Algebra. 4th ed.
HOLMES. Geometric Functional Analysis
and Its Applications.
HEWITT/STROMBERG. Real and Abstract
Analysis.
MANES. Algebraic Theories.
KELLEY. General Topology.
ZARISKIISAMUEL. Commutative Algebra.
Vol.!.

ZARISKIiSAMUEL. Commutative Algebra.
Vol.Il.
JACOBSON. Lectures in Abstract Algebra l.
Basic Concepts.
JACOBSON. Lectures in Abstract Algebra
II. Linear Algebra.
JACOBSON. Lectures in Abstract Algebra
1II. Theory of Fields and Galois Theory.

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HIRSCH. Differential Topology.
SPITZER. Principles of Random Walk.
2nd ed.
ALEXANDERIWERMER. Several Complex
Variables and Banach Algebras. 3rd ed.
KELLEy/NAMIOKA et al. Linear
Topological Spaces.
MONK. Mathematical Logic.
GRAUERT/FRITZSCHE. Several Complex
Variables.
ARVESON. An Invitation to C*-Algebras.
KEMENy/SNELL/KNAPP. Denumerable
Markov Chains. 2nd ed.
ApOSTOL. Modular Functions and
Dirichlet Series in Number Theory.
2nd ed.
SERRE. Linear Representations of Fimte
Groups.
GILLMANIJERlSON. Rings of Continuous

Functions.
KENDIG. Elementary Algebraic Geometry.
LOEVE. Probability Theory I. 4th ed.
LOEVE. Probability Theory II. 4th ed.
MOISE. Geometric Topology in
Dimensions 2 and 3.
SACHSlWu. General Relativity for
Mathematicians.
GRUENBERGIWEIR. Linear Geometry.
2nd ed.
EDWARDS. Fermat's Last Theorem.
KLINGENBERG. A Course in Differential
Geometry.
HARTSHORNE. Algebraic Geometry.
MANIN. A Course in Mathematical Logic.
GRAVERIW ATKINS. Combinatorics with
Emphasis on the Theory of Graphs.
BROWN/PEARCY. Introduction to Operator
Theory I: Elements of Functional
Analysis.
MASSEY. Algebraic Topology: An
Introduction.
CROWELL!FOX. Introduction to Knot
Theory.
KOBLITZ. p-adic Numbers, p-adic
Analysis, and Zeta-Functions. 2nd ed.
LANG. Cyclotomic Fields.
ARNOLD. Mathematical Methods in
Classical Mechanics. 2nd ed.
WHITEHEAD. Elements of Homotopy

Theory.

(continued afier index)

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William A. Adkins

Steven H. Weintraub

Algebra
An Approach via Module Theory

Springer

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William A. Adkins
Steven H. Weintraub
Department of Mathematics
Louisiana State University
Baton Rouge, LA 70803
USA

Editorial Board
S. Axler
Mathematics Department
San Francisco State

University
San Francisco, CA 94132
USA

F. W. Gehring
Mathematics Department
East Hall
University of Michigan
Ann Arbor, MI 48109
USA

K.A. Ribet
Department of Mathematics
University of California
at Berkeley
Berkeley, CA 94720-3840
USA

Mathematics Subject Classifications: 12-01, 13-01, 15-01, 16-01,20-01
Library of Congress Cataloging-in-Publication Data
Adkins, William A.
Algebra: an approach via module theory/William A. Adkins,
Steven H. Weintraub.
p. cm. - (Graduate texts in mathematics; 136)
lncludes bibliographical references and indexes ..
ISBN 978-1-4612-6948-9
ISBN 978-1-4612-0923-2 (eBook)
DOI 10.1007/978-1-4612-0923-2
1. Algebra. 2. Modules (Algebra) I. Weintraub, Steven H.
II. Title. III. Series.

QA154.A33 1992
512'.4-dc20
92-11951
Printed on acid-free paper.
© 1992 Springer Science+Business Media New York
OriginalIy published by Springer-Verlag Berlin Heidelberg New York in 1992
Softcover reprint of the hardcover Ist edition 1992
AlI rights reserved. This work may not be transIated or copied in whole or in part without the
written permission of the publisher Springer-Science+Business Media, LLC, except for brief
excerpts in connection with reviews or scholarly ana1ysis. Use in connection with any form of
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The use of general descriptive names, trade names, trademarks, etc., in this publication, even if
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Production managed by Francine Sikorski; manufacturing supervised by Jacqui Ashri.
Photocomposed copy prepared using TeX.

9 8 7 6 5 4 3 2 (Corrected second printing, 1999)

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Preface

This book is designed as a text for a first-year graduate algebra course.
As necessary background we would consider a good undergraduate linear
algebra course. An undergraduate abstract algebra course, while helpful,
is not necessary (and so an adventurous undergraduate might learn some
algebra from this book).

Perhaps the principal distinguishing feature of this book is its point of
view. Many textbooks tend to be encyclopedic. We have tried to write one
that is thematic, with a consistent point of view. The theme, as indicated
by our title, is that of modules (though our intention has not been to write
a textbook purely on module theory). We begin with some group and ring
theory, to set the stage, and then, in the heart of the book, develop module
theory. Having developed it, we present some of its applications: canonical
forms for linear transformations, bilinear forms, and group representations.
Why modules? The answer is that they are a basic unifying concept
in mathematics. The reader is probably already familiar with the basic
role that vector spaces play in mathematics, and modules are a generalization of vector spaces. (To be precise, modules are to rings as vector spaces
are to fields.) In particular, both abelian groups and vector spaces with a
linear transformation are examples of modules, and we stress the analogy
between the two--the basic structure theorems in each of these areas are
special cases of the structure theorem of finitely generated modules over a
principal ideal domain (PID). As well, our last chapter is devoted to the
representation theory of a group G over a field F, this being an important
and beautiful topic, and we approach it from the point of view of such
a representation being an F( G)-module. On the one hand, this approach
makes it very clear what is going on, and on the other hand, this application
shows the power of the general theory we develop.
We have heard the joke that the typical theorem in mathematics states
that something you do not understand is equal to something else you cannot compute. In that sense we have tried to make this book atypical. It
has been our philosophy while writing this book to provide proofs with a

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vi

Preface

maximum of insight and a minimum of computation, in order to promote
understanding. However, since in practice it is necessary to be able to compute as well, we have included extensive material on computations. (For
example, in our entire development in Chapter 4 of canonical forms for
linear transformations we only have to compute one determinant, that of
a companion matrix. But then Chapter 5 is almost entirely dedicated to
computational methods for modules over a PID, showing how to find canonical forms and characteristic polynomials. As a second example, we derive
the basic results about complex representations of finite groups in Section
8.3, without mentioning the word character, but then devote Section 8.4 to
characters and how to use them.)
Here is a more detailed listing of the contents of the book, with emphasis on its novel features:
Chapter 1 is an introduction to (or review of) group theory, including
a discussion of semidirect products.
Chapter 2 is an introduction to ring theory, covering a variety of standard topics.
In Chapter 3 we develop basic module theory. This chapter culminates
in the structure theorem for finitely generated modules over a PID. (We
then specialize to obtain the basic structure theorem for finitely generated
Abelian groups.) We feel that our proof of this theorem is a particularly
insightful one. (Note that in considering free modules we do not assume the
corresponding results for vector spaces to be already known.) Noteworthy
along the way is our introduction and use of the language of homological
algebra and our discussion of free and projective modules.
We begin Chapter 4 with a treatment of basic topics in linear algebra. In principle, this should be a review, but we are careful to develop as
much of the theory as possible over a commutative ring (usually a PID)
rather than just restricting ourselves to a field. The matrix representation
for module homomorphisms is even developed for modules over noncommutative rings, since this is needed for applications to Wedderburn's theorem
in Chapter 7. This chapter culminates in the derivation of canonical forms
(the rational canonical form, the (generalized) Jordan canonical form) for
linear transformations. Here is one place where the module theory shows its

worth. By regarding a vector space V over a field F, with a linear transformation T, as an F[X]-module (with X acting by T), these canonical forms
are immediate consequences of the structure theorem for finitely generated
torsion modules over a PID. We also derive the important special case of
the real Jordan canonical form, and end the chapter by deriving the spectral
theorem.
Chapter 5 is a computational chapter, showing how to obtain effectively
(in so far as is possible) the canonical forms of Chapter 4 in concrete cases.
Along the way, we introduce the Smith and Hermite canonical forms as well.

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Preface

vii

This chapter also has Dixon's proof of a criterion for similarity of matrices
based solely on rank computations.

In Chapter 6 we discuss duality and investigate bilinear, sesquilinear,
and quadratic forms, with the assistance of module theory, obtaining complete results in a number of important special cases. Among these are the
cases of skew-symmetric forms over a PID, sesquilinear (Hermitian) forms
over the complex numbers, and bilinear and quadratic forms over the real
numbers, over finite fields of odd characteristic, and over the field with two
elements (where the Arf invariant enters in the case of quadratic forms).
Chapter 7 has two sections. The first discusses semisimple rings and
modules (deriving Wedderburn's theorem), and the second develops some
multilinear algebra. Our results in both of these sections are crucial for
Chapter 8.
Our final chapter, Chapter 8, is the capstone of the book, dealing with

group representations mostly, though not entirely, in the semisimple case.
Although perhaps not the most usual of topics in a first-year graduate
course, it is a beautiful and important part of mathematics. We view a
representation of a group G over a field F as an F(G)-module, and so this
chapter applies (or illustrates) much of the material we have developed in
this book. Particularly noteworthy is our treatment of induced representations. Many authors define them more or less ad hoc, perhaps mentioning as
an aside that they are tensor products. We define them as tensor products
and stick to that point of view (though we provide a recognition principle
not involving tensor products), so that, for example, Frobenius reciprocity
merely becomes a special case of adjoint associativity of Hom and tensor
product.
The interdependence of the chapters is as follows:

o
o
o
1

1

1

4.1-4.3

o
1

4.4-4.6

o

1

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viii

Preface

We should mention that there is one subject we do not treat. We do
not discuss any field theory in this book. In fact, in writing this book we
were careful to avoid requiring any knowledge of field theory or algebraic
number theory as a prerequisite.
We use standard set theoretic notation. For the convenience of the
reader, we have provided a very brief introduction to equivalence relations
and Zorn's lemma in an appendix. In addition, we provide an index of
notation, with a reference given of the first occurrence of the symbol.
We have used a conventional decimal numbering system. Thus a reference to Theorem 4.6.23 refers to item number 23 in Section 6 of Chapter
4, which happens to be a theorem. Within a given chapter, the chapter
reference is deleted.
The symbol 0 is used to denote the end of a proof; the end of proof
symbol 0 with a blank line is used to indicate that the proof is immediate
from the preceding discussion or result.
The material presented in this book is for the most part quite standard.
We have thus not attempted to provide references for most results. The
bibliography at the end is a collection of standard works on algebra.
We would like to thank the editors of Springer-Verlag for allowing
us the opportunity, during the process of preparing a second printing, to
correct a number of errors which appeared in the first printing of this book.
Moreover, we extend our thanks to our colleagues and those readers who

have taken the initiative to inform us of the errors they have found. Michal
Jastrzebski and Lyle Ramshaw, in particular, have been most helpful in
pointing out mistakes and ambiguities.

Baton Rouge, Louisiana

William A. Adkins
Steven H. Weintraub

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Contents

Preface

v

Chapter 1

Groups.

1

1.1 Definitions and Examples
1.2 Subgroups and Cosets .
1.3 Normal Subgroups, Isomorphism Theorems,
and Automorphism Groups
......
1.4 Permutation Representations and the Sylow Theorems

1.5 The Symmetric Group and Symmetry Groups
1.6 Direct and Semidirect Products
1. 7 Groups of Low Order
1.8 Exercises .
Chapter 2
2.1
2.2
2.3
2.4
2.5
2.6
2.7

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9

15
22
28
34
39
45

49

Rings

Definitions and Examples
Ideals, Quotient Rings, and Isomorphism Theorems
Quotient Fields and Localization . . . . . . .
Polynomial Rings . . . . . . . . . . . . . .
Principal Ideal Domains and Euclidean Domains
Unique Factorization Domains
Exercises . . . . . . . . . .

Chapter 3

1
6

Modules and Vector Spaces

Definitions and Examples
Submodules and Quotient Modules
Direct Sums, Exact Sequences, and Hom
Free Modules . . . . .
Projective Modules . . . . . . . .
Free Modules over a PID . . . . . .
Finitely Generated Modules over PIDs
Complemented Submodules
Exercises . . . . . . . . . . . . .

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49
58
68
72
79

92
98

107
107
112
118
128
136
142
156

171

174


x

Contents

Chapter 4
4.1

4.2
4.3
4.4
4.5
4.6
4.7

Matrices over PIDs

Equivalence and Similarity
Hermite Normal Form .
Smith Normal Form . . .
Computational Examples
A Rank Criterion for Similarity
Exercises . . . . . . . . . .

Chapter 6
6.1
6.2
6.3
6.4

182

Matrix Algebra . . . . . . . . .
Determinants and Linear Equations
Matrix Representation of Homomorphisms
Canonical Form Theory . . . . . . . .
Computational Examples
.......

Inner Product Spaces and Normal Linear Transformations
Exercises . . . . . . .

Chapter 5
5.1
5.2
5.3
5.4
5.5
5.6

Linear Algebra

Bilinear and Quadratic Forms

Duality. . . . . . . . . . .
Bilinear and Sesquilinear Forms
Quadratic Forms
Exercises . . . . . . . . .

Chapter 7

Topics in Module Theory

7.1 Simple and Semisimple Rings and Modules
7.2 Multilinear Algebra
7.3 Exercises . . . . . . . . .
Chapter 8
8.1
8.2

8.3
8.4
8.5
8.6
8.7
8.8

Group Representations .

Examples and General Results
Representations of Abelian Groups
Decomposition of the Regular Representation
Characters . . . . . . . .
Induced Representations . .
Permutation Representations
Concluding Remarks
Exercises

182
194
214
231
257
269
278
289
289
296
307
319

328
337
341
341
350
376
391
395
395
412
434
438
438
451
453
462
479
496
503
505
507
510
511
517

Appendix
Bibliography . .
Index of Notation
Index of Terminology

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Chapter 1

Groups

In this chapter we introduce groups and prove some of the basic theorems in
group theory. One of these, the structure theorem for finitely generated abelian
groups, we do not prove here but instead derive it as a corollary of the more
general structure theorem for finitely generated modules over a PID (see Theorem
3.7.22).

1.1 Definitions and Examples
(1.1) Definition. A group is a set G together with a binary opemtion
·:GxG-+G
satisfying the following three conditions:
(a) a· (b· c) = (a· b) . c for all a, b, c E G. (Associativity)
(b) There exists an element e E G such that a· e = e . a = a for all a E G.
(Existence of an identity element)
(c) For each a E G there exists abE G such that a·b = b·a = e. (Existence
of an inverse for each a E G)

It is customary in working with binary operations to write a . b rather
than ·(a, b). Moreover, when the binary operation defines a group structure
on a set G then it is traditional to write the group operation as abo One
exception to this convention occurs when the group G is abelian, i.e., if
ab = ba for all a, bEG. If the group G is abelian then the group operation is commonly written additively, i.e., one writes a + b rather than abo
This convention is not rigidly followed; for example, one does not suddenly
switch to additive notation when dealing with a group that is a subset of

a group written multiplicatively. However, when dealing specifically with
abelian groups the additive convention is common. Also, when dealing with
abelian groups the identity is commonly written e = 0, in conformity with

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2

Chapter 1. Groups

the additive notation. In this chapter, we will write e for the identity of general groups, i.e., those written multiplicatively, but when we study group
representation theory in Chapter 8, we will switch to 1 as the identity for
multiplicatively written groups.
To present some examples of groups we must give the set G and the
operation· : G x G ----> G and then check that this operation satisfies (a),
(b), and (c) of Definition 1.1. For most of the following examples, the fact
that the operation satisfies (a), (b), and (c) follows from properties of the
various number systems with which you should be quite familiar. Thus
details of the verification of the axioms are generally left to the reader.
(1.2) Examples.
(1) The set Z of integers with the operation being ordinary addition of
integers is a group with identity e = 0, and the inverse of m E Z is
-m. Similarly, we obtain the additive group Q of rational numbers, R
of real numbers, and C of complex numbers.
(2) The set Q* of nonzero rational numbers with the operation of ordinary
multiplication is a group with identity e = 1, and the inverse of a E Q*
is l/a. Q* is abelian, but this is one example of an abelian group that
is not normally written with additive notation. Similarly, there are the
abelian groups R * of nonzero real numbers and C* of nonzero complex

numbers.
(3) The set Zn = {O, 1, ... ,n -I} with the operation of addition modulo n
is a group with identity 0, and the inverse of x E Zn is n-x. Recall that
addition modulo n is defined as follows. If x, y E Zn, take x + y E Z
and divide by n to get x + y = qn + r where 0 :=::: r < n. Then define
x + y (mod n) to be r.
(4) The set Un of complex nth roots of unity, i.e., Un = {exp((2k7ri)/n) :
o :=::: k :=::: n - I} with the operation of multiplication of complex numbers is a group with the identity e = 1 = exp(O), and the inverse of
exp((2k7ri)/n) is exp((2(n - k)7ri)/n).
(5) Let Z~ = {m : 1 :=::: m < nand m is relatively prime to n}. Under the
operation of multiplication modulo n, Z~ is a group with identity l.
Details of the verification are left as an exercise.
(6) If X is a set let Sx be the set of all bijective functions f : X ----> X.
Recall that a function is bijective ifit is one-to-one and onto. Functional
composition gives a binary operation on Sx and with this operation
it becomes a group. Sx is called the group of permutations of X or
the symmetric group on X. If X = {I, 2, ... , n} then the symmetric
group on X is usually denoted Sn and an element a of Sn can be
conveniently indicated by a 2 x n matrix
2

a(2)

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1.1 Definitions and Examples

3

where the entry in the second row under k is the image o:(k) of k
under the function 0:. To conform with the conventions of functional
composition, the product 0:{3 will be read from right to left, i.e., first
do {3 and then do 0:. For example,

1 2 3 4) (1 2 3 4)
(1 2 3 4)
(3
2413412=4132'
(7) Let GL(n, R) denote the set of n x n invertible matrices with real
entries. Then GL(n, R) is a group under matrix multiplication. Let
SL(n, R) = {T E GL(n, R): detT = I}. Then SL(n,R) is a group
under matrix multiplication. (In this example, we are assuming familiarity with basic properties of matrix multiplication and determinants.
See Chapter 4 for details.) GL(n, R) (respectively, SL(n, R)) is known
as the general linear group (respectively, special linear group) of degree
n over R.
(8) If X is a set let P(X) denote the power set of X, i.e., P(X) is the set
of all subsets of X. Define a product on P(X) by the formula AL.B =
(A \ B) U (B \ A). A L. B is called the symmetric difference of A and
B. It is a straightforward exercise to verify the associative law for the
symmetric difference. Also note that AL.A = 0 and 0L.A = AL.0 = A.
Thus P(X) with the symmetric difference operation is a group with 0
as identity and every element as its own inverse. Note that P(X) is an
abelian group.
(9) Let C(R) be the set of continuous real-valued functions defined on R
and let V(R) be the set of differentiable real-valued functions defined
on R. Then C(R) and V(R) are groups under the operation of function
addition.
One way to explicitly describe a group with only finitely many elements
is to give a table listing the multiplications. For example the group {I, -I}

has the multiplication table

1

1
1

-1
-1

-1

-1

1

whereas the following table

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4

Chapter 1. Groups

e
a
b
c

e
e
a
b
c

a
a
e
c
b

b
b
c
e
a

c
c
b
a
e

is the table of a group called the Klein 4-group. Note that in these tables
each entry of the group appears exactly once in each row and column.
Also the multiplication is read from left to right; that is, the entry at the
intersection of the row headed by a and the column headed by {3 is the
product a{3. Such a table is called a Cayley diagram of the group. They
are sometimes useful for an explicit listing of the multiplication in small

groups.
The following result collects some elementary properties of a group:

(1.3) Proposition. Let G be a group.
(1) The identity e oj G is unique.
(2) The inverse b oj a EGis unique. We denote it by a-I.
(3) (a-I)-I = a Jor all a E G and (ab)-I = b-Ia- I Jor all a, bEG.
(4) IJ a, bEG the equations ax = band ya = b each have unique solutions
in G.
(5) IJ a, b, c E G then ab = ac implies that b = c and ab = cb implies that
a= c.
Proof. (1) Suppose e' is also an identity. Then e' = e'e = e.
(2) Suppose ab = ba = e and ab' = b'a = e. Then b = eb = (b'a)b =
b' (ab) = b' e = b', so inverses are unique.
(3) a(a- I ) = (a-I)a = e, so (a-I)-l = a. Also (ab)(b-Ia- I )
a(bb- I )a- I = aa- I = e and similarly (b-Ia- I )(ab) = e. Thus (ab)-I
b-Ia- I .
(4) x = a-Ib solves ax = band y = ba- I solves ya = b, and any
solution must be the given one as one sees by multiplication on the left or
right by a-I.
(5) If ab = ac then b = a-I(ab) = a-I(ac) = c.
D

The results in part (5) of Proposition 1.3 are known as the cancellation
laws for a group.
The associative law for a group G shows that a product of the elements
a, b, c of G can be written unambiguously as abc. Since the multiplication
is binary, what this means is that any two ways of multiplying a, b, and c
(so that the order of occurrence in the product is the given order) produces
the same element of G. With three elements there are only two choices for

multiplication, that is, (ab)c and a(bc), and the law of associativity says

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1.1 Definitions and Examples

5

that these are the same element of G. If there are n elements of G then
the law of associativity combined with induction shows that we can write
ala2 ... an unambiguously, i.e., it is not necessary to include parentheses
to indicate which sequence of binary multiplications occurred to arrive at
an element of G involving all of the ai. This is the content of the next
proposition.
(1.4) Proposition. Any two ways of multiplying the elements all a2, ... , an
in a group G in the order given (i. e., removal of all parentheses produces
the juxtaposition ala2 ... an) produces the same element of G.

Proof. If n = 3 the result is clear from the associative law in G.
Let n > 3 and consider two elements g and h obtained as products
of all a2, ... , an in the given order. Writing g and h in terms of the last
multiplications used to obtain them gives
and
Since i and j are less than n, the induction hypothesis implies that the
products al ... ai, aHl ... an, al ... aj, and aj+l'" an are unambiguously
defined elements in G. Without loss of generality we may assume that i :::; j.
If i = j then 9 = h and we are done. Thus assume that i < j. Then, by the
induction hypothesis, parentheses can be rearranged so that

and

h

= «al ... ai)(aHl'" aj»(aj+l ... an).

Letting A = (al'" ai), B = (aHl'" aj), and C = (aj+l'" an) the induction hypothesis implies that A, B, and C are unambiguously defined
elements of G. Then
g = A(BC) = (AB)C = h

and the proposition follows by the principle of induction.

o

Since products of n elements of G are unambiguous once the order has
been specified, we will write ala2'" an for such a product, without any
specification of parentheses. Note that the only property of a group used
in Proposition 1.4 is the associative property. Therefore, Proposition 1.4 is
valid for any associative binary operation. We will use this fact to be able to
write unambiguous multiplications of elements of a ring in later chapters. A
convenient notation for al ... an is I1~=l ai. If ai = a for all i then I1~=1 a is
denoted an and called the nth power of a. Negative powers of a are defined

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6

Chapter 1. Groups

by a- n = (a-1)n where n > 0, and we set aD
standard rules for exponents are valid.

= e.

With these notations the

(1.5) Proposition. If G is a group and a E G then

(1) ama n = a m+n , and
(2) (am)n = amn for all integers m and n.
Proof. Part (1) follows from Proposition 1.4 while part (2) is an easy exercise
D
using induction.

1.2 Subgroups and Cosets
Let G be a group and let H <:;:; G be a subset. H is called a subgroup of G
if H together with the binary operation of G is a group. The first thing to
note is that this requires that H be closed under the multiplication of G,
that is, ab is in H whenever a and b are in H. This is no more than the
statement that the multiplication on G is defined on H. Furthermore, if H
is a subgroup of G then H has an identity e' and G has an identity e. Then
e' e = e' since e is the identity of G and e' e' = e' since e' is the identity of
H. Thus e' e = e' e' and left cancellation of e' (in the group G) gives e = e'.
Therefore, the identity of G is also the identity of any subgroup H of G.
Also, if a E H then the inverse of a as an element of H is the same as the
inverse of a as an element of G since the inverse of an element is the unique
solution to the equations ax = e = xa.
(2.1) Proposition. Let G be a group and let H be a nonempty subset of

G. Then H is a subgroup if and only if the following two conditions are
satisfied.

(1) Ifa,bEHthenabEH.
(2) If a E H then a-I E H.
Proof. If H is a subgroup then (1) and (2) are satisfied as was observed in
the previous paragraph. If (1) and (2) are satisfied and a E H then a-I E H
by (2) and e = aa- I E H by (1). Thus conditions (a), (b), and (c) in the
definition of a group are satisfied for H, and hence H is a subgroup of G.
D

(2.2) Remarks. (1) Conditions (1) and (2) of Proposition 2.1 can be replaced
by the following single condition.
(1)' If a, bE H then ab- I E H.

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1.2 Subgroups and Cosets

7

Indeed, if (1)' is satisfied then whenever a E H it follows that e =
aa- 1 E H and then a-I = ea- 1 E H. Thus a E H implies that a-I E H.
Also, if a, bE H then b- 1 E H so that ab = a(b- 1)-1 E H. Therefore, (I)'
implies (1) and (2). The other implication is clear.
(2) If H is finite then only condition (1) of Proposition 2.1 is necessary
to ensure that H is a subgroup of G. To see this suppose that H is a finite
set and suppose that a, b E H implies that ab E H. We need to show that
a-I E H for every a E H. Thus let a E H and let Ta : H ---> H be defined by

Ta(b) = abo Our hypothesis implies that Ta(H) ~ H. If Ta(b) = Ta(c) then
ab = ac and left cancellation in the group G (Proposition 1.3 (5)) shows
that b = c. Hence Ta is an injective map and, since H is assumed to be
finite, it follows that Ta is bijective, so the equation ax = c is solvable in
H for any choice of c E H. Taking c = a shows that e E H and then taking
c = e shows that a-I E H. Therefore, condition (2) of Proposition 2.1 is
satisfied and H is a subgroup of G.
(3) If G is an abelian group with the additive notation, then H ~ Gis
a subgroup if and only if a - b E H whenever a, b E H.
(2.3) Proposition. Let I be an index set and let Hi be a subgroup of G for
each i E I. Then H = niEI Hi is a subgroup of G.
Proof. If a, bE H then a, bE Hi for all i E I. Thus ab- 1 E Hi for all i E I.
Hence ab- 1 E Hand H is a subgroup by Remark 2.2 (1).
0

(2.4) Definition. Let G and H be groups and let f : G ---> H be a function.
Then f is a group homomorphism if f(ab) = f(a)f(b) for all a, bEG. A
group isomorphism is an invertible group homomorphism. If f is a group
homomorphism, let
Ker(J) = {a E G: f(a) = e}
and

Im(J) = {h E H : h = f(a)

for some a E G}.

Ker(J) is the kernel of the homomorphism f and Im(J) is the image of f.
It is easy to check that f is invertible as a group homomorphism if and
only if it is invertible as a function between sets, i.e., if and only if it is
bijective.

(2.5) Proposition. Let f : G ---> H be a group homomorphism. Then Ker(J)
and Im(J) are subgroups of G and H respectively.
Proof. First note that f(e) = f(ee) = f(e)f(e), so by cancellation in H
we conclude that f(e) = e. Then e = f(e) = f(aa- 1) = f(a)f(a- 1 ) for
all a E G. Thus f(a- 1 ) = f(a)-1 for all a E G. Now let a, b E Ker(J).
Then f(ab- 1) = f(a)f(b- 1) = f(a)f(b)-l = ee- 1 = e, so ab- 1 E Ker(J)

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8

Chapter 1. Groups

and Ker(f) is a subgroup of G. Similarly, if f(a), f(b) E Im(f) then
f(a)f(b)-l = J(ab- 1) E Im(f), so Im(f) is a subgroup of H.
0
(2.6) Definition. Let S be a subset of a group G. Then (S) denotes the
intersection of all subgroups ofG that contain S. The subgroup (S) is called
the subgroup generated by S. If S is finite and G = (S) we say that G is
finitely generated. If S = {a} has only one element and G = (S) then we
say that G is a cyclic group.
(2.7) Proposition. Let S be a nonempty subset of a group G. Then

(S)

= {a1a2··· an: n

E

Nand ai or ail E S for 1 ~ i ~ n}.

That is, (S) is the set of all finite products consisting of elements of S or
inverses of elements of S.
Proof. Let H denote the set of elements of G obtained as a finite product of
elements of S or S-l = {a- 1 : a E S}. If a, bE H then ab- 1 is also a finite
product of elements from sus-I, so ab- 1 E H. Thus H is a subgroup of G
that contains S. Any subgroup K of G that contains S must be closed under
multiplication by elements of S U S-l, so K must contain H. Therefore,
H=(S).
0
(2.8) Examples. You should provide proofs (where needed) for the claims
made in the following examples.
(1) The additive group Z is an infinite cyclic group generated by the number l.
(2) The multiplicative group Q* is generated by the set S = {lip: p is a
prime number} U {-I}.
(3) The group Zn is cyclic with generator l.
(4) The group Un is cyclic with generator exp(27riln).
(5) The even integers are a subgroup of Z. More generally, all the multiples
of a fixed integer n form a subgroup of Z and we will see shortly that
these are all the subgroups of Z.
(6) If a = (~~~) then H = {e,a,a 2 } is a subgroup of the symmetric
group S3. Also, S3 is generated by a and (3 = (~ ~ ~).
(7) If (3 =
~~) and, = (g~) then S3 = ((3, ,).
(8) A matrix A = [aij] is upper triangular if aij = 0 for i > j. The
subset T(n, R) ~ GL(n, R) of invertible upper triangular matrices is
a subgroup of GL(n, R).
(9) If G is a group let Z(G), called the center of G, be defined by

n

Z(G)

= {a E G : ab = ba

for all bEG}.

Then Z(G) is a subgroup of G.

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1.2 Subgroups and Cosets

9

(10) If G is a group and x E G, then the centralizer of x is the subset C(x)
of G defined by
C (x) = {a E G : ax = xa}.
C(x) is a subgroup of G and C(x) = G if and only if x E Z(G). Also
note that C(x) always contains the subgroup (x) generated by x.
(11) If G is a group and a, bEG, then [a, bJ = a-1b-1ab is called the commutator of a and b. The subgroup G' generated by all the commutators
of elements of G is called the commutator subgroup of G. Another
common notation for the commutator subgroup is [G, GJ. See Exercise
22 for some properties of the commutator subgroup.
(12) A convenient way to describe some groups is by giving generators and
relations. Rather than giving formal definitions we shall be content to
illustrate the method with two examples of groups commonly expressed

by generators and relations. For the first, the quaternion group is a
group with 8 elements. There are two generators a and b subject to
the three relations (and no others):

We leave it for the reader to check that

For a concrete description of Q as a subgroup of GL(2, C), see Exercise
24.
(13) As our second example of a group expressed by generators and relations, the dihedral group of order 2n, denoted D 2n , is a group generated by two elements x and y subject to the three relations (and no
others):
Again, we leave it as an exercise to check that
2
2
D 2n= { e,x,x,
... ,xn-l ,y,yx,yx,
... ,yxn-l} .

Thus, D 2n has 2n elements. The dihedral group will be presented as
a group of symmetries in Section 1.6, and it will be studied in detail
from the point of view of representation theory in Chapter 8.
(2.9) Definition. The order ofG, denoted IGI, is the cardinality of the set G.
The order of an element a E G, denoted o( a) is the order of the subgroup
generated bya. (In general, IXI will denote the cardinality of the set X,
with IXI = 00 used to indicate an infinite set.)

(2.10) Lemma. Let G be a group and a
(1) o(a) =

00

if and only if an

#

E

G. Then

e for any n > O.

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10

Chapter 1. Groups

(2) If o( a) <
(3) a k

00,

then o( a) is the smallest positive integer n such that

= e if and only if o(a)

I k.

Proof. (1) If an i- e for any n > 0, then aT i- as for any r i- s since aT = as
implies a T- S = e = a s - T, and if r i- s, then r - s > 0 or s - r > 0, which is

excluded by our hypothesis. Thus, if an i- e for n > 0, then I(a) I = 00, so
o(a) = 00. If an = e then let am be any element of (a). Writing m = qn + r
where 0 ::; r < n we see that am = a nq +T = anqaT = (an)qa T = eqaT = aT.
Thus (a) = {e, a, a 2 , ... , an-I} and o(a)::; n < 00.
(2) By part (1), if o(a) < 00 then there is an n > 0 such that an = e and
for each such n the argument in (1) shows that (a) = {e, a, ... , an-I}. If
we choose n as the smallest positive integer such that an = e then we claim
that the powers a i are all distinct for 0 ::; i ::; n - 1. Suppose that a i = a j
for 0 ::; i < j ::; n - 1. Then aj - i = e and 0 < j - i < n, contradicting the
choice of n. Thus o(a) = n = smallest positive integer such that an = e.
(3) Assume that a k = e, let n = o(a), and write k = nq + r where
o ::; r < n. Then e = a k = a nq +T = anqa T = aT. Part (2) shows that we
must have r = 0 so that k = nq.
D

We will now characterize all subgroups of cyclic groups. We start with
the group Z.
(2.11) Theorem. If H is a subgroup of Z then H consists of all the multiples
of a fixed integer m, i.e., H = (m).
Proof. If H = {O} we are done. Otherwise H contains a positive integer
since H contains both nand -n whenever it contains n. Let m be the
least positive integer in H. Then we claim that H = {km : k E Z} = (m).
Indeed, let n E H. Then write n = qm + r where 0 ::; r < m. Since n E H
and m E H, it follows that r = n - qm E H because H is a subgroup of
Z. But 0 ::; r < m so the choice of m forces r = 0, otherwise r is a smaller
positive integer in H than m. Hence n = qm so that every element of H is
D
a multiple of m, as required.

We now determine all subgroups of a cyclic group G. Assume that G =

i- {e}. If H contains a power
a -m with a negative exponent then it also contains the inverse am, which
is a positive power of a. Arguing as in Theorem 2.11, let m be the smallest
positive integer such that am E H. Let as be an arbitrary element of Hand
write s = qm + r where 0 ::; r < m. Then aT = a s - qm = as(am)-q E H
since as and am are in H. Thus we must have r = 0 since r < m and
m is the smallest positive integer with am E H. Therefore, s = qm and
as = (am)q so that all elements of H are powers of am.
If a is of finite order n so that an = e then n must be divisible by
m because e = an E H so that n = qm for some q. In this case, H =

(a) and let H be a subgroup of G such that H

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1.2 Subgroups and Cosets

11

{e, am, a 2m , ... ,a(q-l)m}. Therefore, IHI = q = n/m. However, if the order
of a is infinite, then H = {e, a±m, a±2m, ... } = (am) is also infinite cyclic.
Thus we have proved the following result.

(2.12) Theorem. Any subgroup H of a cyclic group G = (a) is cyclic. Moreover, either H = (e) or H = (am) where m is the smallest positive power of
a that is in H. If G is infinite then m is arbitrary and H is infinite cyclic.
If IGI = n then min and IHI = n/m. If m is any factor of n then there is
exactly one subgroup H of G of order n/m, namely, H = (am).
The above theorem gives a complete description of cyclic groups and
their subgroups. From this description, it is easy to see that any two cyclic

groups of order n are isomorphic, as well as any two infinite cyclic groups
are isomorphic. Indeed, if G = (a) and H = (b) where IGI = IHI = n then
define! : G -+ H by !(a m ) = bm for all m. One checks that f is a group
isomorphism. In particular, every cyclic group of order n is isomorphic to
the additive group Zn of integers modulo n (see Example 1.2 (3)), and any
infinite cyclic group is isomorphic to the additive group Z.

(2.13) Definition. Let G be a group and H a subgroup. For a fixed element
a E G we define two subsets of G:
(1) The
The
(2) The
H}.

left coset of H in G determined by a is the set aH = {ah : h E H}.
element a is called a representative of the left coset aH.
right coset of H in G determined by a is the set H a = {ha : h E
The element a is called a representative of the right coset H a.

Remark. Unfortunately, there is no unanimity on this definition in the mathematical world. Some authors define left and right cosets as we do; others
have the definitions reversed.
A given left or right coset of H can have many different representatives.
The following lemma gives a criterion for two elements to represent the same
coset.

(2.14) Lemma. Let H be a subgroup of G and let a, bEG. Then
(1) aH = bH if and only if a-1b E H, and
(2) Ha = Hb if and only if ab- I E H.
Proof. We give the proof of (1). Suppose a-1b E H and let b = ah for some
hE H. Then bh' = a(hh') for all h' E Hand ah l = (ah)(h-Ih l ) = b(h-Iht}

for all hI E H. Thus aH = bH. Conversely, suppose aH = bH. Then
b = be = ah for some h E H. Therefore, a-Ib = h E H.
0

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12

Chapter 1. Groups

(2.15) Theorem. Let H be a subgroup of G. Then the left cosets (right cosets)
of H form a partition of G.
Proof. Define a relation L on G by setting a "'L b if and only if a- 1b E H.
Note that

(1) a"'La,
(2) a "'L b implies b "'L a (since a- 1b E H implies that b- 1a = (a- 1b)-1 E
H), and
(3) a "'L band b "'L c implies a "'L c.
Thus, L is an equivalence relation on G and the equivalence classes of L,
denoted [alL, partition G. (See the appendix.) That is, the equivalence
classes [alL and [b]L are identical or they do not intersect. But
[alL

= {b E G : a "'L b}
= {b E G : a- 1 b E H}
= {b E G : b = ah for some

h E H}

=aH.

Thus, the left cosets of H partition G and similarly for the right cosets.

0

The function ¢a : H --+ aH defined by ¢a(h) = ah is bijective by
the left cancellation property. Thus, every left coset of H has the same
cardinality as H, i.e., laHI = IHI for every a E G. Similarly, by the right
cancellation law the function 1/Ja(h) = ha from H to Ha is bijective so that
every right coset of H also has the same cardinality as H. In particular,
all right and left cosets of H have the same cardinality, namely, that of H
itself.
(2.16) Definition. If H is a subgroup of G we define the index of H in G,
denoted [G : H], to be the number of left cosets of H in G. The left cosets
of H in G are in one-to-one correspondence with the right cosets via the
correspondence aH +-+ H a -1 = (aH) -1. Therefore, [G : H] is also the
number of right cosets of H in G.
(2.17) Theorem. (Lagrange) If H is a subgroup of a finite group G, then
[G: H] = IGI/IHI, and in particular, IHI divides IGI.
Proof. The left cosets of H partition G into [G : H] sets, each of which has
exactly IHI elements. Thus, IGI = [G: H]IHI.
0

(2.18) Corollary. If G is a finite group and a E G then o(a)

IIGI.
o

Proof.

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1.2 Subgroups and Cosets

(2.19) Corollary. If IGI

= n,

then an

=e

for all a E G.

Proof.

(2.20) Corollary. If IGI

13

D

=p

where p is prime, then G is a cyclic group.

Proof. Choose a E G with a i- e and consider the subgroup H = (a). Then

H i- {e}, and since IHI I IGI = p, it follows that IHI = p, so H = G.
D

(2.21) Remark. The converse of Theorem 2.17 is false in the sense that if
m is an integer dividing IGI, then there need not exist a subgroup H of G
with IHI = m. A counterexample is given in Exercise 31. It is true, however,
when m is prime. This will be proved in Theorem 4.7.
(2.22) Definition. If G is any group, then the exponent of G is the smallest
natural number n such that an = e for all a E G. If no such n exists, we
say that G has infinite exponent.
If IGI < 00, then Corollaries 2.18 and 2.19 show that the exponent of
G divides the order of G.
There is a simple multiplication formula relating indices for a chain of
subgroups K <:;; H <:;; G.

(2.23) Proposition. Let G be a group and H, K subgroups with K <:;; H. If
[G : KJ < 00 then
[G: KJ = [G: H][H : KJ.
Proof. Choose one representative ai (1 ::; i ::; [G: HJ) for each left coset of
H in G and one representative bj (1 ::; j ::; [H : KJ) for each left coset of
K in H. Then we claim that the set

consists of exactly one representative from each left coset of K in G. To
see this, let cK be a left coset of K in G. Then c E aiH for a unique
ai so that c = aih. Then h E bjK for a unique bj so that c = aibjk for
uniquely determined ai, bj k. Therefore, cK = aibjK for unique ai, bj , and
we conclude that the number ofleft cosets of Kin G is [G : H][H : KJ. D

(2.24) Remark. If IGI < 00 then Proposition 2.23 follows immediately
from Lagrange's theorem. Indeed, in this case [G : KJ = IGI/IKI =

(IGI/IHI)(IHI/IKI) = [G : H][H : KJ.
(2.25) Examples.
(1) If G = Z and H = 2Z is the subgroup of even integers, then the
cosets of H consist of the even integers and the odd integers. Thus,

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14

Chapter 1. Groups

[Z : 2Z] = 2. Since Z is abelian, it is not necessary to distinguish
between left and right cosets.
(2) If G = Z and H = nZ, then [Z : nZ] = n where the coset m+H consists
of all integers that have the same remainder as m upon division by n.
(3) Let G = 8 3 = {e, a,a 2 , (3, a(3, a 2 (3} where a =
and (3 =
(~
If H = ((3), then the left cosets of H in G are

nii)

i ; ).

H={e,(3}

aH = {a, a(3}

while the right cosets are

H = {e, (3}
Note that, in this example, left cosets are not the same as right cosets.
(4) Let G = GL(2, R) and let H = SL(2, R). Then A, B E GL(2, R) are
in the same left coset of H if and only if A-I B E H, which means that
det(A -1 B) = 1. This happens if and only if det A = det B. Similarly,
A and B are in the same right coset of H if and only if det A = det B.
Thus in this example, left cosets of H are also right cosets of H. A set
of coset representatives consists of the matrices

{[~ ~]

:aER*}.

Therefore, the set of cosets of H in G is in one-to-one correspondence
with the set of nonzero real numbers.
(5) Groups of order :::; 5. Let G be a group with IGI :::; 5. If IGI = 1, 2, 3, or
5 then Corollary 2.20 shows that G is cyclic. Suppose now that IGI = 4.
Then every element a =1= e E G has order 2 or 4. If G has an element a
of order 4 then G = (a) and G is cyclic. If G does not have any element
of order 4 then G = {e, a, b, c} where a 2 = b2 = c2 = e since each
nonidentity element must have order 2. Now consider the product abo If
ab = e then ab = a 2 , so b = a by cancellation. But a and b are distinct
elements. Similarly, ab cannot be a or b, so we must have ab = C. A
similar argument shows that ba = c, ac = b = ca, bc = a = cb. Thus, G
has the Cayley diagram of the Klein 4-group. Therefore, we have shown
that there are exactly two nonisomorphic groups of order 4, namely,
the cyclic group of order 4 and the Klein 4-group.
The left cosets of a subgroup were seen (in the proof of Theorem 2.14)
to be a partition of G by describing an explicit equivalence relation on G.
There are other important equivalence relations that can be defined on a

group G. We will conclude this section by describing one such equivalence
relation.

(2.26) Definition. Let G be a group and let a, bEG. Then a is conjugate to
b if there is agE G such that b = gag-I. It is easy to check that conjugacy

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