Introduction to Modern Abstract Algebra

This book is in the ADDISON-WESLEY SERIES IN MATHEMATICS

Ocnuultiftl/

Editor: LYNN H. LoOMI8

D A V I D M. BUR TON, University

of New

Hampshire

INTRO-DUCTION

TO

MOO'ERN ABSTRACT

ALGEBRA

ADDISON-WESLEY PUBLISHING COMPANY Menlo PIITk, California

.

London

.

Amatflfdem

•

ReedIng, M8ItIaChuIItItIt Don MIlls. 0nfarl0 S)IdIIey .

COPYRIGHT ® 1967 IIY AIlIII~{)N-Wt:~Lt:Y l'URLI~IIING COMrANY, INC. ALL RIGIITS Rt:St:RV}:/). TIII~ 1I00K, on l'AltT~ Tlnau:(w, MAY NOT III': Rt:I'IUJIlUCt:1> IN ANY FOUM WITHOUT WIUT'n:N I't:nMI~HION OF Til}: l'UIILISln:n. 1'liINT}:/) IN TIn: UNITt:!) STATES OF AMt:RICA. I'UIILlsllt:1> SIMULTANt:OUHLY IN CANAI>A. LIBRAUY OF CONGIlESS CATALOG CAIlII NO. 67-19426. , _ 0·201.00722·3 IJKLMNOPO-MA 19818

PREFACE

This book has been written with the intention of providing an introduction to model'll aht:!t.ract algebra for mathematics major8. The r('.ader is not presumed at thc Outllct to possess any previout:! knowledge of the concepts of modem algebra. Accordingly, our beginuing is somewhat elementary, with the exposition in the earlier sections proceeding at a leisurely pace; much of this early material may be covered rapidly on a first reading. An attempt. has been made to keep the book as self-contained as possible. To smooth the path for the unexperienced reader, the fir8t chapter is devoted to a review of the basic facts concerning sets, functions and number theory; it also serves as a suitable vehicle for introducing some of the notation and terminology used subsequently. A cursory examination of the table of contents will reveal few surprises; the topics chosen for discussion in COUr8es at this level are fairly standard. However, our aim has been to give a presentation which is logically developed, precise, and in keeping with the spirit of the times. Thus, set notation is employed throughout, and the distinction is maintained between algebraic systems as ordered pair8 or triples and their underlying sets of elements. Guided by the principII! thnt 0. st(~l"ly diet. of definitions and cxullIples SOOIl 1)(~c()llIel'l unpalatable, our eiTOl-ts are directed towards establishing the most important and fruitful results of the subject in a formal, rigorous fashion. The chapter on groups, fOl' example, culminates in a proof of the classic Sylow Theorems, while ring and ideal theory are developed to the point of obtaining the Stoile Representation Theorem for Boolean rings. Ell route, it is hoped that the, reader will gain an appreciation of precise mathematical thought and t.he current standards of rigor. At the eud of each section, there will be found a collection of problems of varying degrees of difficulty; these constitute an integral part of the hook. They introduce a variety of topics not treated in the main t.('xt, as well as impart much additional detail ahout material covered earlier. Home, especially in the latter seet.ions, pl'Ovide slIbl'ltantial extensionl'l of t.he'theory. We have, on the whol(', resist.('d t.h(' t.('mptation to lise the exercises to develop results that will be ncedl'd HllbsequC'nt.ly; aN a 1·(,~lIlt., the reader need not work all the problems in order t.o read the reHt of the hook. Problems whose solut.ions do not appear particularly straiJ!;htforward arc accompanied by hints, Besides the general index, a glossary of !!pecial Hymhol!! iN also included. v

vi

PREFACE

The text is not intended to be encyclopedic in nature; many importaut topics vie for inclusion and some choice is obviously imperative. To this end, we merely followed our own taste, condensing or omitting altogether certain of the concepts found in the usual first course in modem algebra. Despite these omissions, we believe the coverage will meet the needs of most students; those who are stimulated to pursue the matter further will have a finn foundation upon which to build. It is a pleasure to record our indebtedness to Lynn Loomis and Frederick Hoffman, both of whom read the original manuscript and offered valuable criticism for its correction and improvement. Of our colleagues at the University of New Hampshire, the advice of Edward Batho and Robb Jacoby proved particularly h~lpful; in this regard, special thanks are due to William Witthoft who contributed a number of incisive suggestions after reading portions of the galley proofs. We also take this occasion to express our sincere appreciation to Mary Ann MacIlvaine for her excellent typing of the manuscript. To my wife must go the largest debt of gratitude, not only for her generous assistance with the text at the various stages of its development, but for her constant encouragement and understanding. Finally, we would like to acknowledge the fine cooperation of the staff of Addison-Wesley and the usual high quality of their work.

Durham, New H amp8hirc March 1967

..D.M.B.

CONTENTS

Chapter 1 1-1

Preliminary Notion.

The Algebra of Sets

1-2 Functions and Elementary Number Theory Chapter 2

2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2-9 Chapter 3

3-1 3-2 3-3 3-4 3-5 3-6 Chapter 4 4-1

1 13

Group Theory

Definition and Examples of Groups . Certain Elementary Theorems on Groupf! Two Important Groups Subgroups Normal Subgroups and Quotient Groups Homomorphisms The }<'undamental Theorems . The Jordan-Holder Theorem . Sylow Theorems

27 41 52 64

75 89 103 117 128

RI. . Theory

Definition and Elementary Properties of Rings Ideals and Quotient Rings Fields Certain Special Ideals . Polynomial Rings Boolean Rings and Boolean Algebras

141 156 '172 183 196 219

Vector Spec..

The Algebra of Matrices

4-2 Elementary Properties of Vector Spaccli 4-3 HaMes a.nd Dimension 4-4 Linear Mappings

235 249 263 277

Selected Reference.

297

Inde. of Special Symbol. and Nototion. ,

299

Inde.

305 vii

CHAPTER 1

PRELIMINARY NOTIONS

1-1 THE ALGEBRA OF SETS

This ehaptcr hrieAy summarizes IIOme of the bll.llic notions concerning sets, functiollH, and number theory; it alllO serves Itl! It vehicle for establishing COIlventions in notation and terminology used throughout the text. Inasmuch as this material is intended to serve primarily for background purposes, the reader who is already acquainted with the ideas in this chapter may prefer to cmbark directly 011 the next. Within the confines of one section, it is obviously impossible to give complete coverage to set theory or, for that matter, to achieve a logically coherent exposition of such a formalil!tic diseipline. The subsequent presentation should thus be regarded simply as a summary of the fundamental aspects of the subject, and not as a sYRtematic development. Rather than attempt to list the undefined terms of set theory and the various axioms relnting them, we shall take an informal or naive approach to the subjeet. To thil! cnd, the term set will be intuitively understood to mean II. collection of objects having some common characteristic. The objects that make up a given sct arc called its elements or members. Sets will generally be designated by capital letters and their elements by small letters. In particular, we shall employ the following notations: Z is the set of integers, Q the set of rational numbers, and R' the set of real numbers. The symbols Z+, Q+, and R~ will stand for the positive elements of these sets. If x is an element of the set A, it is customary to use the notation x E A and to read the symbol E as "belongs to." On the other hand, when x fails to be an element of the set A, we shall denote this by writing x I;l A. There arc two common methods of specifying a particular set. First, we may list all of its elements within braces, as with the set {-I, 0,1, 2}, or merely list some of its elements and use three dots to indicate the fact that certain obvious clements have been omitted, as with the set {I, 2, 3,4, ... }. When such a liHting il! not practical, we may indicate instead a characteristic property whereby we can determine whether or not a given object is an element of the set. IHore specifically, if P(x) is a statement concerning x, then the set of all elementR x for whieh the J;tatcmcnt P(x) is true is denoted by {x I P(x)}. For example, we might have {x I x is an odd integer greater than 2I}. Clearly, 1

vi

PREFACE

The text is not intended to be encyclopedic in nature; many important topics vie for inclutlioll and some choice ill obviously imperative. To this end, we merely followed our own taste, condensing or omitting altogether certain of the concepts found in the usual first course in modem algebra. Despite these omissions, we believe the coverage will meet the needs of most students; those who are stimulated to pursue the matter further will have a firm foundation upon which to build. It is a pleasure to record our indebtedness to Lynn Loomis and Frederick Hoffman, both of whom read the original manuscript and offered valuable criticism for its correction and improvement. Of our colleagues at the University of New Hampshire, the advice of Edward Batho and Robb Jacoby proved particularly ht'lpful; in this regard, special thanks are due to William Witthoft who contributed a number of incisive suggestions after reading portions of the galley proofs. We also take this occasion to express our sincere appreciation to Mary Ann Ma.eIlvaine for her excellent typing of the manuscript. To my wife must go the largest debt of gratitude, not only for her generous assistance with the text at the various stages of its development, but for her constant encouragement and understanding. Finally, we would like to acknowledge the tine cooperation of the staff of Addison-Wesley and the usual high quality of their work. Durham, New Ifampshire March 1967

..n.M.B.

CONTENTS

Chapter 1

Preliminary Notion.

1-1 The Algebra of Sets 1-2 functions and Elementary Number Theory Chapter 2

2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2-9 Chapter 3

Group Thoory

Definition and Examples of Groups . Certain Elementary Theorems on GroupR Two Important Groups Subgroups Normal Subgroups and Quotient Groups Homomorphisms The l!'undamental Theorems . The Jordan-Holder Theorem . Sylow Theorems

Definition and Elementary Properties of Rings Ideals and Quotient Rings Fields Certain Special Ideals. Polynomial Rings . 3-6 Boolean Rings and Boolean Algebras

4-1

27 41 52 64

75 89 103 117 128

Rlnt Theory

3-1 3-2 3-3 3-4 3-5

Chapter 4

1 13

141 156 '172 183 196 219

Vector Spac..

The Algebra of MatriceR

4-2 l!~lementary Propertie.'l of Vector Spaces 4-3 U8l!C1I and Dimension 4-4 Linear Mappings

235 249 263 277

Selected Reference.

297

Indox of Spoeial Symbol. and Notation. .

299

Index

305

vii

2

I-I

PRELIMINARY NOTIONS

certain sets may be described both ways: {O, I} = {x / x E Z and x 2 = x}.

It is customary, however, to depart slightly from this notation and write {x E A / P(x)} instead of {x / x E A and P(x)}.

Definition 1-1. Two sets A and B are said to be equal, written A = B, if and only if every clement of A is an element of B and every element of B is an element of A. That is, A = B provided A and B have the same elements. Thus a set is completely determined by its elements. For instance,

{I, 2, 3}

=

{3,I,2,2},

since each set contains only the integers 1, 2 and 3. Indeed, the order in which the elements are listed in a set is immaterial, and repetition conveys no additional information shout the ad. An empty set or null set, represented by the symbol 0, is any Bet which has no elements. For instance,

o=

{x E R' I x 2

< O}

or

0= {X/XFX}.

Any two empty sets arc equal, for in a trivial sense they both contain the same elements (namely, none). In effect, then, there is just one empty set, so that we are free to speak of the empty 8et 0. The set whose only member is the element x is called 8ingleton x and it is denoted by {x}: {x} = {y I y = x}. In particular, {O} F 0 sincll 0 E {O}.

Definition 1-2. The set A is a BUb8et of, or is contained in, the set B, indicated by writing A ~ B, if every element of A is also an element of B. Our notation is designed to include the possibility that A = B. Whenever ~ B but A F B, we will write A C B and say that A is a proper BUbset of B. It will be convenient to regard all sets under consideration as being subsets of some master set U, called the universe (universal 8et, ground set). While the universe may he diffC'rent in different contexts, it will usually be fixed throughout any given difol(~llllllion. There arc several immediate (:onscquenc(~s of the definition of sct inclusion.

A

TheOrem 1-1. If A, B, and C are subsets of some universe U, then a) A ~ A, 0 ~ A, A ~ cr, b) A ~ 0 if and only if A = 0, c) {x} ~ A if and only if x E A; that is, each clement of A determines a subset of A,

1-1

THE ALGEBRA OF SETS

d) if A 5;; Band B 5;; C, then A 5;; C, e) A 5;; Band B 5;; A if and only if A

=

3

B.

Observe that the result 0 5;; A follows from the logical principle that a false hypothesis implies any conclusion whatsoever. Thus, the statement "if x e 0, then x e A" is true since x e 0 is always false. The last assertion of Theorem 1-1 indicates that a proof of the equality of two specified sets A and R is generally presented in two parts. One part demonstrates that if x e A, then x e B; the other part demonstrates that if x e B, then x e A. ' An illustration of such a proof will be given shortly. We now consider several important ways in which sets may be combined with one another. If A amI B are subsets of some universe U, the operations of union, intersection, and difference arc defined as follows. Deftnition 1-3. The union of A and B, denoted by Au B, is the subset

of U defined by A U B = {x I x

e

A or x

e

IJ} •

The intersection of A and B, denoted by A n B, is the subset of U defined by

AnB= {xlxeAandxEB}. The difference of A and B (sometimes called the relative complement of B in A), denoted by A - B, is the subset of U defined by

A - B

=

{x I x E A but x ~ B}.

In the definition of union, the word "or" is used in the "and/or" sense. Thus the statement "x E A or x E B" allows t.he possibility that x is in both A and B. It might also he Jlot(ld par(mthetiml.\ly that, utilizing this new notion, we could define A to be a proper subset of B provided A ~ B with B - A ~ 0. The particular difference U - B is called the (absolute) complement of B and designated simply by -B. If A and B are two nonempty sets whose intersection is empty, that is, A n B = 0, then they are said to be disjoint. We shall illustrate these concepts with an example. Examp.e 1-1. Let the universe be U

and the set B

=

= {O, 1,2,3,4,5, 6}, the set A = {I, 2, 4},

{2, 3, 5}. Then

A uB = {1,2,a,4,5},

A

n JJ

=

{2},

A - B

and

B - A

=

{3,5}.

Also, -A = {O, 3, 5, 6},

-B

Observe that A - B !lnd B - A are disjoint.

=

{O, 1,4, 6}.

=

{t,4},

4

1-1

PRELIMINARY NOTIONS

In the following theorem, some simple consequences of the definitions of union, intersection, and complementation are listed. Theorem 1-2. If A, B, and C are subsets of some universe U, then a) A U A = A, AnA = A, b) A uB = B u A, A nB = B nA, e) A U (B u 0) = (A u B) u A n (B nO) = (A n B) n C,

c,

d) A u (B nO) = (A u B) n (A A n (B u C) = (A n B) u (A e) A U 0 = A, A n 0 = 0, f) A u U = U, A n U = A.

u C),

nO),

We shall verify the first equality of (d), since its proof illustrates a technique mentioned previously. Suppose that x E Au (B n C). Then, either x E A or x E B n C. Now, if x E A, then clearly both x E Au Band x E A U C, so that x E (A u B) n (A u 0). On the other hand, if x E B n C, then x E B and therefore x E A u B; also x E C and therefore x E A u C. The two conditions together imply that x E (A U B) n (A u C).

This establishes the inclusion, A

u (B n 0) ~ (A u B) n (A u 0).

Conversely, suppose x E (A U B) n (A u 0). Thcll both x E A U Band x E AU C. Since x E A U B, either x E A or x E Bi at the same time, since x E A U C, either x E A or x E C. Together, theMel conditions mean that x E A or x E B n C; that is, x E A u (8 n C). This proves the opposite inclusion, (A U B) n (A u 0) ~ A u (B n C). By part (e) of Theorem 1-1, the two inclusions are sufficient to establish the equality, A u (B n C) = (A u 8) n (A u C). If A, B. and C are sets such that C ~ A and C ~ B, then it is clear that A n B. Thus it is possible to think of A n B as the largcst set which is a· subset of both A and B. Analogously, A U B may he interpreted as the smallcBl set which contains hoth A Rnd B. The next t1worcm relates the operation of complementation to other operations of set theory.

C

~

1-1

THE ALGEBRA OF SETS

5

Theorem 1-3. Let A amI B be subsets of the universe U. Then

a) -(A U B) = (-A) n (-B), b) -(A n B) = (-A) U (-B), c) if A ~ B, then (-B) ~ (-A), d) -(-A) = A, _o~ = U, -V = 0, e) A u (-A) = V, A n (-A) = 0.

To give the reader a little more familiarity with set-theoretic argument, we shall establish the first of the above assertions. For the proof, let x be an arbitrary (!Icmcnt of -(A U B). Then x ~ A U B. Hence x is in neither A nor n. Thill implicH t.lmt x E - A ami x E n, from which it follows that x E (-A) n (-B). Thus -(A u B) ~ (-A) n (-B). Conversely, if x E ( - A) n (-B), then x belongs to both - A and -B. In other words, x ~ A and x ~ B. This guarantees x ~ A U B, that is 0_0

xE-(AUB). We consequently have the inclusion (-A) desired equality holds.

n (-B)

~

-(A

u B)

and the

The first two parts of the above theorem are commonly known as DeMorgan'8 rules. There will be occasions when we wish to consider sets whose elements themselves are sets; in order to avoid the awkward repetition "set of sets," we shall frequently refer to these as families of sets. One family which will prove to be of considerable importance is the so-called power set of a given set. Definition 1-4. If A iii I1.n nrhit.mry l'Ict, thcn t1w !let WhOHC clements are nl\ the HuhRI't.H of A ill known nil the power 8(~t of A allli dmlOtcu by P(A):

peA)

=

{B I B ~ A}.

A few remarks arc in order before considering a specific example. First, since 0 ~ A and A ~ A, we always have {0, A} ~ peA) no matter what the nature of the set A. (If A = 0, then of course peA) = {0}.) The next thing to observe is that if x E A, then {x} ~ A, hence {x} E peA). From this, we infer that the power set of A has, at the very least, as many elements as the set A. Indeed, it ean be shown that whenever A is a finite set with n elements, then peA) is itself It finite set having 2 n elements. For this reason, the power set of A iH oft!'11 represented hy the Hymbol 2A. Example 1-2. Huppol'll! t.he l'Id A ,~ (fl, Ii, c}. Tht· aI:I

POWC1'

tid of A, whirh has

its c11:mclItH all the subl:lCts of fa, b, c}, is then

P(A) =

{0, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, A}.

6

1-1

PRELIMINARY NOTIONS

It is both desirable and possible to extend our definitions of union and intersection from two sets to any number of sets. Assume to this end that a is a nonempty family of subsets of the universe U. The union and intersection of this arbitrary family are defined by,

ua = na =

{x / x E A for some set A E a}, {x / x E A for every set A E a}.

At times we will resort to an indexing set to define these notions. To be more precise, let 1 be a set, finite or infinite, and with each i E 1 associate a set A,. The fCl!ulting family of sets,

a=

{Ad i E I},

is then said to be indexed by the elements of I, and the set 1 is called an index Bet for a. Wh(ln t.hifl notation is employed, it is customary to denote the union and intcrHCction of the fnmily a by and

n{Ai / i E f}.

If the nature of the index set 1 is clearly understood or if the emphasizing of

it is inessential for some reason, we simply write,

-

and Example 1-3. If A"

=

{x E R' / -lin ~ x ~ lin} for n E Z+, then

u{A .. / n E Z+}

=

{x / x E A .. for some n E Z+}

=

n{A .. / n E Z+}

=

{x / x E A .. for every n E Z+} = {O}.

At,

In passing, we should note that by a chain of sets is meant a nonempty family e of subsets of some universe U such that if A, BEe then either As; B or B S; A. For instance, the family in Example 1-3 constitutes a chain of sets. From our definition of set equality, {a, b} = {b, a}, since both sets contain the same two elements a and b. That is, no preference is given to one element over the other. When we wish to distinguish one of these elements as"being the first, say a, we write (a, b) and call this an ordered pair. It is possible to give a purely set-theoretic definition of the notion of an " ordered pair as follows: Definition 1-5. Tbe ordered flair of el(!ments a and b, with its first component a and lK!cond component b, denoted by (a, b), ill the set

(a, b) = {{a,b}, {a}}.

I-I

THE ALGEBRA OF SETS

7

Note that according to this definition, a and b are not elements of (a, b) but rather components. The actual elements of the set (a, b) are {a, b}, the unordered pair involved, and {a}, that member of the unordered pair which has been selected to be first. This agrees with our intuition that an ordered pair should be an entity representing two elements in a given order. For a ~ b, the sets {{a, b}, {a}} and {{b, a}, {b}} are unequal, having different elements, so that (a, b) ~ (b, a). Hence, if a and b are distinct, there are two distinct ordered pairs whose components are a and b, namely, the pairs (a, b) and (b, a). Ordered pairs thus provide a way of handling two things as one while losing track of neither. In the next th(!Orem, a useful criterion for the equality of ordered pairs is obtained; the proof ill subtle, but simple, relying mainly on Definitions 1-1 and 1-5. Theorem 1-4. Two ordered pairs (a, b) and (e, d) are equal if and only if = e and b = (l.

a

Proof, If a

=

e and b

=

{a}

d, then it is clear from Definition 1-1 that

=

{e}

and

{a, b} = {e, d},

This in tum implies {{a, b}, {a}} = {{e, d}, {e}}, whence (a, b) = (e, d). As for the converse, suppose that {{a, b}, {a}} = {{e, d}, {en. We distinguish two possible cases: 1) a = b. In this case, the ordered pair (a, b) reduces to a singleton, since (a, b)

=

(a, a)

=

{{a,a}, {a}}

=

{{a}}.

According to our hypothesis, we then have

{{an = {{e, d}, {en, which means {a} = {e, d} = {e}. From this, it follows that the four elements a, b, e, d are all equal. 2) a ~ b. Here, both {a} ~ {a, b} and {e} ~ {a, b}. If the latter equality were to hold, we would have e = a and e = b, hence the contradiction a = b. Now, by virtue of the hypothesis, each member of the set (e, d) belongs to (a, b); in particular, {e} E {{a,b}, {a}}. This means that {e} = {a} and accordingly a = e. Ap;nill hy KIIPPOKitioll, (a, II} E f fe, (l}, fr.}} with {a, b} ~ {e}. It may thus be inferred that la, b1 = {c, d} lUlll therefore b E {e, d}. As b cannot equal e (this would imply that a = b), we must conclude that b = d. In either case the desired result is established.

8

1-1

PRELIMINARY NOTIONS

Having faced the problem of defining ordered pairs, it is natural to consider ordered tripl('s, ordered quadruples and, for that matter, ordered n-tuples. What simplifies the situation is that these notions can be formulated in terms of ordered pairs. For instance, the ordered triple of a, b, and c is just an ordered pair whose first component is itself an ordered pair: (a, b, c)

= (a, b), c).

Assuming that ordered (n - I)-tuples have been defined, we shall take the ordered n-tuple of alt a2, ... ,an to mean the ordered pair (a" a2, ... ,a,.-I), a,.), abbreviated by (a" a2, ... ,an), It should come as no surpriMC that two ordered n-tuples equal whenever their corresponding components are equal; in other words, (at. a2, .•. , a,.) = (b" b2, .•. , bn )

if and only if ak

=

bk for k

=

1,2, ... ,n.

Definition 1-6. The Carte8ian product of two nonempty sets A and B, designated by A X B, is the set

A XB

=

{(a, b) I a E A and b E B}.

Whenever we employ the Cartesian product notation, it will be with the understanding that the sets involved are nonempty, even though this may not be explicitly stated at the time. Observe that if the set A contains n elements and B contains m elements, then A X B has nm elements, which accounts for the use of the word "product" in CartcRian product. Example 1-4. Let A

=

{-I,O, I} and B

=

{0,2}. Then,

A X B = {(-I, 0), (-1,2), (0,0), (0, 2), (1,0), (1, 2)},

while B XA

=

{CO, -1), (0,0), (0, 1), (2, -1), (2,0), (2, I)}.

Clearly the sets A X Band B X A are not identical. In fact, A X B = B X A if and only if A = B. By a (binary) rrlatinn in a nOllf'lIIpt.y Ret A if:! meant a subset R of the CartcRian A X A. H t.h(~ e1elllent (a, b) E R, wc KUY thllt a iH related to b with respect to the relat.ion R and write aRb. For instance, the relation < in R' consists of al\ points in the plane lying above the line y = x; one usually writes 3 < 4 rather than the awkward (3,4) E <. Frankly, the concept of a relation as defined is far too general for our purposes. We shall instead limit our attention to a specialized relation known as an equivalence relation. prod\l(~t

1-1

THE ALGEBRA OF SETS

9

Definition 1-7. A rl'lllt.ion Il in It liet, A iii 8nid to he 11I1 (!quivalence relation ill A provided it Ilatislied the threc propertie8, 1) refll'xive propl'rt.y: alla, for eaeh a E A, 2) Kyn\llletril~ proJ)('rt,y: if allb for IiOmc a, b E A, thcn blla, :i) tram!itive property: if allb and bUe for IiOllle a, b, t: E A, thcn aRc.

Equivalence relations arc customarily denoted by the symbol,.., (pronounced "wiggle"). With this change in notation, the conditions of Definition 1-7 may be reHtatcd in a more familiar form: J) 0"" 0, for I'l\I'h 0 E A, 2) a ,.., b implic8 b ,.., a, 3) both a ,.., band b ,.., r. imply a ..., c.

In tJlI~ following (lxnmpII'K, WI' h~lwc to t.ho rt'ndor t,h(l t.ILl'Ik of verifying that each rdlltion dCH(~ril)(!(1 Ilet,unIly iii lUI cfJuivulmwc relation. Example 1-5. Let A be an arbitrary nonempty set and define for a, bE A" a..., b if nnd only if a = b (a = b is tacitly interpreted to mean that a and b are identical clements of A). This yields an equivalence relation in A. Example 1-6. Consider the set L of all lilies in a fixed plane and let a, bEL. Then,.., is an cquivalenee relation in [J provided a,.., b means that a is parallel to b; let, us agrce thnt any line is parallel to itl«)lf. Example 1-7. Take Z to be the set of integers. Given a, b E Z, we define an cquivalclwc reln'tion ,.., in Z by requiring thnt a ,.., b if nnd only if a - b E Z., the He!. of (wen int,l'gerM. Example 1-8. All a linal illustration, suppose A = Z+ X Z+ and define (a, b) ,.., (e, d) to mean ad = be. A simplc cldculation reveals this is an equivalence relation in A.

One is frequently led to conclude that the reflexive property is redundant in Definition 1-7. The argumcnt goes like this: If a,.., b, then the symmetric property implies b ,.., a; since a,.., band b,.., a, using the transitive property, it follows that a ,.., a. Thus, there appears to be no necessity for the reflexive eondition at all. The ftl~W in t.his reasoning IiI'S in t,he faet that for some clement a E A, tllI'rl' llIay not, I'xisl. any b E A Slid, I.hat, a ,.., h. AI~I~(mlil\j!;ly, we would 1101. IlIlv(' a ,.., (t fol' I'VI'I'Y 1I1I'lIilu·1' of It, liS llw I'l'fI('xiv(' propl'l'1.y l'("quin'M. I'(!rllltps lh(, prilwipal ,'('aSOIl for ('ollsidl'l'illK ("qllivall"If"" rclat.i()m; in a IiCt A is that they separat.e A illt.o certain (!onvcnient suhliets. To be more precise, supposp ,.., is a KivclI I'quivall'lIl'e relation ill A. For each a E A, we let [aJ denote the suhset of A I'onsisting of all clements which are equivalent to a:

[aJ = {x E A I x ,.., a}. This set [aJ is referred to as the equivalence class determined by a.

10

PRELIMINARY NOTIONS

1-1

Some of the baKil: properties of equivalence classes are listed in the next theorem. Theorem 1-5. J.A:~t - he an equivalence relation in the set A. Then,

1) for each a E A, [a] ¢ 0, 2) if b E [a], then raj = [I)]; that is, any clement of the equivalence class [aJ determines the cllUlS, 3) for any a, b E A, with faJ ¢ rbI, [a] n [b] = 0, 4) U{[a]laE A}

=

A.

Proof. Clearly, a E [a], since a-a. To prove (2), let bE [a], RO that b - a. Now, SIIPPOKI' ;r E [a). whil'" impIiI'R.1" - a. UKinK t.he KYlllmctril~ and transitive propcrt.il's of -, it follmvH that. :c - b, h('nl'c x E fbI. This estnhlislwH the inclusion [a] s;; [b]. A Himilllr argument. yil'lds the opposite in('lusion and thus the equality [a] = fbI. We derive (3) hy assuming, to the contrary, that there is some element C E [a] n fbI. Then by statement (2), which has just been verified, [a] = [e] = rbI, an obvious contradietion. Finally, since each clll.SS [a] s;; A, the inclusion U{[a] I a E A} s;; A ill apparent. To obtain the reverse inclusion one need only demonstrate that each clement a in A belongs to some equivalence c1aSllj but this is evident: if a E A, then a E [a].

We next conned. the idea of an equivalence relation in A with the concept of a partition of A. Definition 1-8. A partition of a set A iH a family {Ai} of nonempty subsets of A with the properties 1) if Ai ¢ Aj, then Ai n Aj = 0 (pairwise disjoint),

2) UAi

=

A.

Briefly, a partition of A is a family {Ai} of nonempty subsets of A such that every element of A belongs to one and only one member of {Ai}. The integers, for instance, have a partition ('onsisting of the sets of odd and even integers: n Zo = 0. Another partition of Z might be the sets Z+ Z = Z. u Zo, (positive integ<'rH), Z_ (nl'Kntive integers), and {O}. Theorem 1-5 may be viewed as asserting that if - is an equivalence relation in A, then the fnmily of nil l'(Juivalent'e classes (with respect to the relation -) forms a partition of A. We now reverse the situation and show that a given partition of A indu('l's a natural equivalence relation in A.

z.

Theorem 1-6. If {Ai} is a partition of the set A, then there is an equivalence

relation in A whose cquivalen('e classes are precisely the sets A j. Proof. For clements a, b E A, we take a - b if and only if a and b belong to the same subset A j . Th(' reader may check that the relation -, so defined, is

1-1

THE ALGEBRA OF SETS

11

actually an equivalence relation in A. Now suppose the clement a E Ai. Then b E Ai if and only if b "'" a, that is, if and only if b, E raj. This demonstrates the equality A. = raj. In summary, the nhove ClitlCUHHiOIl Hhows that there is no C14scntial distinction betwccn partitions of a set and equivalence relatiolls in the set; if we start with one, we get the other. Example 1-9. Let

A

=

R' X R' and define the relation"", by

(a, b) "'" (c, d) if and only if a - c

=

b - d.

TIII'II - is Ilil I'Iluivll1c-lIc'c- rc-laLion ill A. Thl! cquivlLlmwe dlLHH determined by the clement (a, 11) iii !limply [(a, b)J

=

{(c, d) I a - c

=

b -

d}.

This set may be represented geometrically as a straight line with slope 1 passing through the point (a, b). Therefore, the relation"", partitions A into a family of parallellinel!.

PROBLEMS

In the following excrciscl! :I, B, and C are subset!! of some universe U. I. Provo t.hat. A n B ~ .. I U B. 2. Suppose ,1 ~ B. Hhow that \ a) A n C ~ B n C, b) A U C ~ B U C. 3. Prove that A - B = A n (-B), and use this result to verify each of the following identities:

0 = :I, 0 - A = 0, ..t - .1 = 0, b) A - B = A - ( ..1 n B) = (11 U B) - B, c) (11 - B) n (B - A) = 0.

a) A -

4. Simplify the following expressions to one of the symbols A, B, II U B, An B, A -B: a) A n (A U B), b) A - (11 - B), c) n B) U (-.1».

-«..I

5. Prove that A n (B U C) = (A n B) U (A n C). 6. Establish the following reKults on differences: a) (A - B) - C = A - (B U C), b) .1 - (B - C) ... (II - B) U (A n C), c) A U (B - C) = (:I U B) - (C - A), d) .t n (B - C) = (A n B) - (.1 n C).

12

1-1

PRELIMINAlty NOTIONS

7. The notion of set indw;ion may be expressed either in terms of union or intersection. To see this, prove that a) AS:;; B if and only if .1 U B = B, b) ..l s:;; R if and only if .1 n R = :1. 8. a) If .1 s:;; /I nllli .1 ~ -II, prove I.hat :I .., tl. b) If A !; Hand -.1 !;; H, prove that B - ll.

9. Establish the two absorption laws: A U (A () B) = A,

n (A

A

U B) = A.

10. ASRume that :1, H, and Carll Ret.s for which AUB=,IUC

and

Prove t.hat B = C. [Hin.t: H = B () (B U .\).1

a = {.It, .h, ...} be a family of subsets indexed by the positive integers Z+. Define a new family m = {BI, B2, ... } as follows:

]1. Let

Bl = "h;

Bn

=

An - U{.h I k

1,2, ... , n - I} for n

=

>

1.

Show that a) the member" of Hare di!ljoint. !lets,

b)

ua

=

um.

12. For any thrlle ~ets .. t, JJ and C, establish t.hat a) .1 X (HUe) = (.1 X H) U (:1 X C), b) 11 X (B n C) = (.1 X B) n (.t X C), r) A X (B - e) = (ot x B) - (A X C), d) A X B = U{A X {b} I bE B}.

13. Classify earh of the following relations R in the set Z of integers as to whether they do or do not have the properties of being reflexive, symmetric, and transitive: a) aRb if and only if a < b, b) aRb if and only if a - b is an odd integer, c) IJRb if and only if ab ~ 0, d) aRb if and only if a 2 = b2 , e) aRb if and only if la - bl < I.

]4. Let S be a finite set, but ot.herwise arbitrary. Determine if the relations defined below are equivalenre relations in P(S): a) J,.., B means .1 s:;; H, b) .1 ,.., B meanll ..\ and B have the same number of elements. 15. How many distinct equivalence relations are there in a. set of 4 elements?

16. Prove that the following relations,.., are equivalence relation!! in the Cartesian product R' X R': a) (a, b) ,.., (e, d) if and only if b - d = m(a - c), m a fixed real number, b) (a, b) ,.., (e, d) if and only if a d = b c, c) (a, b) ,.., (e, d) if and only if a - c E Z, b = d.

+

+

1-2

l"UN(''TiONS ANn ELl!lMENTAltY NUMBER THEORY

13

1-2 FUNCTIONS AND ELEMENTARY NUMBER THEORY

Let us turn next to the concept of a function, one of the most important ideas in mathematics. We shall avoid the traditional view of a function as a "rule of ('Orrl~HJI(IIlClcfl(:e or nfl(ll!;ivI' inHt,\lad II. definit.ion ill U1rfllli of ordered pn.irs. What t.hili Inl.t.l'l· npprllueh l:Leks in f1utumluel!..'i is lIIorc thull I:ompemmted for by its clarity II.lId precision.

f is a set of ordered pairs Buch that pairs have the same first component. Thus (x, Yl) Ef and (x, Y2) E J implies Yl = Y2.

Definition 1-9. A Junction (or mapping)

no two

distill(~t

TIll! ",)Jl(~(:t.illfl of all lirlit ('WIIJlUfI(!IItli IIf II. fuuetion f iii ealled tho domain of the fuuetioll nlld is denoted by D" while the collection of all second components is called the ran!le of the funetion and is denoted by R,. In terms of set notation,

= R, =

D,

{x I (x, y)

EJ for some y},

{y I (x, y) EJ for some x}.

If J is a function and (x, y) E J, then y is said to be the functional value or image of f at x and is denoted by f(x). That is, the symbol f(x) represents the ullique second component of that ordered pair of f in which x is the first component. We lihall oCC8.:-;ionally ohscrve the cOllvention of simply writing fx for f(x). Example 1-10. If the function f is the finite set of ordered pairs,

f = {(-1,0), (0,0), (1, 2), (2, I)}, then D, = {-1,0, 1, 2}, and we write f( -1)

R,

=

{O, 1, 2},

= 0, f(O) = 0, J(I) = 2 and J(2) =

1.

It is often convenient to descrihe a function by giving a formula for its ordered pairs. For instanl:e, we might have

J=

{(x,x 2 +2) IXER'}.

Using the functional value not.ation, onll would then wriw f(x) = x 2 + 2 for each x E R'. Needles!! to say, there are funetions whose ordered pairs would be difficult-if IIOt impoliliible-to express in terms of a formula. The diseerning reader is advised to keep in mind the distinction between a function and its values or, as the case may be, its formula; although the notation sometimes leads to confusion, these concepts are obviously quite different. Definition 1-10. COII;;ider a funetion J ~ X X Y. If /), = X, then we say t.hat J iii a fUflct.inn fmlll X int() }', or t.hat f mall'~ X into Y; thiH Kitua.tion is expressed symboIieally by writing f: X ----> 1".

14

1-2

PRELIMINARY NOTIONS

The function I is said to be onto Y, or an "onto" function, whenever I is a function from X into Y and RI = Y. Thus I is onto Y if and only if for each ye Y there exists some xeD, with (x, y) e/, so that y = I(x). Since functions are sets, we have a ready-made definition of equality of functions: two functions I and g are equal if and only if they have the same members. Accordingly, I = g if and only if D, = Dg and I(x) = g(x) for each element x in their common domain. Suppose I and g are two specific functions whose ranges are subsets of a system in which addition, subtraction, multiplication and division are permissible (one may think of functions from R' into R'). The following formulas define functions 1+ g, 1- g, I· g and I/g by specifying the value of these functions at each point of their respective domains:

+

(f gHx) = I(x) + ,,(x), ) (f - g)(x) = f(x) - g(x), (I· g)(x) = f(x)(j(x), (f/g)(x) = f(x)/g(x) ,

where xeD, n Dg where

x e (D, n Dg) -

{x e Dill g(x)

= OJ.

We term f + (I, f - g, f· g and fig, the pointwise sum, difference, product and quotient of f and g. Examp'e 1-11. SUppOtlC

f =

{(x,

V4 -

x2)

I -2 5

x

5 2}

and

g

=

((x,~) I R' ~ {O}},

so that f(x)

Then for x e D, n Da

=

= "D,

-

v4 -

2 g(x) - -. x

X2,

{OJ,

(f + g)(x)

=

v4 -

x2

2 + x-,

(/ - g)(x)

=

v4 -

x2

-

_IT.r 2 )

(J. (J)(.r) -- (v <1 (f/g)(.r)

=

v4-x 2 2/x

2 -, x

2 x' x_~

= 2 v4 -

;1:2.

The function operations just eonsidered plainly depend on the algebraic properties on the range; the domain merely furnishes the points for these pointwise operations. The most important operation involving functions, functional composition, is independent of such algebraic structure and relies only on the underlying set.s.

1-2

/

15

FUN<-'TIONS AND ELEMENTARY NUMB Ell THEORY

Definition 1-11. The composition of two functions / and 0, denoted by 0, is the function 0

/0 g =

{(X,7/) I for some z, (x, z) E g and (z,7/) E/}.

Written in terms of functional values, this gives

I«(/(x»),

(f (I)(X) = 0

where

This last notation serves to explain the order of symbols in f • 0; the letter g is written directly beside x, since the functional value O(x) is obtained first. It is apparent from the definition that, so long as RII n D, 'F- 0,/. g is meaningful. Also, Dlo ll ~ DII and Rlo ll ~ RI .

Examp'e 1-12. IA't

I =

{(x,

vi) I x

E

R', x

~ O},

and

u= so that/ex) =

vi, o(x) (fog)(x)

= 2x

{(x, 2x

+ 3.

+ 3) I x E R'} ,

Then,

= /(g(x») =

+ 3) =

f(2x

v2x

+ a,

where [x E DII I g(x) ED,}

D"II =

=

{x E R' I 2x

=

{x

I 2x + 3

+ 3 ED,}

~ O}.

On the other hand, (0 o/)(x)

=

o(f(x»)

=

g(v'x)

=

2v'x + 3,

where D llo,

Olin ObRf'rVeH t.Jmj,

=

{x E D, I/(x) E D II }

I· (I

iH dilTl'rC'lIt from

=

{x ~ 0 I v'x E R'}

= {xlx~O}. (f Ii indeed, mn!ly 0

does

I· (f =

g

0

f.

The next theorem concerns some of the basic properties of the operation of functional eompositioll. Its proof is an exereise in the use of the definitions of thiR S()ctioll. Theorem 1-7. If /, g and h are functions for which the following operations are defined, then . 1) 2)

3)

(f. g) • Ii - 1 (y • It), (f y) • It = (f. h) + (g. It), (f. g) • h = (f 0 h) . (g 0 h). 0

+

16

1-2

PRELIMINARY NOTIONS

Proof. We establish here only property (3). The other parts of the theorem are obtained in a similar fashion and 80 are left as an exercise. Observe first that DU'II)%

= = =

{x E D/o I hex) E DI

=

D,o/a

{x E D/o I hex) E DI"}

n D II }

{x E D" I hex) ED,} n {x E D/o I hex) E D II }

n

Vll0/o

=

DUo/a)'(II0/a)

Now, for x E D("II)o/a, we have [(f. g)

0

h](x) = (f. g)(h(x»

= /(h(x» . g(h(x»

=

(f h)(x) . (g h)(x)

=

[(f h) . (g h)](x),

0

0

0

0

whidl, Recording t.o UlP d(!finit,ion of equality of functions, Hhows that (f. g)

0

h

=

(f 0 h) . (g

0

h).

Once again, consider an arbitrary function f: X ~ Y. While no element of X can be mapped under / onto more than one element of Y, it is clearly possible that several (perhaps, even all) elements may map onto the same element of Y. When we wish to avoid this Hituation, the notion of a one-to-one function is useful. The formal definition follows. Definition. 1-12. A function / is termed one-ta-one if and only if XI, X2 E D" with XI '" X2, impli(!s/(xl) '" /(X2)' That is, distinct clements in the domain have distinct functional values.

When establiilhing one-f,o-oncness, it will often prove to be morc convenient to use the contrapollitive of Definition 1-12:

In terms of ordered pairs, It function I ill onc-to-onll if and only if no two distinct ordcred pairs of / have the same second component. Thus the collection of ordered pairs obtained by interchanging the components of the pairs of / is also a funct.ion. This oh8(·rvat.ion indicates the importance of such functions. More specifically, the inverse of a one-to-one function /, symbolized by /-1, is the set of ordered pllirs,

/-1 = fey, x) I (x, y)

Ef}.

The function /-1 has the properties (f-I • I)(x) ~ x for xED"

(fo/-I)(y)

= yforYEDrt = RI.

1-2

FUNCTIONS AND ELEMENTARY NUMBER THEORY

17

To state this result a little more concisely, let us introduce some special terminology . Definition 1-13. Given a nonempty set X, the function ix: X -+ X defined by ix(x) = x for each x E X is called the identity [unction on Xj that is to say, ix merely maps caeh element of X onto itself.

ExpreRRed in terms of the identity funetion, what was just seen is that for any function j: X -+ Y which is both one-to-one and onto Y, and

[0[-1

= iy.

It migftt also be mentioned at this point that the identity function ix is itself a one-to-one mapping onto the set X such that iXI = ix. Example 1-13. The fUlwtioll [ = {(x,3x - 2) I x E nil is on(,>..to-one, for 2 ~ ax:.! -- !! impliml x. = X2' Cont:I(Jqucmtly, the inVllftj(J of j exists and is the set of ordered pail~[-I = {(3x - 2, x) I x E R'}. It is preferable, however, to have[-t defined in terms of its domain and the image at each point of the domain. Observing that

ax. -

{(3x-2,x)lxER'} we choose to write

r

1

=

{(x,!(x+2»lxER'},

= {(x,!(x + 2) I x

In terms of functional vl\lueM, [-I(X)

E R'}.

= !(x + 2) for each x E R ' .

An important situation ariscs when we consider the behavior of a function on a subset of its domain. For example, it it! frequently advantageous to limit the domain 80 I,hnt t.he fUlldion beeomcM one-to-one. Suppose, in general, that j: X -+ Y it! an nrbitrary function and the subset A ~ X. The composition j i ... : A -+ Y is known as the restriction of [ to the set A and is, by established cllstom, denoted by f I A; dually, the funetion [ is referred to as an extension of f I A to all of X. For the reader who prllfers the ordered pair approach, flA= {(x,y)l(x,Y)E[andxEA}. 0

In any event, if the clement x E A, then (f I A)(x) = [(x) so that both [and coineide on the set A. It is well worth noting that while there is only one restriction of the given function [ to the subsct A, [ is not necessarily uniquely determined by [I A. The particular restrietion ix I A = i A , when viewed as a funetion from A into X, is termed the inclusion or injection map from A to X. Thl! lwxt ddinitinn 1'lIIilodil'fl lL fn~CJUlmtly employed uotational device. Observe that despite the use of the symbol j-1, the function [ is not required to be one-to-one.

[ IA

18

1-2

PRELIMINARY NOTIONS

Definition 1-14. Conllider a (unction f: X -+ Y. If A !;; X, then the direct image of A, denoted by f(A), is the subset of Y defined by

f(A)

=

{f(z} I z e A}.

On the ot.her hand, if B !;; Y, then the inverse image of B, denoted by f-l(B}, is the subset of X defined by f-l(B)

=

{z /f(z)

e B}.

It shall be our convention to omit unnecessary parentheses whenever possible. In regard t.o RinglctonR, for iIlRt,allce, we Rhl\lI write dirc(~t l\n<1 inverse images lUI f(:z:) ami J- I (x), fnUlI'r UIILII J( :r; ) nncl I ( (x:). TIII~ HI,II(hmt who worriuH about notlltion nmy fcelllOnll'what UIICIJ.'W about this double usc of the symbol f- 1• The abulIC of notation Ilhould not cauRe any confusion, howev
r

Theorem 1-8. For each subset B

!;;

Y,

J(rl(B»)

!;;

B.

ProoJ. If b eJ(j-I(R», then b = f(a) f~r some element a in f-t(B). From this, it follows that f(a) e B, and conscquclntly b e B. Corollory. If, in addition, j maps onto the sct Y, then

Proof. In view of the indusion proved in the theorem, we need only establish that, under the existing hypothesis, B !;;f(j-I(B»). For this, let be B; then, as j is by supposition an onto function, b = f(a) for some choice of a in X. Since a ej-l(B), 80

that

1-2

FUNCTIONS AND ELEMENTARY NUMBEU THEORY

19

Theorem 1-9. For ea.ch subset A k X,

Proof. The proof is almost obvious, for, if a e A, thenf(a) ef(A); hence,

Corollary. If, in addition, f is a one-to-one function, then

Proof. H Jllninly HllfIlI'I'H 1.0 I'HI.nhliflh t.hn 0111' il",hlllicm , - I
Before terminating this section, it may be well to review, quickly, some of the facts from number theory which we shall require later. Most of these results depend on the HC)-(~nlll!(l Wcll-Ordnring Prirwiplc: Well.Orderlng Principle. Every nonempty subset B of nonnegative integers contains a smallest element; that is, there exist..~ some (unique) element a e S with a ::; b for all b E S.

Let us start with the following result.

Theorem 1-10. If a, be Z, with b q and r such that a= qb+ r,

>

0, then there exist unique integers

0::; r <

b.

Proof. We begin by proving that the set,

=

S is not empty. Since b

~

{a - xb I x e Z; a - xb ~ O},

1, lalb

~

lal, and

a - (-Ial)b = a + lalb

~

a + lal

~ O.

Hence, for x = -Ial, a - xb E B. By the Well-Ordering Principle, S contains a smallest integer, say r. In ot.her words, there is somc q e Z for which r

We now show that r

<

=

a -

qb,

r ~ O.

b. In the contrary case, r

~

band

a -- (t}l- I)b = (a -- qlJ) .- I, = r - b ~ 0,

which implics r - b E 8. Since r - b < 7', t.his contradicts the choice of r as the HmItJ1<'Ht elenl('nt in 8; therefore, luwing reached a. contradiction, r < b.

20

1-2

PRELIMINARY NOTIONS

To prove the uniqueness of the integers q and r, suppose that

=

o where 0 S

r < h, 0 S r' < h.

Adding the inequalities -b -b

Hence, blq -

q'l <

b,

qIJ + r Then

q'h

+ r',

r' - r =

b(q - q'), consequently

Ir' - rl = blq - q'l. < - r S 0 and 0 S r' < b, we obtain

< r' - r < 80

=

b,

Ir' - rl < b.

or

that

o S Iq - q'l < 1. Since Iq - q'l is a nonnegative integer, it follows that q = q', which in tum gives T = r'. Corollary. (Division Algorithm). If unique integers q and r such that a

=

0,

qr+ b,

b E Z, with b

OS

T

¢

0, then there exist

< Ibl.

Proof. It suffices to consider the case where b is negative. Then the theorem yields unique integers q' and r for which

a Since

Ibl =

=

q'lbl + r,

Ibr> 0 and

OS r < Ibl.

-b, we may take q = -q' to get

a

=

qIJ + r,

OS

r

< Ibl.

Let us now make the following definition. Deftnitlon 1-15. Let a, b E Z, with 0 ¢ O. Thc integer a is said to divide b, or 0 is a divi81w of h, in JoIymhols a I h, provid<.'<.i there exists HOme c E Z such that b = ac. If a does not divide b, then we write arb.

When the notation a I h is employed, it is to be understood (even if not explicitly mentioned) that a ¢ O. Some immediate consequences of this definition are noted below; the reader is asked to verify each of them. Theorem 1-11. Let a, b, c E Z. Then

1) 0 10, 1 I a, a I a, . 2) a I ±1 if and only if a = ±1, 3) if a I b, then ac I be,

1-2

21

4) if a I band b I c, then a I c, 5) a I band b I a if and only if a = ±b, 6) if c I a and c I b, then c I (ax + by) for every x, y E Z.

From (1) above, we see that every integer a ¢ 0 is divisible by 1 and a, diviflOrH whidl ar(! fn'IJIlI'IIUy f(·f(·rn,d t.o II.H illlpropl'r lliviH()fH. An int.eger a > 1 having no divisol'H other than the improper ones is said to be a prime number; all integer a > 1 that is not prime is termed composite. Thus, according to our definition, 1 is neither prime nor composite. In particular, an integer a > 1 is composite if and only if there exist integers b, c with a = be, 1 < b < a,

l
d I b. Also,

Definition 1-16. Let a and b be integers, not both of which are zero. The r()mmon tiiviR01' of a I\lId b, dell()u,d by Ked (a, b), is t.he positive integer tl Ilueh t.hnt

{JreairHt

1) d I a and d I b, 2) if c I a and e I b, then e I d.

Briefly, ged (a, b) is the largest integl1r in the set of nil common divisors of a alld b.

A naturu.l qU(!lltion to ask is whether the integers a Ilnd b can posseS!! two different greatest eommon divisors. For an an!!wer, suppose there are two positive integers d and d' whieh satisfy the conditions of Definition 1-16. Then by (2), we must have did' as well as d' I d, whence d = ±d' [Theorem 1-11(5)]. Since tl and d' are hoth positive integers, it follows that d = d'. Thus, the greatest COlIllllOII diviHor of a and b is unique, when it exists. The following theorem will prove that any two integenl, which are not both zero, actually do have a greatest common divisor.

Theorem 1-12. If a, b are integers, not both of which are zero, then ged (a, b) exists; in fnet, t.here ('xiRt integerR x II.l\d y such that gcd (a, b)

=

ax + by.

Proof. First, define t.he set S by S

=

{au -I- bv I u, v E Z; au

+ bv > O}.

+

This set S is not empty. For example, if a ¢ 0, the integer lal = au bO will lie in S, where we ehoose u = lor -1 according as a is positive or negative. By the Well-Ordering Principle, S must contain a smallest element d > 0; that is to say, there exist x, y E Z for which d = ax + by. We assert that d = gcd (a, b).

22

1-2

PRELIMINARY NOTIONS

From the Division Algorithm, one can obtain integers q and r such that + r, ~ r < d. But then r will be of the form

°

a = qd

r

=

a - qd

=a= a(1

q(ax -

qx)

+ by) + b( -qy).

Were r> 0, this representation would imply rES, and contradict the fact that d is the least integer in S. Thus r = 0, so that d I a. A similar argument establishes that d I b, making d a common divisor of a and b. On the other hand, if e I a and e I d, then by Theorem 1-11(6), e I (ax + by), or rather, c I d. From these two statements, we conclude that d is the greateflt common divisor of a and b. It may be well to record the fact that the integers x a.nd y in the representation gcd (a, b) = ax + by are by no means unique. More concretely, if a = 90 and b = 252, then ged (90, 252)

=

18

(3)90 + (-1)252.

=

Among other possibilities, we also have 18

=

(3

+ 252)90 + (-1 -

90)252

=

(255)90

+ (-91)252.

There is a special case of Theorem 1-12 which will play an important role in the future; while it is, in effect, a corollary of the foregoing result, we shall single it out as a theorem. But first, a definition: two integers a and b, not both of which are zero, are said to be relatively prime (or prime to each other) if and only if gcd (a, b) = 1. For instance, the integers 8 and 15 are relatively prime, although neither is itself a prime. The.Nm 1-13. Let a, ,) E Z, lIot both zero. Th(m a Ilnd b are relatively prime if and only if there exiflt integers x and 11 such that 1

=

ax

+by.)

Proof. If a and b are relatively prime, 80 that gcd (a, b) = 1, Theorem 1-12 guarantees the existence of x and y satisfying 1 = ax + by. Conversely, suppose 1 = ax + by for suitable x, y E Z and that d = gcd (a, b). Since d I a, d I b, Theorem 1-11(6) implies d I (ax + by), or rather d 11. Because d is positive, this forces d = 1 [Theorem 1-11(2»), as desired.

In light of Theorem 1-13, one may easily prove Theorem 1-14. (Euclid's Lemma). If a I be, with a and b relatively prime, then a I e. . Proof. Since gcd (a, b) = I, there exist integers x and y for which 1 MUltiplying bye, wc obtain e

=

(ax

+ by)e =

a(cx)

+ (be)y.

=

ax

+ by.

1-2

FUNL'TIONS ANI) ELI!:MENTARY NUMB.m TIIEOItY

23

Now a I a trivinlly ILlld a I be by hypothcNi8, 80 that a IlIU8t divide thc sum acx + bey; hence a I e, as asserted. Corollary. If p i8 a prime and pi (ala2' .. a,,), then p I at for some k, 1 :::; k:::; n.

Proof. OUf proof is by induction on n. For n = I, the result obviously holds. Supposc, al:! the induction hypothesis, that n > 1 and that whenever p divides a product of less than n factors, then it divides at least one of the factors of this product. Now, let pi (ala2' .. an). If p divides at, there is nothing to prove. In thll contmry I!MC, p and al arc rdativcly prime; hence, by the theorem, p I (a2' .. an)· Since the product a2' •. an contains n - 1 factors, the induction hypothe8is implies p I at for some k with 2 :::; k :::; n.

Having developed the machinery, it might be of interest to give a proof of the Fundamental Theorem of Arithmetic. Theorem 1-15. (Fundamental Theorem of Arithmetic). Every positive integer a > I I~nll hll (lxpreHllCd u..'\ a P1"lltluct of prime!! i thi!! rcpretIClltu.tioll i8 unique, apart from the order in which the factors occur.

Proof. The first part of the proof-the existence of a prime factorization-is proved by induction on the values of a. The statement of the theorem is trivially true for the integer 2, since 2 is itself a prime. Assume the result holds for all positive integers 2 :::; b < a. If a is already a prime, we are through; otherwise, 4 = be for suitable integers b, e with 1 < b < a, 1 < e < 4. By the induction hypothesis, b = PIP2" • p"

with P.,

p, nil prime!!.

Hut theil, a = be = PI' ..

PTP~

.•.

p!

is a product of prinles. To establish uniqueness, let us supposc the integer a can be represented as a product of primes ill two ways, say

where the Pi and qi are primes. The argument proceeds by induction on the integer n. In the case n = 1, we have a = PI = ql(q2' •. q...). Since PI is prime, it possesses no proper factorization, so that m = 1 and PI = ql. Next, assume n > 1 and that whenever a can be expressed as a product of less than n factors, this representation is unique, except for the order of the factors. From the equality PIP2 ... Pn = qlq2' .• q... , it follows that PI I (qlq2' •• q...). Thus, by the preceding corollary, there is some prime qk, 1 :::; k :::; m, for which PI I qki relabeling, if Iwccssary, we may suppose, PI I ql. But then PI = qt,

24

1-2

PRELIMINARY NOTIONS

for ql has no divisors other than 1 and itself. Canceling this common factor, we conclude 1'2 .•. PA = q2 •.. qm' According to the induction hypothesis, a product of n - 1 primes can be factored in essentially onc way. Therefore, the primes Q2, .•• , qm are simply a rearrangement of the primes 1'2,'" , PA' The two prime factorization!'! of a are thus identical, (~ompleting the induction step. An immediate consequence of this theorem is the following: Theorem 1-16. (Euclid). There are an infinite number of primes.

Proof. Assume the Rtatement is false; that is, assume there are only a finite number of primes PIr 112, ... ,1'". Consider the positive integer a

=

(1'11'2' .. PA)

+ 1.

None of the primeR Pi divides a. If a were divisible by PIr for instance, we would then have p, 1(0 - P,1'a'" 1'A) by Theorem 1-11(6), or PI 11; this is impossible by part (2) of the same theorem. But, since a > 1, the Fundamental Theorem asserts it must have a prime factor. Accordingly, a is divisible by a prime which ifol not amonK our list of all primes. ThiR lugument shows there is no finite listing of till! prillle int<~gerH. This comJlI(!t<~H our Hurvc~y of HOmel of till! fUluJu.numtlLl notion!!! c:onceming sets, functions, and arithml'tie in thll int,egers. Although the treatment was purposely sketchy, it is hoped that the reader did not find it too superficial. In the subsequent chapters of the text, we shall utilize the foregoing concepts by applying these ideas to certain specific! situations.

PROBLEMS 1. Let I: X ~ }' be an arbitrary function. Define a relat-ion in the Ret X as follows: for any two elementR x, y E X, x,." y if and only if I(x) = I(Y). Verify that ,." is an equivalence relation in X and describe its equivalence c19.!l8e8.

2. Show that the ordered triples (a, b, c) and (a', b', c') are equal if and only if a = a', b = b', c = e'. 3. Give an example of two functions I and g from R' into R' with I ~ g for which

log

=

go/.

4. Determine log, g I, and their respective domains, given that a) I = {(x, x 2 x) I x E R'}, g = {(x, ax 4) I x E R'}, b)/= {(x,x/(x 2 1)) IxER'}, g= {(x,l/x)lxER'-{O}}. 5. Let I, g and h be runction!; such that Rio ~ D. and R. ~ D,. Verify the law 0

+

+

+

1

0

(g • h) =

(I. g)

0

h.

1-2

FUNCTlO!IIS AND ELEMENTARY NUMnER THEORY

6. Let I, g and h hE' fundionl! from R' into R' with Rio ~ D, (fIg)

0

n

/)/1'

25

PrO'!e that

h = (f 0 h)/(g 0 h).

7. Determine which of the function14 below are one-to-one. In thollC cases in which the fund inn I ill not (llll'-tn-Onl', "xhihit. two pl1irH (:tl, YI), (X2, Y2) EI such that Xl ;o! X2 but YI = Y2. a) I = {(x,:t 2 + 1) I :t E R'}, b)/= {{x,lx-Jl)I-2:5:t:52},

c)

I

I

= {(x, 11x) x E R~}.

8. For every pair of real numbers a and b, define a function I",,: R' -+ R' by the formula/ob{x) = a:t b for each x E R'. a) Show t.hat. /I. /'.0 = lab. b) For a ;o! 0, prove that I"" ill both one-to-one and onto. c) For a ;o! 0, obtain 1.. 1• 9. Using fllnl'tions/: R' -- R', give an example of a function which ill a) one-to-one but not onto, b) onto but not one-to-one. 10. For functionR g: X -+ }' and I: Y -+ Z, flhow that the following statements are true: a) If I iH all nnto funl!tioll, then I ill 8\1111. b) If log i,.. 8 (1ll1...t~l-Illl(l fllnl't.ion, th(\n g ill aillo. n) If I nlld 0 urI' hnth nnn-tn-nlle fUllnt.jllllll, t.Iwn log iH I~IHCI nnu-tn-one and (jo g)-I = g-I 0/-1.

+

0

0

(/

11. Establish the following characterizations for any function I:

x. -- Y:

ill onto Y if and only if for all functions g, h: Y -+ Z, g. I = hoi implies that g = h. b) I is one-to-one if and only if for all functions g, h: Z -+ X, log = , 0 h. implies that g = h.

a)

I

12. If I: X -- Y,

g: Y -

Z and A ~ X, prove that

13. Given I: X -- }' and .1, lJ

I) I A

= go

(g

0

~

X, show that

(I I A).

a) 1{.1 U B) = I(A) U I(B), b) I(A n B) ~/(A) n/{lJ), c) I(A) - I(B) ~/(A - B),

d) if A

~

B, then/(A)

~/(B).

14. Given I: X -+ Yand A, B ~ Y, show that a) ,-1(.-\ U B) = 1- 1(11) U/-I(B), b) 1- 1(.,1 n B) = 1- 1 (:1) n,-I(B), c) I-I (..t) - 1-1 (B) = I-I (.1 - B), d) if .1 ~ B, t1ll'IlJ 1(.1) ~rl(B). Comparing th(l1lC resultH with those of Problem 13, one can I!CC that inverse images are much better behaved than direct images.

26

1-2

PRELIMINARY NOTIONS

15. If

I: X

-+

Y, prove that

I

is a one-to-one function if and only if

I(A

for all set.'! .1, B

~

n B)

= I(A) nf(B)

X.

16. Given intt'.j/;ers a and b, which arc not hoth zero, establish the following facts concerning gcd (a, b): a) p;cd (a, -b) = gcd ( -a, b) == ged (-a, -b), b) whenever a ;o! 0, p;ed (a, 0) == lal,

c) p;ed (a, b) = lal if and only if a I b, d) p;ed (ea, eb) = lei ged (a, b), provided c ;o! 0, e) ged (a, b) = ged (a, b ea), for every c E Z.

+

17. Prove that. if a, b, e arc integers, no two of which arc zero, then 11:1'" (gl'" (a, b),

c) ... g('d (a, p;1:d (b, c» = gcd (I/:cd (a, c), b).

18. Prove the two 8I!.'lCrtions below: a) If lI;ed (a, b) = 1I;"d (a, c) == 1, then gcd (a, be) ... 1. b) If gcd (a, b) = 1, a I c and b I e, then ab I c. ]9. Let a and b be intep;erR, not both zero. The least common multiple of a and b, denoted by lem (a, b), iH the positive integer e such that

I

I

1) a e and b e; that iH, e is a multiJlle of both 2) if a I e and b I c, tht'n e I c.

a and b,

Show that the least common multiple of a and b is related to the greatest common divisor of a and b by

(11:m (a, b) )(gl:d (a, b»

==

labl.

20. Let a, b, c E Z, with a and b not hoth zero, and let d = ged (a, b). Verify that there exiMt intcgcrM x and 1/ Hueh that

ax if and only if die.

+ by

= c,

CHAPTER 2

GROUP THEORY

2-1 DEFINlnON AND EXAMPLES OF GROUPS

In this I!hapter, nlld t.hroughout the renllLind(~r of thc text" we shall deal with mathemntielll HYHtemK whidl ar(~ d(~fined hy a preserihcd list of properties. Emphasis will be on deriving theoremll that follow logically from t.he postulates and which, at the same time, help to describe the algebraic structure of the particular system under consideration. This axiomatic approal!h not only penn its us to concentrate on essential ideas, but also unifies the prescntation by showing the basic similarities of many diverse and apparent.1y unrelated examples. We first confine our attention to systems involving just one operation, since they are amenable to the simplest fornlal description. Despite this simplidty, the axioms permit the construction of a profuse and elegant theory in which one encounters many of the fundamental notions common to all algebraic systems. Before beginning, however, it iH necessary to arrive at some understanding conceming the UKe of the equivalClwc relation =. We will henceforth take the equality Hign t.o mean, intuitively, "is the same as." In other words, the Hymhol = aHK(~rt.K thnt the two pllrtieular expreHllions involved arc merely differcnt nameH for, or descriptions of, one and the HaOle objcct; jUHt onc objed is being conllidered, and it ill named twice. To indicate that a and b arc not the same object we shall, naturally enough, write a F- b. As a fimt steJl in our program, we introduce the concept of a binary operation. This idea is the cornerstone of all that follows.

Definition 2-1. Given a nonempty set S, any function from the Cartesian produet S X S int.o S is "ailed 1\ binary operation on S. A biliary op('rntion Oil S thuH 1l..'i."IigIlK to ('ad I ordl'J'('(\ pair of clements of S a uniquely determined third element of the same sct S. For instllnee, if P(A) denotes the power set of a fixed set A, then hoth U Ilnd n are binllry operations on P(A). In praetice, we shall generally use the symbol * to represent a binllry openlt.ioll lind write a * II, inst.ead of *(a, IJ)), for its vlLlue nt the ordered pair (a, b) E S X S. While this eonvcntion is at varimlee with the functional notation developed in the previous ehapter, its use in the prescnt 27

28

W({JUl'

2-J

TH~;OHY

situation is dietated by long-standing mathematical tradition. At the very least, it has t.he Ildvantage of avoiding some rather clumsy notation. From tim<.' to timl~, we shall permit ourselves to make such informal statements us "('ombine a with b" or "form a * b." In a precise sense, what is really meant of ('ours!' is to upply the fUlwtion * to the ordered pair (a, b). The most uscfulaspcet of a binary opcmtion is that, having once formed the clement a * b, we may ill turn combine it with other members of S; the result of all sueh ealeulatiolls again lies in S. Needless to :;ay, the particular notation used for the abstract product of two clement:; is of no great importance. On occasions ROme other symbol, as equally nOlICollllllittal as *, will he employed. Specifically, we will frequently ('hooSt· to write a IJ in phwe of a * b (in this context, the symhol is not intended to hav(' allY Sll('pial (·onlleet.ion with fUlwtiQlml eomposition). In gelleral, a and IJ will havl' 110 llulllI'J'ieal valul' hut will simply Iw arbitrary element.!! in our lIIulpriyilll!; SP!. S, whatever this set may be, while * may well be some law of ('ompo:;itioll whid! bear:; no resemblance to the usual operations of !'1('lnl'lltllry :dp;I'I>rIt. Closely aliiI'd to the notion of It binary opl'ration is the so-called closure condition. For It formal statement of this property, suppose that * is a binary operation 011 the set S nnd A ~ S; the subset A is sllid to be closed under the opcmtion • Jlmvided a • b E A whenever a and b are in A. The desirable feature here is that when A is closed under the operation ., the restriction of • to the subset A is Il binary operation on A as well as S. 0

0

Example 2-1. Ordinary slIbtra!'tion is dearly a binary operution on the set Z of illti!g!'l'H; the SUhH!'t. Z+ of pOtlitive integer!!, however, is not closed under

sllbtr:wtion. When thl' lid S heing ('oll:;id('reu IUI.,'! u relatively small number of elements, the results of applying the operation • to its members may be conveniently represented in what mij!;ht be called un operation or multiplication table. Wo eonstrUl·t this tahle by first listing the members of S in the same order both vertically and horizontally. The result a • b then appears in the body of the table at the intersection of the row headed by a and the column headed by b. Conversely, such a table could equally well serve to define a binary operation on 8, for the re8ult, of ('ombining any pair of elements of S would be displayed somewhere in tllP tahle. Example 2-2. A binal'Y ·operation • may be defined on the three-element set S = (1,2, 3} by means of the operation table below: •

2

3

1

1 2 3

2

3

1 2

3

2

3

1

2-1

In:FINITION AND

~:XAMJ>I..:I'\

OF

<moups

29

According to the t.able, the product 2 • 3, for inst.ance, iR equal to the clement 2, located at the interRection of the row marked 2 and t.he column marked 3. Given an arbitrary binary operation ., there iH ('ertninly no ren.son to expect that a • b will be the Rllllle IlS 11 • a for nil a Ilnd b. In faet, it can be seen in the above example that 1 • 2 = 2, whereas 2. 1 = a. One must consequently take care to refer to a • b as the product. of a and b and to b • a as the product of b and a; the distinction is quite importallt. We should also point out that it is obviously possible to combine an element with itself. Thnt is to say, a • a can be defined. Deflnition 2-2. By a mathematiml .~y.~lcm (01' malhrmatical structure), we shldl mean Il IlOTwmpty s!'t of (·Ienwnls t.og('ther wilh olle or more biliary operat.ions d!'fined on t.his set..

A maf,Ju'mal iC'al syslpm ('onsist.ing of Uw set S ILnd It single binary operation. will be d('llot\'d hy tlw ordered pair (8, .); analogously, a system eonsi:4ing of the Ret S lllld two operat.ions • awl. will hI' I'l~pr<'Hcnl,('(l by the ordered triple (8, ., 0). Example 2-3. The pair (8, .), where the set S = {I, -1, i, -i} and the operation is that of mdinary mult.iplication, is a mltth('nmtieall'!ystem provided one defines i 2 = -1. Example 2-4. If Ze and Zo denote the even and odd int<'gers, respectively, then (Z., +, .) ('onstit.utes a mathematical system, while (Z., .+, .) cloes not.. In the latter caRe, Ow set Z. ill not. ('Ios(>d undN addition, sin('l' t.he sum of two odd integerl'! is lweessarily even.

The systemH to be studied Hlloscqtwntly arc dall8ificd aecording to the properties they possess or, to put it anot.her way, accordillg to the axioms they satisfy. Our object will be to present a sequent.ial development of the principal mathematical Hystems of modern algebra, beginning with t.hose involving relatively few axioms and progressing to systellls sat.isfying more detailed hypotheses. The axioms whieh form the starting point of the abstraet theory can be, by nature, ratlwr varied. The growing t.endeney of modern mathematies is to isolate almost. any convenient set of propl'rtie~ from its original context, to define a parti(,ular system, llnd to develop the eorrcspollding Ilbstract theory through logical d(~dul'tion. ROllle of tlH'se forlllal :txiolllat ie theorie~, sueh as the notion of It group, have It fundamentlll irhportalH'(' to the whole of mathematics and have been instrullll'ntal in IInifyillg various apPllrently unrelated branches; other I heories, while sat.isfying the esHwt i(' and inquisitive needs of the mathemati('ian, are limited in the extent of their applicability. We do not mean to create the impression that it, is the usmil praetiec for one to define a new system by arbitrarily (apart from logical (,oll~iderntions) writing down axioms. Although thpr(' is no part.icular np('('ssity for the model to precede the thcorct~al development, in most cases the axioms nrc the abstract realization

30

2-1

GlWUP THEORY

of the properties common to a variety of specific examples. With these general remarks out of the way, let us get down to work. A set on which It Hingle unrestricted binary operation is dcfined does not by itself yield a Htructure rich enough for our purposes; the concept, being too general, ill poor in (~ont<mt. Certain rcQ.HOnahlll limitations must be impo8(~d on the opcmt,jon if one itJ to obtain utlCful rctJulttJ. In the following pllragrnphH, !lOme of the mOM) bailie requirements arc named and brieRy examin(.'
tive if a * (b • c)

=

* defined

(a * b) * e

on the set 8 is said to be lJ88Oeia(associative law),_

for every triple, distinet or not, of elements a, b, and e of 8. Example 2-5. The operation of subtraction on the set

R' of real numbers is

not associative, since in general a -

(b -

e) F (a - b) -

e.

* may be defined on Z, the set of integers, by taking a * b = a + b + abo (We shall frequently delete the dot and write the product of a and b under ordinary multiplication simply as ab.) Then a * (b * e) = a * (b + e + be) = a + (b + e + be) + a(b + e + be),

Example 2-6. An associative operation

while (a. b) • c = (a =

(a

+ b + ab) * e + b + ab) -I- e + (a + b + able.

The equality of these two expressions follows in part from the fact that addition and multiplicat.ion Ilrc themselves associative in Z. When del~ling with a Hystem who8() operation is defined by a mUltiplication table rather than II. formula, it is generally quite tedious to establish the associa-

2-1

1l1<:1<'INITlON ANI> I<:XAMI'LES 01<' OROUPS

31

tivity of the operation, for one must compute all possible threefold products. On the other hand, it may be fur easier to show thltt the operation is not a.'lROciativc, as all we need do in thi::; ea.'lC is find three pnrticular clements from the underlying Ret fOf whieh the nS!lO(~iative law fails. Example 2-7. Conl'lid(!f the I)JI(!MLI.iuli • t11!fiw,t1 on the set S = {l, 2,' 3} by the operntion tlLllle:

•

1 2 3

1 2

1 2 3 3 1 2 2 :J

a

From t.his I.allll!, we = ]; I.hnt, is,

3 *3

!j(!('

that 2. (1 • :J) = 2.:J = 2, whereas (2. 1) •

a=

2 • (1 • a) ~ (2. 1) .3.

The associative law thus fails to hold in the system (8, .). The mathematical system which we shall use to build up more complieated nlgehmil! llt.rudufCS ill known 11."1 a semigroup. Definition 2-4. A

Bemi(JTOUl1 ill a paif (8, .) (!onllillting of a nOnCml)ty scI. 8 together with an associative (binary) operation. defined on S.

Let us stress t.hat it is an abuse of lnnguage to say a certain sct alone is a semigroup without also specifying the operation involved, as it may be quite possible to equip the same set with several associative operations. For this reason, wc have utilized the ordered pair notation to indicate both the operation and the underlying set of elements. Observe that since any three elements from the set of a semigroup always associate, there is no particular reason for parentheses. Consequently, when dealing with sueh It system, the I!ymbol a • ~ • c has meaning in the sense that we are frL'C to interpret it either as a • (b • cj or as (a * b) • c. More generally, the notlttion al • a2 • •..• am is unambiguous, for it can be shown that all ways of inserting parentheses 80 as to give this expression a value yield the same felmlt, (Theorem 2-4). An opemtion whieh il! not as80ciative has the dedded disadvantage thtLt the notation for multiple-factored products can be(!ome quite unwieldy 11." a result of the eonstltnt need for parentheses. In order to ROlidify the notion of a semigroup, we present several examples. Example 2-8. There Itrl! severnl semigroups wit.h whi('h t,he reader is already familiar. If, for insta.nce, Z+ denotes t.he set of ull positive integers, then both the pairs (Z +, +) and (Z +, .) form semigroups. Similar statements hold for the sets Z, Q, and R'.

2-1

(mOUI' 'l'llI':Olty

Example 2-9. Define the operation

a

*b =

* on

the real numbers by the rule

max {a, b},

a, b E R'.

That is, a * b is the larger of the two numbers a and b, or either one if a Here, we have a * (b * c) = max {a, b, c} = (a * b) * c, so that (R', *) satiRficR the requirempnts of a scmigroup.

=

b.

•

Example 2-10. For IlIlY XC!!. X, eueh of UIIl HYH!.mnH (I'(X), u) Ilnd (I'(X), n) cOIlt!titutCR It 1-!!'luigrouJl (Tlworem 1-2). Example 2-11. 1,<'1. X bl! It lIollcmpty R(~f. ulld S he t,l1P (!olledioll of all funetion8 f: X -+ X. If· dl'lIot.I'1-! flllU'tiollal 1'()1IIJ10I-!itioll, t.hell the pair (8,0) provides another iIIustrat.ion of It semigroup (Problem !i, Seetion 1-2).

As we Rhall Rubs<''1uentiy SI'I', the relevance of the semigroup concept lies in the fact that many important systems contain the semigroup structure as a subsystem. We have already indicated that the order in which elements occur in a product is quite essential. If it is pos..-;ible to interchange the order of eombining any two elenwnts from our set without affecting the result, then the operation is termed commutativ('. Definition 2-5. The operation

* defined

if

on the set S is called commutative

(commutative law),

for every pair of clements a, b E S. Examples 2-8, 2-9 and 2-10 arc of commutative semigToups (semigroups whose operation is commutative), while in Example 2-11 functional composition is not, in grneral, a commutlttive operation. Although the commutative law may fail to hold throughout an entire system, it may still be valid for particular pairs of elemrnts; accordingly, it will be convenient to make the following drtinition. Definition 2-6. Two clt'mrllts a and b are said to commute or permute (with each other) provided a * b = b * a.

Employing this terminology, we observe that the opemtion of the system (8, *) is commutative if and only if every pair of clements of S commute.

Once an operation has been defined on a set, one finds that certain elements play special rol(,l-!; t1H'rl' may ('xist identity clements and inverse elements. Definition 2-7. The system (S, *) is said to have a (two-sided) identity elelllrnt for the oppmf.ioll * if there exist~ an elenwnt e ill S such that

a*e=e*a=a

2-1

33

UJo.WINITION ANU EXAMPLES OF GROUPS

for every a E S. An element e having this property is called an identity element (unit clement, Iwutml clement) for (8, *). An identity element thus causes each element of the set S to remain stationary under the operation. In particular, notice that e * e = e. Of course, for a given system, an identity element mayor may not exist; in case an identity does exist, it must be unique, as the theorem below shows. Theorem 2-1. A mnthcmat.ieal Rystem (8, *) hIlS ut, most one identity c1cllwnt. Proof. For the pro()f, let Ul:! lIupP()lIe that (S, *) Iml:! two identity clements and 1". Sil\l"e e * n = a for el\l~h a E S, then in JllLrti(~lIlnr e * e' = e'. But Oil the other hUIIII, e' ill ulH() an iuent.ity clement, til.) Wt~ 1IIII1It have e * e' = e. We thus ohtain e = e * e' = e' anu consequently e = e'; t.hat is, if the system actually has an iuelltity, then there is precil:!ely OIle clement with this property.

I'

It follows from Theorem 2-1 that, whenever (S, *) has an identity, we are justified in using the expression "the identity element of (S, *)"; the symbol e will be reserved exclusively for this identity. Definition 2-8. A semigroup (8, *) is said to be a semig1'Oup with identity if there exists a (unique) identity element for (S, *). Example 2-12. The semigroup (Z +, .) possesses an identity clement, namely, the positive integer I. On the other hanu, the semigroup (Z+, +) has none, since 0 It: Z +. Example 2-13~ Bot,h the semigroups (P(X), U) and (P(X), n) have identities. Here, the empty lid 0 is the iuelltity element for the union operation, since

Au0=0uA=A

for each set

A s; X.

As is easy to see, the universal set X acts as the identity element for the operation of intersection, since AnX=XnA=A

for each set

A s;;; X.

Example 2-14. To record one more example of a semigroup with identity, consider the set of numbers

S = {a + bv'2 I a, b E Z}, and the operation of ordinary multiplication. First, one is obliged to check that S iR actually closed under multiplication; this is fairly clear, for if a bV2 and c + dv'2 are arbitrary members of S, then

+

(a

+ bV2)(c + dV2) =

(ad

+ 2bd) + (ad + bc)V2 E S.

34

2-1

GROUP THEORY

It is not particularly difficult to establish that the pair (8, .) is a commutative semigroup with identity clement 1 = 1 + ov'2; we omit the argument. When working with an operation which has an identity element, it is natural to inquire which elements of the underlying set, if any, have inverses. Definition 2-9. Let (8, *) be a mathematical system with identity element e. An I'II'II1Pllt a E 8 iH Haiti to have 11 (tW()-Hitlfld) inverse under UlCl operation * if there exil:lts ROme member a' of 8 SUdl that

a

* a' =

a'

*a =

e.

An element a' having thiH property iH called an inverse of a and is customarily denoted hy a-I. An inverse hall the effect of reducing a given clement, under the operation, to the identity element. In particular, since e * e = e, we may infer that e- I = e. It will be established shortly that, for a semigroup with identity, each element has at most one inverse relative to the unique identity clement. (The reader might try to work out the proof for himself.) Thus, when dealing with such a system, there is no ambiguity of meaning in the symbol a-I and, if it exists, we are free to speak of "the invefHC of an element. " Example 2-15. Let S be the set of all ordered pairs of nonzero-real numbers and * the binary operation defined by (a, b)

* (e, d) =

(ae, bd).

Then the system (S, *) forms a (commutative) semigroup with identity, with the pair (1,1) serving as its identity element. For (a, b) E S, we evidently have (a, b)-I = (l/a, lib), since (a, b)

* (l/a, lib) =

(a(l/a), b(l/b»

=

(I, 1).

Example 2-16. Let X be a nonempty set and 8 be the collection of all functions f: X -+ X. It is easy to I:lOO that the system (8, 0) is a semigroup with identity, having as it.'! iJl'ntity the identity map i x . A function f E S will possess an

inverse relative to t hI' operation of composition if and only if f is a one-to-one mapping from X onto itself; in this event, the inverse of f (under 0) is the usual inverse function 1-1: Example 2-17. As a further illustration of these ideas, let us return to the semigroup (P(X), U) of Example 2-13. In this case, just the empty set 0 possesses an inverse; for if A E P(X), with A ~ 0, there is no subset A - I of X such that A U A - I = 0. Likewise, in regard to the semigroup (P(X), n), the only member of P(X) which has an inverse is the universal set X.

2-1

DEFINITION AND EXAMPLES OF GROUPS

There is a mnt,hematieai system, known the proJlm'til~s we have so fur dil:!cUMll(lll.

1\.'1

35

a group, which displays most of

Definition 2-10. The pair (G, *) is II. group if and only if (G, *) is Ii semigroup with identity in which each element of G has an inverse. ,

While the above definition is perfectly acceptable, we prefer to rephrase it in the following lIlor(! ddaiJl'd forllJ, nwrl'ly as It /lmtt<'r of eonvllni(,ncc. Definition 2-11. A group is a pair (0, *) consisting of a nonempty set G and a binary operation * defined on G, satisfying the four requirements: 1) G is closed under the operation *, 2) the opl'mt.ion • is ItRsoeiativll, 3) (J contllinl:! lUI identity clement e for the opel'll.tion *, and 4) each element a of a has an inverse a-I E a, relative to *.

This definition calls for several remarks. For one thing, the first of the requirements cited above could easily have been omitted, since any set is dosed with respect to a binary operation defined on it. (We merely wish to emphasize that one must always ('heck the dosure condition.) Observe particularly that eommutativity is not required in the definition. If it happens that the group ol><'ration satisfies this additional hypothesis, then (G, *) is referred to Itl'! 1\ ('ommulalillc or aIle/ian group. Let us also point out that it is possible to give a Il'ss redundant version of the group I\xioms from which the present axioms follow IUllogical consequences; for t.his, we refer the reader to Problem 14 at the end of the section. When the group operation is clearly understood. one often identifies the group with its underlying set of elements and refers to the group as G rather than as (G, *). For furt.her simplieity, many authors drop the star notation and write ab in place of a * b. While we shall continue to adhere to the a * b convention, we will nonetheless adopt some of the terminology of ordinary multiplication and talk of forming products, multiplying elements together, etc. At times, we shall also be somewhat impreeise and speak loosely about the elements of the group (G, *), when we re:dly mean the clements of the underlying set G; however, this should cause no particular confusion. In order that the reader fully appreciate the generality of the concept of a group and at the sume time guin l>Ome familiarity with this idea, we pause to offer u selection of examples. Further examples appear in the exercises. Example 2-18. Let a be allY nonzero real number and consider the set G of integral multiples of a: G= {naJnEZ}.

The pair (G, -H, wher<', as usual, + indil'llt.es ordinary addition, forms a commutative group. III this eaS(', t.he identity is 0 = Oa E G, while the inverse of an arhitrary e1pm(,lIf, na of 0 is - (na) = (-n)a E O.

36

onoup

2-1

TIIEOHY

Example 2-19. COIlHider till! H!'!' of ordered pnil's, G = {CO, 0), (0, 1), (1,0), (1, I)},

and the operation • defined by Table 2-1. In this group, the identity element is the pair (0,0), and every clement is its own inverse. Here the verification of the associative law becometi a process of detailed enumeration of all possible cases that could arise. Table 2-1

(0,0) (0, I) • ---_.

- - .-(0, II) (0, I) (1,0) (I, I)

(II, II)

(0, I) (I, 0) (I, 1)

(0, I) (0,0) (I, I)

(1,0)

(1,0)

.

(I, I)

(1,0)

(I, I)

(I, I)

U,O) (0, I) (0,0)

(0,0) (0, 1)

Since the entire table is symmetric about the main diagonal (upper left to lower right), the group operation. is commutative. Note that each element of G appears once and only once in each row and column of the table. Indeed, any multiplication table for a group has this feature. Example 2-20. Let P(X) be the power set of somc fixed nonempty set X. As we have SI.'Cn, the systcms (P(X), u) and (P(X), n) possess identity elements and X, respectively, but neither system has invel'SCs for any clements other than their respective identities. Consequently, P(X) does not constitute a group with regard to the formulation of either unions or intersections. It is possible, however, to remedy this deficiency by defining another operation on P(X) in terms of union and intersection which will fCsult in a group structure. More specifically, conHi
o

A

~

B

=

(A - B) U (B - A),

for A, B E P(X).

This operation, known as the symmetric difference of A and B, yields the set which is represcnted by the shaded area in Fig. 2-1. We shall leave as an exercise the v(>rificatioll t hat the symnlC'tric differenc(> operation is commutative and associative. It is easy to H(,(, that fot· any Hl't A s;;; X (that iH, fot' any clement of P(X»,

A

~

0=

(A -

0) u (0 -

A) = A U 0 = A,

which proves that the empty 8('t 0 serves ns un identity element for~. Moreover, A ~ A

=

(A -

A) U (A _.- A)

= 0 u 0 = 0.

2-1

1>";)o'lNITION ANIl

l~XAMPL"~'i

OF GUOUPS

37

This implicH t.hllt, e:wh dl'lIWllt, of J>(X) ill it.!l own invc1'IIC. Conscquently, the mllthclllatil'nl HYHtl'l1l (f'(X), a) is n I:OJlllllut,nt.ive gnlUp.

Examp'e 2-21. As 1\ silllple exampl(! of a noneollllllutative group, let the set G consist of all ordered pail'R of real numbers with nonzero first component: G= {(a,b)la,bER',a~O}. Define the operation • on G by the formula

(a, b) • (e, d)

=

(ae, be + Il).

Thn nSH()(:intivit.y IIf t.he oJlI~mt,ioll flllluWH frolll tho f,ulli/inr properties of the real numbers, for wc have

[(a, b) • (e, tl») • (e,f)

+

=

(ae, be (l) • (e,J) = «ae)e, (be d)e f) = (a(ee), b(ee) (de f) = (a, b) • (ee, de + f) = (a, b) • [(e, d) • (e, f)].

+ + + +

It is readily verified that the pair (1,0) serves as the identity clement, while the inverse of (a, b) EGis (lla, -bla). To see that the group (G, *) is not commutative, merely consider the elements (1,2) and (3,4) of G: • (1, 2) • (3, 4) = (3, 10)

~

(3,6)

=

(3,4) • (1,2).

Example 2-22. For another example of a noneommutative group, take the set G as consist,iug of the six fUllctions f" /2, ... .J6, where for x E R' - {O, I}, we define l2(x)

x-I

x,

f3(X) = 1 -

= !, x

14(x) = - - ,

x

X

f6(x)

i5(X) = .r - 1 '

=

1

-1-'

-x

I.et the group operation \)(' that of functional composition. Thus, as an illustration, we have (/2 • is)(.')

= I2Us(x» = 1-

=

12

x = !a(x),

C~ x) =

1/(11_ x)

38

2-1

GROUP THEORY

whieh implies that. J2 hand, (f6 • !2)(x)

=

•

J6 = h. On the other

Table 2-2

it /2 fa !4

16(12(x»)

=16

G)

=

" ""

l/x

X

= - - = 1,,(x)

x -

I

fo

/2 fa !4 f6 fo f6 fs f4 fa /2 /2 fa fa f4 /l /2 f6 fs f4 f4 fa f5 f6 /2 it fll f" fo f4 fa /2 fo fo fs /2 it fa f4

1 1-

f6

"

'

so that 16 ·12 = is, which shows that the operation • is not commutative. The multiplication for (G, .) in this case is given by Table 2-2. Since functional composition is associative (Theorem 1-7), the system (G,.) is certainly a semigroup. The operation table shows that It is the identity element and the respective inverses are

It l

=

it.

1"41 = 16,

1"21 =!2, fil

=11'"

ri

l

= fa,

1;1 =14.

To encompass all the different groups above in a single concept obviously requires the formulation of the underlying group concept in the most general terms. This is preeisely the point we hope to convey to the reader; the value of contemporary mathematics lies in its power to abstract and thus to lay bare the structurally essential relations between superfieially distinct entities. Historically, thc notion of 11 group aro~ early in the ninet.L·(mth century out of attempts to solve polynomial equations. Galois waH the first to use the word "group" in any teehnical sense when he considered the group of permutations of the roots of such equations. A major achievement in the evolution of the theory was Klein's cln.."sification, in the IS70's, of the various branehes of geometry according to groups of transformations under whieh certain geometric. propertics remain invariant. It remained some time, however, before satisfactory group postulates, free of redundancy, were stated. Definition 2-11, first formulated in 1902, is attributed to the American mathematician E. V. Huntington. In the twentieth century, group theory has embraced all branches of mathematics and, incl(~ed, a wid" vari('ly flf ot.her fields. It is difJieult to give exampltJ8 without beeoming too tedlllical, but the theory of groups is now employed in the study of quantum mechanics, general relativity, and erystallography. In these areas, group theory is not only a tool with which calculations arc made but also a souree of coneepts and principles for the formulation of new theories. A recent example can be found in the physics of fundamental particles with the discovery of a new "elementary partide" whose cxistenee had been predicted from a e1a."sifil'ation scheme hased on groups. It is certainly appropriate to begin our investigation of mathematical systems with this concept.

2-1

DEFINITION AND EXAMPLES OF GROUPS

39

PROBLEMS

J. Determine which of the following binary operations on the set Q are associative and which are commutative. a) a * b = 0 b) a * b = l(a b) c) a * b = b d) a * b = a b- 1 2. Rllppose Lhe system (S, *) haR an identity element; show that if the equation

+ +

(a * h) * (c * d) = (a * e) * (h

* d)

holds for all pOH"ible choices of element!i a, h, c and d of S, then the operation i!! both aH:-Iociative and commutative. 3. Prove that the set of ordered pairs of real numbers together with the operation defined on 8 by (a, h) * (e, d) = (a e, b d 2bd)

+

4.

* *

+ +

constitute!! a commutative !lemigroup with identity. l,(~t 111'1 define a hinary operation * on the !!et S = {I, 2, 3, 4, 6} as follows: a

*h

= gcd (a, h).

For example, 6 * 4 = 2, 3 * 4 = J, etc. Show that (8, *) is a commutative semigroup. [Hint: Problem 17, Rection 1-2.) 5. ConRi!\er the three-element !let S = {a, h, e} and the operation mlllt.i"li(~ation table helow:

*

a h e

a h

abe h a e

c

e c e

* given

by the

Verify that the pair (8, *) is a commutative semigroup with identity, but not a group.

6. In the following illlltanceR, determine whether the systemR (G, *) described are (~ommul.ative groups. For tholie systems failing to be so, indicate which axioms are not satisfied. a) (} - Z+, (I * II - max {(I,ll:, h) a = Z, tl * b = mill {fl, bl (the Hlllall(!r of a and b), e) (} = n' , a * b = a h - ab, d) (J = Z+, a * b = Illax {a, b} - min {a, b}, e) (J = Z X Z, (a, b) * (e, d) = (a e, b d), f) G = R' X R', (a, b) * (e, d) = (ae bd, ad +. bd), g) (J = R' X R' - (0,0), (a, b) * (e, d) = (ae - bd, ad be). 7. RUJlPol'l(l that a E R' - {O, I} Ilnd conRider the set G of integral powers of a: G -= {a k IkE Z}. If· denotes ordinary multiplication, prove that (G,') is a group.

+

+

+

+

+

40

2-1

GROUP THEORY

+

8. u-t G = {I, (-1 iV3)/2, (-1 - iV3)/2}, where i 2 = -1. Show that the lIytltem (a, .) is a group. 9. Con!!idl'r the set a consisting of the fOllr functions/J,i2,Ja,/4: /I(x)

1

= x,

! 2(X) = -x'

hex)

=

-x,

x

with x E R' - {O}. Prove that (G,o) is a commutative group, where functional composition. 10. Prove that the symmetric difference opt'ration

:1

aB

= ( ..1

- B) U (B -

0

denotes

A)

discussed in Example 2-20 may also be defined by the formula

.\ a B

= (:I U B) -

(.\

n B).

11. Granting the associative law, show that the following two operation tables define groups:

...

-

.

I :i

5 7

I

3

5 i

-_._---

5 7 7 5 5 7 I 3 7 5 3 1 I

:~

a

I

...

a

b c

d

a b

a b c b a d c d b d c a

d c a b

c d

12. If G = {a E R'I -1 < a < I}, verify that (G, >II) form!! a commutative group, with the operation'" givl'n by a+b a>llb = - I fib

+

for a, bEG.

13. Let (S, >II) Ill' a sl'migroup wit.hout illl'nt.ity and e hl\ any ('It~ment not in 8. Define the olll'ration on the t«'t"" = S U {e) by ml'SnH of I.he ruleH 0

for all a, b E 8, aoe=a=eoa

for all

a E 8'.

Show that the pair (8',0) is a semigroup with the identity element e; (8',0) is said to be obtained by adjoining an identity. 14. Prove that the following weakened Het of axioms are actually equivalent to the classical axioms for a group as given in Definition 2-11: A group is a mathematical system (G, >II) for which 1) >II is an aN80ciative oper&tion, 2) there exists an element e in G such that a >II e = a for all a E a (existence of a right identity), 3) for ('.sch a E G, thero ('xists an ('lement a-I in G such that a'" a-I = e (existence of right inverse.'1).

2-2

CERTAIN ELEMENTAltY TUEOItEMS ON GROUPS

41

2-2 CERTAIN ELEMENTARY THEOREMS ON GROUPS

AI! wc rmllllrke
Proof. Theorem 2-1 may be used to establish that the identity is unique. To show Omf. an element has exaetly one inverse in G, we proceed as in Theorem 2-1 by showing that two sUPJlosedly distinct inverses are actually equal. To this end, lu!sumc thc clement a E a has two invt·rscs, a; and a~. Then according to the definition of inverse, a

* a~

= a~

*a

* a~ =

a

= P,

a~

*a =

e.

But, the identity element is the same in both eases (there is only one identity as we have Illr('lldy 8('l'n), so that

a

* a~

=

a

* a~.

!\I uitiply both I!ides of this equation on the left by a; (or by a~) to get a~

* (a * aD

Using the associative law, we have (a;

= a~

* (a * a~).

* a) * u; =

(' * a~ = e * a~ whi('h proves that a has ollly one inverse.

or

a~

(a~

=

* a) * a~, a~,

and so

42

2-2

OROUP THEORY

An inspection of the proof Hhows that we have established a little more than is indicated by the statement of the theorem. We have, in fact, shown that if an element of a semigroup with identity has an inverse, then it must be unique. Corollary. Each clement of a semigroup with identity has at most one

inverse. A further useful conclusion to be drawn from Theorem 2-2 is that (a-I)-I = a. This stems from the observation that (a- 1)-1 is an element of G for whidl

Since a itself 1m.'! t.his propcrty, and sin(!c invcl'8CH have jUtit been scen to he uni
=

(a * b) * (c * d)

a * «b * c)

* d).

Proof. L('t UII t('lllpomrily dpn()t.(! t.lw produet c * d by x. opemtion * is I\.'!.'«l(!iat.iv(!, WI! hnvn

a * «b * c)

Then, Killce the

* d) = a * (b * (c * d» = a * (b * x) (a * b) * x = (a * b) * (c * d).

Theorem 2-3. If (G, *) is a group and a, bEG, then (a * b)-J = b- I * a-I.

That is, the inv('rRC of a prodm·t of group clements is the product of their inverses in reverse ordl·r. Proof. A(!cording to the dpfinition of inverflC, nil we nc(ld to show is that (a

* b) * (11- 1 * a-I) =

(b- I * a-I) * (a * b)

=

e,

where e is the group id!'nt.ity. From t.he uniqueness of t.he inverse of a * b, we would then conclude that Using the above lemma, we have (a * b) * (b- I * a-I)

= "'''

* b- 1 ) * a-I) a* (e*a- I ) = a*a- 1

a * «b

= e.

A similar argument ('stnhlishcs that (b- I

* a- I ),* (a * b)

= e.

2-2

I~U;MENTAltY

C.:ltTAIN

1'HEOREMS ON OROUPS

43

Corollary. If a and b are invertible elements of a semigroup with identity, then

To be sure, if the operation

* is commutative,

then we do have

how(wl'r, in t.!1(' ahM
Example 2-23. Lpt. r: dC'lI11l.ll tim HI·1. (If nil urdtln~1 pnil"!! of renl 1lI11nbers with nonzero first. ClolllpOIwnt. If t.he hinary operation • is defined on the sct G by the rule (a, b) * (e, cl) = (ae, be d),

+

t.hen (a, *) is n nOllc'omlllutntive ~roup [Example 2-21). The ici<'ntity clement of t.he group iK t.Iw pnir (1,0); the inverKll of lUI eleml'lIt, (a, b) EGis (1Ia, ·-l)la). A din'l·t c'()JlIIHllat.ioll KIIOWI'I t.hat,

«1, a) • (2,4»-1

=

(2,10)-1

(!,

-5),

while

(2,4)-1 • (1,3)-1 =

(l,

-2) • (1, -3)

= (l,

-5).

Thus «1,3). (2,4»-1 = (2,4)-1. (1,3)-1, 1\8 is guaranteed by Theorem 2-3. However, computing the product of the inverses in the order

we obtain (1, -:I) •

(l,

-2)

= (1, -i),

so that

«1,3) • (2, 4»-1

~

(1,3)-1. (2,4)-1,

I,'or this ~rollP thl'lI, t.he inverl'lc of a pmchl<'t of I'lements if! not equal to the procllld of t.hC'il· rC'sJlc'diVtl illVClrKC'H ill clirC'1't ordm', ThiH Hhoulcl lint. be plLrticularly surprising inasmueh as the group (G, *) is noncommutativc. The previous Il'mma is Iwtually a special case of a result mentioned carlier whil'h in C'fTC't'1. aSKI'rts t.hat, pan'ntheRCs nrc superfluous in 1\ product of group e\C'IIl('nt 1'1, and C'OIlH('qU(·lIt.1y t.hat t.Iwil' olllisHioll ('lUI Il'ILd t.0 110 misunderstanding. TI\(' exa(·t story is told ill thl' npxt t.hcort'lll. First., howevl'r, let us int.n>duI'" Homc ll.lIxiliary notut.ion: Impposc (8, .) is a Hcmigrollp I1l1cl \.lie ,,11'IlIPlltH a I t a2t ' , • , a,. h('\ollg 1.0 S (whcre 11 > 1), Their

44

2-2

mWlll' TIIJ<:OIlY

.~tandarrl produrl,

lIymholized by al • a2 • , , , • an, ill defined recursively as

a, ...

Whl'n n

=

(l~ ... ,

, , ...

an

'I, for imll.iuH'l', u. • aI

•

a2

(12 • a:1 • a4

will t.urn

0111,

t.o he

* aa ... a4 = (al • a2 • a3) ... a. «al ... a2) • aa) • a4'

In essence, we are usill!!; induct.ion to define It pltrt.icular grouping of parentheses and thereforl' It Jlnrtil'lliar clemcnt of S. With t.hill Ill'finit ion in mind, WI' 1I0W lIhow thl\t IL produI~t iii d(!t.erminoo lIolely by the on\c'r of it II fal'torll, and 1I0t, on the manner of distributing parent.ht,!!(,!!, Theorem 2-4. ((;",If'ralizrr/ AS80datillr ',aw). I'(!t. (8, .) be n. HCmigroup and a" a2, ' , , ,all ht' 1\ lid of n ~ ;~ e1em('nhl of S. Theil all possible produ
shall employ is to prove all products of the elements * ... - a"i our argument proceeds by induction on n, where n ~ 3. In the case n = 3, the result follows direetly from the assQ('iativity of -: WI'

a" a2, ' , , , a" arc equal to their standard product at - a2

Next, assume that the assertion holds for all products of m: factors, where m: < n, and let x be any product involving n factors. Sinccx is obtained by slleceflsive applications of the operation -, there must be a final multiplication between two expreflsions having less than n factors. By the induction hypotheses, each of t.hese two expressions equals the standard . product, flO that x may be written as

a :5

l:5k
whence

The usual associative law now yieldfl

2-2

CEHTAIN Jo:LJo:MENTAHY TIIEOHJo:M8 ON GUOUPS

45

Applying first our induction assumption to the expression in brackets (which ('I'rl,uilll.v ('OIlt.Hilll~ II'i'lM I,hnn It fnl'loTM) nllll HII'll Hw dl'finil,ioll of I,hl' stundanl pJ'(}(llII~t. Ilgnin, WI' inrI'" tlml,

This completes the induction stcp and therefore the proof of the theorem. Given four elements aJ, a2, aa, a4 E S, parentheses can be legally inserted and the elements mult.iplied (in the given order) five different ways, Theorem 2-4 Jl(~rmjt.H \If'! to cO/u'lude that all theHC produnts nrc cqual: «(al * a2) * a3) * a4)

=

(al * (a2 * aa» * a4 = a, * «a2 * aa) * a4) al * (a2 * (a3 * a4» = (al

* a2) * (aa * a4)'

As It matter of notation, it will bc our tendency to omit parentheses in writing products; the exception to this will be found 011 those occasions when we wish to emphasize the associative law or a certain grouping of elements. Theorem 2-5. (Cancellation Law). If a, b, and c arc elements of a group (G, *) such that either a * c = b * c or c * a = c * b, then a = b.

Proof. Sinee c E G, c- I exists in G. on the right Mid(· hy c-', we obtnin (a * c) * c:- I

l\lultiplying the equation a

=

*c = b *c

(b * c) * c-'.

Then, by the associative law, this becomes

or a * e = b * e. Therefore a implies a = b.

=

b. Similarly, we can show that c * a

Corollary. The only solution of the group equation x

*x =

x is x

=

=

c*b

e.

Pmof. Tlw eoneiusioll iM lUI illllllediate eonsequenee of the cancel1n.tion 1n.w and the fact that .c * x = x * e.

In n system (8, *), all element xES is said to be idempotent provided x * x = x. Whnt we have jU!5t shown is thut u group possesses exactly one idempot('nt clement, namely the group identity. For an illustration of a system in which eVl'ry element is idempotent the student is referred to Problem 4, Section 2-1.

46

2-2

GROUP THEORY

This last theorem allows us to cancel, from the same side, in equations involving group clements. Wc cannot conclude, however, that a * c = c * b implies a = b, unless the group is known to be commutative. An arbitrary binary operation need not satisfy the cancellation law. To see this, we consider the set G = {I, 2, 3} under the following multiplication table:

*

I

1 2 3

I 2 3 2 I 2 3 2 I

2 3

On examining the table, we observe t.hat 2 * 1 = 2 * 3; but obviously 1 ~ 3. The failure of the cancellation law in this instance results from the fact that when we mUltiply both sidcN of the equation 2 * 1 = 2 * 3 by 2- 1 = 2, the element 2 does not associate with the product 2 * 3; that is, 2

* (2 * 3)

~ (2

* 2) * 3.

Theorem 2-6. In a group (G, *), the equations a * x have unique solutions. Proof. First, x

=

a-I

=

band y

* b satisfies thc group equation a * x =

b, since_

a

* (a- 1 * b)

*a =

b

* a-I) * b = e *b = b. =(a

This shows that there is at least one solution in G; it remains for us to show that there is only one. Suppose there is some other element x' E G such that a * x' = b. Thcn a * x' = a * (a- 1 * b), so that by the cancellation law, x' = a-I *b,

or

x'

= x.

The second part. of til<' t.limr!'111 may be proved in an analogous manner. Corollary. In a multiplication table for a group, each element appears exactly

once in each row and column. Proof. For a proof by contradiction, suppose that the element b occurred twice

in the row headed by a. such that

There would then exist elements

Xl ~ XII.

a

* Xl =

b

and

a

* XII =

b.

Xl

and

XII,

with

2-2

CERTAIN ELEMENTARY THEOREMS ON GROUPS

47

However, this situation is plainly incompatible with the above theorem which IlSserts that there is one and only one solution of the equation a * x = b. A proof for columns can be obtained by imitating the argument for rows. It can be shown that all groups with fewer than six elements are commutative; thus a noncom mutative group must necessarily contain at least six • of thill faet ill ROmewhat, lengthy, although the actual clements. The proof details are not by themselves particularly difficult. The subsequent lemma will serve to isolatc the most tedious aspect of the theorem.

Lemma. If a and bare noncom muting elements of a group (G, *)-that is, a * b ¢ b * a-then the clements of the set,

{e, a, b, a * b, b * a}, are all distinct. Proof. The basic idea of the proof is to examine the members of the set {e, a, b, a * b, b * a} two at a time, and show that each of the ten possible equalities leads to a contradiction of the hypothesis

a*b¢b*a. On several occasions, the cancellation law is used without explicit reference. The argument runs as follows:

= a implies a * b = e * b = b = b * e = b * a, 2) e = b implies a * b = a * e = a = e * a = b * a, 3) e = a * b implies a * e = e * a = (a * b) * a = a * (b * a), so that e = b * a or a * b = b * a, 4) e = b * a implies e * a = a * e =.a * (b * a) = (a * b) • a, so that e = a. b or b * a = a * b, 5) a = b implies a * b = a * a = b * a, 6) a = a * b implies e = b, reducing to case (2), 1) e

7) a 8) b

= b * a implies e = b, reducing to case (2), = a * b implies e = a, reducing to case (1),

9) b

=

10) a

b • a impli(,H e = a, reducing to case (1),

*br

b * a contradicts the hypothesis.

The proof of the lemma is now complete. Theorem 2-7. Any Jloneommutative group has at least six clements. Proof. If (G, *) is a noneommutative group, it must have a pair of noncommuting c1emcnts a and b. According to the lemma, the set fe, a, b, a * h, b • a} then consist.s of distinet member.;. We now proceed to establish that one of tho group ('I('men/s a * a or a * h * a is distinct from these five; however, it is not possible to lSJlceify ablStraeily whether it is a * a or a * b * a.

48

2-2

GROUP THEORY

With the aid of the lemma, we first show that a '" a is different from each member of {a, b, a '" b, b '" a}. To start with, observe that a) a '" a = a implies a = e, reducing to case (1) of the lemma, b) a '" a = b implies a '" b = a'" (a'" a) = (a'" a) '" a = b '" a, c) a '" a = a'" b implies a = b, reducing to case (5), d) a'" a = b '" a implies a = b, reducing to case (5). Thus, either a '" a ~ e, in which case a '" a is the sixth distinct element of 0 or else a '" a = e. In this latter instance, we ('an show a '" b '" a to be distirwt from each of e, a, b, a '" b, b '" a and (~onS(·CJ\wl\t.Iy to be th(~ required sixth clement. The reasoning IlI'/'/' ,I"I)('nclH on the fud that a'" (a'" b '" a) = (a'" a) '" (b '" a)

c) a '" b '" a = case (7), f) a'" b '" a = g) a '" b '" a = h) a'" b '" a = i) a'" b '" a =

=

e '" (b '" a)

(' implies b '" a = a '" (a '" b '" a)

a implies a '" b = b implies a '" b = a'" b implies a = b '" a implies a =

=

a '" e

=

=

b '" a. a,

reducing to

e, reducing to case (3), a '" (a '" b '" a) = b '" a, e, reducing to case (1),

r, reducing t.o ease (1),

whi('h cstllblishcH till' n~HIlIt. Anltlt.crnative proof will be presented in Acetion 2-fi. The generalized associative law insures that the product a '" a"" .. '" a has I\n unarnhigtlouR mel\ning irreHpectivc of how the factors arc parenthesized. Designating the foregoing product hy the symbol ale (assuming there arc k factors), we may int rodUl~e the notion of th(~ pOHitive powers of a. The next definition extmlllH t.hiH i
a E 0 are defined by

a" aO a-Ie

= a'" a"" .. '" a = e,

=

(k factors),

(a-I)",

where k EZ+. With t.hese convent.ions, t.he customary laws of exponents have their counterpart in group theory. Theorem 2-8. Let (G, "') be a group, a E G, and m, n E Z. of a obey the following laws of exponents:

2) (an)'" = a nm = (am)", 4) e"

= e.

The powers

2-2

49

CEHTAIN Jo;Lt:MENTARY THEOHEMS ON GROUPS

The detailed proof of these statements rcquires a brcltkdown into "cu..cres" and (!UII Ilafdy be left ItS an exercille. We caution the reader that the property (a * b)n = an * btl is not to be expected in an urbitrnry group (It monwllt.'s r(~A(l(:t,ion Ilhould V ('onvillee olle of this), DI~ Wc shall 1I0W ('ollchule t.his l'(:ctioll with t.wo (7.'..--_-+-_ _ / partieularly important, examples of groups, since 1',: ,/2 we shall have frequcnt oeeasion to refer to them ___ --~-..l.~-Cl H in t.hp fut.ure. ,.,/': '" '\..,-

c..p

/

j,

___

~/

,~

:

It

Example 2-24. Tlw J,!;rollp to bl' illt.ro
*

Ruo

R lxo

R2iO

R:u;o

II

V

1>J

D2

Ruo RpHl R270 R:u;o

Rlso R2iO Rano Ruo

R:!(Io RHO RIHO

lluo

Il IIHl

DI V

D2 11

V /)2 II

II DI V

II V

1)2

ll2iO RallO RHO lllNn V /I

1)1

11 I'

Ih

/)1

1127o

1)2

Ih lh V

Ih

J/

R2iO

/)2

Ih

R:u;o /I

/I

/)1

Ih

R 270

RIIO

V

R!IO

112711

1)1

llll'o ll!JO

V RINO llano R270

RulO

Ih

R270

Ra(lO RI80

R:wu

RUII

RallO

The ('olllpll'le llIultipli('alion talll!' for the op!'ratioll * is shown ill Table 2-a, Note t.hut /lallo 8('rVCH liS the idcntity elCIIH'lIt ami cl1l'h of R ISI )' /l3so, H, V, D" alHI [)2 il:l itl:! own illv!'rse, whereas Roo lind U 270 ar<~ inverses of mteh other.

50

2-2

GROUP THEOUV

The assoc~iativc law also holds, but this is not immediately obvious. We shall sec later that thc symmctric!! of the !!quare are equivalent to a group of perm utatiom:! of the Het {I, 2, :l, 4} (obtICrve that a HymmetrY ill complctely dcscribc(~ by its effed on the vcrticlcs) and asso(~iativity therefore follows from the associativity of funetional composition. Granting this for the moment, the proof that (G, *) constitutes a group is complete. Similar groups may be defined for other geometric figures; in fact, for any regular n-sided polygon. Problem 14 at the end of this section den.ls with the group of symmetriel!l of the equilatcro.l trianglc.

Example 2-25. Let (G, *) bc an arbitrary group. For a fixed clement a E G, defin a by f,,(x) = a

*x

for eltch

xE G.

That ill, fa multiplil~s (or tranHlates) eanh element of G by a on the left. x E G, then x = a * (a-I * x) = fa(a- I * x),

If

so that fa maps G onto itself. Moreover, fa is one-to-one, for if x, Y E G with la(Y), then a * x = a * y. From the cannellation law, we conclude that X= y. Suppose we combine two of thesc mappings, say la and Ib, under the usual composition of fUllctions. For any x E G, we see that

I ..(x) =

= 1,,(Jb(X)) = fa(b * x) = a * (b * x) = (a * b) * x = I ... b\x), .Ib = lub, SO that the tlCt of all such functions

(fa ·fb)(X)

This means that fa is closed under the operation of functional eomposition. For thl) lIake of 1I0tl1t.ioll, tlCt /<'0 = {fa I a E GJ. Our aim is to show that the pnir (Po, 0) iH Iwtunlly a p;roup. Indeed, if e ill tile iilpnt.ity demcnt for (a, *), then f. a/li!! lUI the identity for (FG • • ), since fa f. = fa •• = fa = f .... = f. ola. 0

l\-Iorcover, (j,,)-I

=

1.. -1, for we have

f .. • f,,-I = fa ...- I

=

f.

=

fa-I ... = la-I. I ..·

We alrendy know t.hat. (:oll1position of fUllctiolls is associative (Theorem 1-7), so it folIowl:I that (Fa, .) formll It group. • PROBLEMS 1. Given that a, b, c,

Illltl

d arp l'\cmcnts of thc scmi!!:roup (G, *), prove that

«a

* b) * c) * d

= a

* (b * (c * d».

2-2

51

CERTAIN ELEMENTARY THEOREMS ON GROUPS

2. Complete the proof of Theorem 2-8. 3. Prove the theorem: A group (G, *) is commutative if and only if (a * b) -I

=

a-I. b- I for every a, bEG.

4. Given a anc! bare element.s of a group (G, *), with a. b = b * a, show that (a. b)k = a k • bk for every integl'r k E Z.

5. Let (G, .) be a group such that (a. b)2 = a 2 * b2 for every a, bEG. Prove that the group is commutative. 6. Given a 2 = e for every element a of the group (G, .), I!Ihow that the group must be commutative. 7. A group (G,.) is said to be cyclic if there exists an element a E G such that every element of G ill of thl' form a k for Home integer k (pollit.ive, negat.ivl', or zero). Hueh an ,~Imllen t a ill ,~alll'd a generator of Ul(l grou JI. a) Prove that any cyr.lic group is commutative. Il) Given a = {I, -I,i, -i}, with i 2 =-1, show that Whieh of its elements are generators?

(a,·) is a

,~yelic

group.

8. Prove that if a and b are elements of a group (G,.) with the property and then a 4 = e = b4 • 9. Prove that if (G,.) is a group having more than two elements, then there exist a, bEG, with a "" b, a "" e, b "" e, such that a. b = b. a. 10. If (8,.) is a semigroup with identity and G the set of all elements of 8 having inverseK wit,h rC!lpect to the operation *, verify that the pair (G, *) ill a group. II. Prove that a group may alternativl~ly he defined as a semigroup (G,.) in whieh, for all a, bEG, ea(~h of the equations a • x = band y. a = b has a KOlution in G. [Hint: Use the eharacteriza~ion of a group given in Problem 14, Section 2-1.) 12. Hhow that any semigraul) (."I,.) with a finite number of element!! possesses an iil,·mJlot.ent e\em('nt. II/int: For 11 > I, proceed hy induction OIl n, the number of elements of ."Ii givon a E S, lot o't = {ak I k = 2,3, ...}, and argue according to whether or not .tI containll the clement a.) 13. For allY syKtem (B, .), define the Het

.t = {a E

I * (b'" r)

S a

=

(a'" b) '" c for all b, c E B}.

If A "" t\, prove tlmt. 1.111' pair (.1, *) ill a "ellligroup. 14. 1.1'1, 1.111' set G ,·onsist or ,·,·rtain rigid 1II0t ions of an '·lJuilal.·ml Iriall~I,·. P"rlllit""d lIIotions I~n' thn'" .'IIlI'kwi,,1' rotations 1l12!t, R24't, and R;wu ahoul the el'nll'r t hrou~h 1.I1Ijt;h·s or 120, 240, and :i(i() dl'grt·(·s, rl'~pe,·ti\"I'I.\", and Ihr,'(' )"I'fil'..tions /,1, /'2, 11.1\11 '-:I I1hout lillI'S II, 12, 1I1111/a 11" illllh·all'd (Fig. 2-:J) . .\s U"III1I, dl'lill" I III' opl'rntion * Oil (J 10 h,' onl' motion follow,"I hy nllothl'r. Prove thnt th,' sysh'm (G, *) is a group.

Figure 2-3

;')2

2-3

GROUP TtU;ORY

15. Let (0,.) ancl (11,0) he two Cartellian product

ax

difltin~t

II =

groups. Define a hinary operation' on the

{(g, II)

Ig E

(I,

II E II},

all followK : for

(gl, 1£\) • (g2, h2) = (g\. g2, h\

0

h2)'

Prove that (0 X H,') ill a group, called the direct product group of (0,.) and (H, o)j show further that this group is commutative whenever the original groups are ('ommutativc. 2-3 TWO IMPORTANT GROUPS

This SC(·t ion is tI('vot~,tI 10 an eXllIllilHlI ion of I,wn important and frequently m«,d grllllpH: til(' gl'ClllJl of illlC'g(,r!I Illoclul" ft allcl Ihe 1!;11111P of l><'rmutntionH of the I'lellwllts of a 1'1'1 (tllC' s(J-I'alled HynlllH'1 ri(' group). Whilc thcl!C group!! nrc of ROIlW int('rC',,1 pl'l' 1;(', our mnin purpo"C' ill illt.rochlC'inp; them ill to provide two more ('oncrete examplelS 10 iIIustml.e eonccplK which will be developed subsequently. W(! hegin wit h all illvl'KI igal.ioll of t.he not.ioll of ('ongrtwne(', in t.erms of whieh the group of illt.cW·r!I modulo n will hI' forlllulat(!d.

Definition 2-13. Let n be a fixed positive integer. Two integers a and b are said to be congrent modulo n, written a

== b (mod n),

if and only if the difference a - b is divisible by n. That is, a if and only if a - b = 1m for some integer k. For instance, if n

=

== b (mod n)

7, we hnve

== 24 (mod 7), -!i == 2 (mod 7), -s == -50 (mod 7), etc :J

If a - b is not divisiblc by n, we Ray that a is incongruent to b modulo nand, in this cast', write a of h (mod n). It ill noh'wort hy I hat, C'vC'ry pair of int.I'p;('fII nre ('onp;ru<mt. mociulo 1, while II. 1':1 il' of illl C'g(,l"~ al'" ('OIlI(I"IIC'II1. lI\o(hllo '! pnlViclt,t! I.ht,y an' hot.h (WI'II or hot,h odd. Our firHt thc'orl'lI\ pnlvicll's a useful .. Iuu·nd.('rizltt,ion of (~ongrll('JI('(l modulo n in tl'rms of rCIlHliIHlerl! 011 division hy n.

Theorem 2-9. I..(It 11 Iw a fixed J>o~itive intl'gl'r nnd a, b be arbitrary integers. Then a == b (mod n) if alld only if a and b have the same remainder when divided by n,

2-a

TWO IMI'OItTANT GIlOUPS

Proof. SUppOIIC firMt a IE b (mod n), 110 that a = II On division by n, b lelwes a certain remainder r: II = qn I r,

+ kn

53

for HOme integer k.

o :5 r < n.

whef(l

ThUI!, a = b '1- kn = ('1 1- k)n 1 1', whi"h HhowH a has t.he Maille remainder I\!! b. On the other hand, let a = qln l' and b = q2n r, with the same remainder r (0 :5 r < n). Then

+

with

ql -

+

q2 all int.cgcr. Hence, n is a factor of a - b I\nd

HO

a

== b

(mod n).

COlIgruC'nc'(' JIIay hn vi(!wC'cI 1t.H Il t.ype of equalit.y ill tlw KI'IIK1! that itH beImvior with rC'l!p(!d t.o acldit ion ILnd lIlult.ipli(!nt ion iH rl'lllillil'!eC'lIt of orciillary (·qllnlit.y. HOlllu of U\(' (!Ienwllt.nry prop(!rt.im-l of (!qlllLlit.y whieh all'!O cnrry over to congrucrW(!1! nrc liNt,c'u in the next thoor(·IIl. Theorem 2-10. J..ct n be a fixed positive integer and a, b, c be arbitrary integers. Then 1) a == a (mod n), 2) if a == b (mod n), then b "'" a (lIIod n), 3) if a == b (mod n) and b == c (mod n), then a == c (mod n), 4) if a == b (mod n) und c == d (mod n), th('n a -/- e == b d (mod n), ac == db (mod n), !i) if a == b (mod n), thcn aC == be (mod n), () if a == b (mod 11), then ak == hk (mod n) for every positive integer k.

+

Proof. For any int.cp:er a, a - a = 011, so that a == a (mod n) by Definition 2-13. If a == b (mod n), !.Iwn a -- b = kn for sOllle int.C'ger k. Henc(~

h

a -. (

k)n,

wherc -k is nn integ('r. This yields (2). To obtain (:3), Mllppose that a == b (mod n) and b == c (mod n). Then

a - b = kn for

HOllie

and

b-c=hn

intl'l(C'rM k, It. The rdo I"I! , a-- c -

(a - ,)

1 (II -- c) --

whidl implies a == c (mod n). Similarly, if a == I) (mod n) ancl c Much that a - b = kIn

== d

k,t

+ /", =

(k I It)n,

(Jllod n), thcn th('re exiHt integm's k .. k2

and

54

(lItolTI' Tln:o!!y

COI\I!('q UCII t.ly,

(a -I- c) -

+ d) =

(b

+ (c + k 2 )n,

(a - b)

= (k l

d)

=

kin

+ k2n

or a

+ c == b + d (mod n).

AIl:!O,

ac ... btl (mod n). This ('stahli!!lw;! (4). Prolwrty (:;) follow;! dirt'cUy from t1w second part, of (4), since c ... c (mod n). Finally, WI' prove (Ii) hyan inuudiw argumcnt. The Ktatement is certainly tmc for k = 1. AssulIJing it holds for an arbitmry k, we must show that it also holds for k -I I. Bill, this is imnl<.'uiate from (4), since ale == ble (mod n) and a == b (m(1I1 11) imply aka ... bkb (mod n), or a k+ 1 == 1I1e +1 (mod n).

In the fOl"('goillg t.h('())·('m wc !il\W that if a == b (mod n), then ea == eb (mod n) for any int.eger e. It is interesting to note that the converse of this statement fails to be truc. For instance, 2· 4 == 2· 1 (mod 6), yet 4 ~ 1 (mod 6). To put. it anotiwr way, one cannot, lllll"est.rictedly apply the cancellation law in t.he alg!'hru of congrll('IIC('S. 1'11<' only posit.ive as.'I<'rtion that call be made in t.his r('glll"li is ('mhodil'd in Ole following th{,'Orlllll. Theorem 2-11. If m "'" (,1, (mod 11) and c i;! rl'lat.iv('ly prime to n, th(m == b (mod n).

a

Proof. By hypoOwsis, e(a - b) = kn for some integer k. Since e is prime to n, it. follows that II must. divid!' a - b [Theorem 1-14); hel\(~c, a == b (mod n). Definition 2-14. For lUI arbitrary integer a, let [a] dellot
r,c E Z I x ... a (mod n)} {x E Z I x = a

+ kn for some integer k}.

We ('alI la) t.he cOlly/'umce dass, modulo n, detemlinoo by a and refer to a as n l'epresenlatiz'e of this cIa!!!!. By way of illilst mt ion, HIIPPOS(' thaf. WI' an' d('aling wit.h 1'11<'11 [01 :.1' E Z I .1'· ak fOl' :-lUll"! k E Zl I

\

... ,

!I,

Ii, -

('ongrll('I1(!(~

a, (I, a, Ii, !I, ... l.

modulo 3.

TWO IMJ'OltTAN'I' WWIIJ'S

AIHO [I) =

{X

EZ

IX

= 1

+ ak for som<' k E Z}

{... , -8, -5, -2, 1,4,7,10, ... }. Similarly,

[2) =

[ ... , -7, -4, -1,2,5, X, 11, ... }.

Obs<'l"Ve that every integer lies in olle of these three classcs. Int.egers in the sanle eonp;rucn(~c dass af(~ eongl"lll'nt Il\odulo 3, while integers in different c-Iassl's an' iIH" 0 IIJ?; r \1('11 I lIIodulo :t A particular ('()ngru('n('(~ (,lass Illay be d('sigllal.(~d in a variety of ways by merely changing its reprcRentativ(-. I II the above illustration, for instance, [-7) = [2] = [11] = [35] = ...

It sulli(~cs 1.0 rl'lIIllrk t.hat the eharaeteristic feat.nrc of t.lwsc variou!! represcntatives is that., in I his I'as(', tlH'Y all difTer from clwh other by multiples of 3, and in p;(,IH'ml, difTer by multiples of n. For ('onveni(mee, onc often selects the slllllll('st nOIlIlI'p;at.ive int.cp;pr from ('tu·h congruence dass t.o represcnt it; in pr:wt.i(·I', WI' shall :ulh('f(- 1.0 t.his notat.ional (·onv('nl.ioll. To return 10 t.11e general ease of congruence modulo n, let Z,.

=

{[OJ, [1], [2], ... , [n - II}.

Several properties of t.he collection Z,. which we shall lat.er requirc appear in HII' lwxt, t.hl'or(,llI. Theorem 2-12. Ld, n he

It

1) for !':lch [a] E Z", [a] ..,&

pOHitiv(~

illt('p;l'rllllll Z" he

fIJi

(Iefillcd above. Then

0,

2) if [a] E Z,. and b E [a], then [b] = [a]; that is, any clement of the congruence class [a] determines the class, 3) for any lal, [1/] E Zn where raj ..,& fIJ], [a) n [b) = 0, 4) Uaa] I a E c--

Z:

z.

I'roof. The first three assert.ions of Theorem 2-10 indieat.c that. the relation a"" IJ (mod n) forms an equivalence relat.ioll ill !.lIP s('l. Z of illt('gc~rs. Indeed, the

eongruenee daSiSes as defined in Definilion 2-14 are >limply the equivalence dasscs for this equivnlenee relation, Vi('w('(1 in t.his light., the statement of Theorem 2- 12 is a I ran>ll:ttion of Theorem 1-5 into t.he lalll!:uage of "eongruellce modlllo n." TtlP set. Z", whosl' element.s arc tlH' ('onl!:rlwn('c da>l~-s lllodulo n, is tradit.ionally known liS t.hl' sd of iflfl'fIl'1',~ 1/111//U/1l n. It, llIay sl.rikl' t.hl' relul!-r Ural. I.his 1,('rIllillol0lt.v is HOIIH'wllllt irmppr'ol"'iatl' fOI', 11I·(·(·is(.J.v sp!'akilll!:, Ow dmllellt.s of Z" art' 1101 l'ingll' inl.l'gl'l·s, hili, mlitl'r illfinit.(' ~'IH of int'('l!:l'rH. l\(on-ov('r, t.Iw :l1't. Z" is 1101. illli"il(~, lik(, I hI' illll'It('I'S, bill. is LL lillil.!- SI'1. wit.h /I. (~I('IlWIlI.H. Whil(~

56

2-:1

GROUP TIIBORY

t.his i:> not quite in accnr'll with our intuition, we bow to long-standing custom eOllt,illll!' 1.0 n~fl'r 1.0 Z" aH t.h(, illt,l'lI;l'rloi IIIOdliio n. Bya partition of t1w set S, we mean a family of nonempt,y subsets of S which are pairwise disjoint and whose union is all of S. It, follows from Theorem 2-12 that, for eaeh n E Z+, the integers modulo n {~onst.itute a partition of the set Z.

I1IHI Hindi

+"

Definition 2-15. A hillary operation may be defined on Zn for ('ad, [al. [II] E Z", 1«'1, [a] -f .. [II] ~' [alii].

11.'1

follow:>:

D{'finition 2-1;; ass!'!'ls I,hat. Hw lIlodular sum of t.wo {~ongruellcc (,lasses [a] and [b] is tl\(' Uni'll\(' mellliwr of Z" whil'h ('olltaills the ordinary sum a II. Ilow(w(~r, th!'r!! is It ImbUe prohlelll (·olllll'et.l'd wit.h t.his delillit.ioll. IlIlllollllud, as addit iOIl of ('ollll;rUl'n('e c:lItsses in Z,. is detin{'d in teruls of reprel:lCntatives from th(,sl' dusses, we must show that the operation +n doeM not depend Oil the two l'I'P!'l's('lItativ{'s dloS<1JI. It mllst be prov('d formally, that if [a'l = [a) alld W) = [/1), tlH'II [a'] +n [b'] = [a] +n [b], or rather, [a' b') = [a b]. ;\Iowa' E [a'] ~ [a] alld 1/ E W] = [IJ], whil'h impii!'s

+

+

a'

==

a (mod n)

b'

and

+

== b (mod n).

By virt 11(' of TlrPOI'!'II' :!-1O(4), it, follows t.hat.

a' -I h' =_0 a

II (mod n),

-j

or

a'i b'

E

tal

bl

+

Th('orl'm 2-12(2) thl'lI ill!ii('at!'s that [a' 1/) = [a I /1), IlS dl'sired. Thus the opernt.ion i:> IInllmbig;uously defined, independent of the arbitrary choice of r('pr('sent at iv(·s.

+"

Example 2-26. Rllppose we {'onsider ('onJ!;I'UI'IH'!l modulo 7 and the typical mldit ion [a) -I- 7 [(j] = [3 n] = [9].

+

SinN! [:1] = [10] and [0] used

=

[101

[--I!)], the same allliWl'r should he obtained if one

+7 [-I;;] =

[10 - 15] = [-5].

While these results appear superficially different, both congruence classes [9] and [-!)] may be !'xpressed more simply as (2). Thus, although written in terms of different repreRellt.atives, either modular addition gives the same sum, [:2). Other pORsihl1' ('hoi(~es [-4] -h [-X] = [-121. [17) +7 [0] = (23), f3] +7 [13) = [Hi], also yil'ld the sum (2). We nt·(· 1I0W in a posit ion t.o prove one of UI(' prindpnl theorems of this section.

Theorem 2-13. For elwh posit.ive inl!'ger n, t.l1(' mathematical system (Z", t-,,) forms n ('omlllutative group, known as t.he (/1'OUp of inieyel's modulo n. J'l'()of. TllP ass()('iat ivily and ('ommut.ativity of t.\w opl'ralioll -1-,. nre 1\ direct consequenee of til(' sanl(' properties of the intl'geriS under ordinary addition.

2-:1

Indeed, if [a],

TWO IMI'OHTANT GROUPS

rbI,

57

[e] E Zn, then [all .. ([bj-I .. [c)) =Iaj I" [b I

r.J

+ (b -/ c)] = [(a + b) + c] = [a + b] +.. [r] = [a

([ll) I" [II)) I .. If').

Similarly, [a] 1-" [b] = [a

+ b] =

[b

+ a] =

[b) -1-.. [a).

By d('fiuition of -I,,, it is clear Um!. [0) iH the identij,y elmnellt. [a] E Z", then In - a] E Zn Ilnd [a]

+n [n

thlLt [a)-I = [n -- a]. mutative group.

110

- al

=

[a

+ (n

- a)]

=

Filll1l1y, if

[n] = [0],

Thill l'OIl1pl(lh~s the proof that. (Z",

+..) ill

n com-

InddmltnIly, Thron~m 2-Ja nlHO shows thnt for HV(' ..y positive integer n t1l1're ex iH\.l'I at. 11'11.,{1. 0111' eommut.at,ivl' gnmp with n "ll'Irll'lIl.s. If WI! adopt Uw c'ollvlmt.ioll of dl'lIiglmt,illg c~a"h I'ulIgru()IW(! dUIlI'; by it.H smaIlI'!;t. lIonn(·gnt.iv(~ I'(·pn,sellt.utiv(', 011'11 the ol)('ratioll t:lhle for, say (Z4' -h), looks like

+.

[0] [1]

[2] [a]

[0] [1]

[0] [1]

[1] [2]

[2] [3]

[21

[21 [a]

(a]

[0]

[3] [0] [1]

[0]

[1]

(2]

[3]

For simplir.it.y, it is convenient to remove the hrnckets in the designation of the eongruen(!e dallses of Zn. Thus we oft.1'1l write Zn = {O, 1, 2, ... , n - I}. With this notation, thc above operation table assumes the form

+4

0

0

0

1 2

3

1

2 3 2

:~

1

2

0

2 3

;~

3 0

0

1

2

I

For the seeollu of our I.wo exnmpleH, let us turn to the study of permutation groups. To this clld, supposc Umt N is n finite set having n clements which, for simplieit.y, we tnkl' to be t.he first n natural numbers; that is to SI\Y,

N= {1,2, ... ,n}.

58

2

GROUP THEORY

Definition 2-16. By a permutaliun of the set N is meant any one-to-oll mapping of N onto itself. In what follows, the totality of all permutations of the set N will be deno!.,·, by the Rymhol S". RiTl{'e the number of differ('nt, permut.ations of n object,!; i nl, the fift;t thing to note is that Sn ill itself a finite set with n! distinct element, Next, any permutation f E Sn may be described by

f= {(l'/(l», (2,/(2», ... , (n,/(n»)). While this is the Il(!eeptable funct.ional notation, it will prove to be more veni('nt to r('prcll('nl. f ill a two-line form

('Oil

where till' (~nrrl'Rpollding imag(,R uppear 1)('low eaeh integer. Clearly, the ord,·, of the ('ICIIII'lIts ill th(' t.op row of thiR lIymhol ill inulIllwriul, for th(l columll: may lw rl'arrnllged without affectillg t.he natufC of the function. Prccil:ll'l) speaking, if (J iR Illl arbitrary p(!rmutation of the integers, 1, 2, ... ,n, thcn .' could equally well b(' given by f -

(g(l)

g(2)

f(g(t»

f(g(2»

...

g(n»)

... f(g(n»)

From thill, WI' illf('r UIIlt. (,Iwh of the n! {>I'rmutntions in' 8" may be writ,tA'I' in n! ditTl'fCnt way!!. For illlltarU'e, the following two symbols both reprcsclIl the same clement of 8.:

(1 2 3 4) 2

4

:1

I

( 214:i).

'

4 2

1 3

Permutation!'!, h('ing fum'tions, may be multiplied under the operation of functional compol'litioll. Thus, for lK.·rmut!ltions I, U E S",

fog

=

(1 2 f(t)

( =

=

(1(\)

n) f(n)

f(2)

g(2)

l(g(1»

f(g(2) )

C(g;l)

2 l(g(2»

(1 2 ... 0

g(1)

g(2)

g(n) ) f(g(n»

f(g~»

)-

0

...

( 1 g(1)

g~) 2 g(2)

g~)

2-3

TWO IMPORTANT GROUPS

59

What we have done is to rearrange the columns of the first (left) permutation until its top row is thc same as the bottom row of the second (right) permutation; thp product /0 g is then the permutation whose top row is the top row of the se(!OII(I factor and whose bottom row il:l the bottom row of the first factor. With a little pmet.iee one can evaluate produet.l:I without luwillg to w.rite out. this intermediat!' preparation. }lany authors prefer to carry out the multiplication of P(·"llllll.ations in the oppo:-;it,(! onl"r (that ill, they apply the factors in It prouud frolll )(·ft. to right.), alld t.he read('r :-;hould h,· parl,il'tllllrly wlltehful for this, Before stating a theorelll whieh indieutell the algebmic nature of Sn under I his 1lI,,1 h()(1 of (!Ollll)()sit ion, we hop(~ to dnrify sOllie of till' foregoing "ointH Wit.ll IlII ('xILlllple,

Example 2-27. If 1.111' s(·I. N ('onsiHls of t.he illt.(·/.t(·rs I, 2, a! Ii Jll'rlllllf.nl.ions ill Sa, 1I1L1l1dy, 2

2 2

tlllln t.hcn! nrc

=C :;) , .ra=Ca :J) , fs =C :a)J , .f6 =C ;) .

=C :~) , f4 =C :) ,

II

a,

/2

:~

2

2

:l

2

2

2

2

:l

3 2

2

A t.ypical lIlultiplieation, say f4 0/6, proceeds as follows:

.f4 f6=C 0

=G On the other hand,

16 ·f4

WI!

2 :~

h:wl!

=Ca =G

flO

a)'·Ca D a 2

2 2

a

2

2

2

:)·G

2

:;) ·c ~)oC

2

:J)= C I

2

2 :l

:)=12,

2

:~)

a

2

2

~)=G :) =fa,

3

2

1

that. lIIult.ipli('ntion of pl'l'mutatiolls is not ('ol1lmutntivl',

Theorem 2-14. 1'1)(' pair (S", .) forms It /.troup, kllown as the symmell'ic f/l'OUP lin n Hymbo/s, wlli('" is IIOJJl:OIIUllutut.ivc fOJ' n ~ :t The proof of f his fad. is OIllil.t.('(1 in:l.'lIl1ll('h as !l mom gllllCral version of the thcorem will be l!;iv('11 shortly. III Jlus:sing, it is only 11I..'Cc8I!ary to note that the

(iO

2-:\

I:llOl'1' 'I'll EOHY

identity e1cmcnt for (,sn, 0) is the permutation

whil<' thc

1lI111t.iJllieativ(~

2

n),

2

n

inverse of any permutation f E ,sn is described by

l'arf'ntlH'tic'ally, we might observe that t.he gnlllp of symmetrieK of ttw 2-24) I'Iln he Imbsurncd uuder the theory of permutatiou groups, for these symmetries really do nothing more than map the four vertices of the square in It one-to-one fallhion onto t.hemselves, This particular group is ohviously not ('qual to the symmetric group (R., 0), sinee the set S. mllst. (,(lIItaill -1! c :'!·I ('ll'lIl1'lIt.s; illdl'eli, one ('lUI I'Ilsily lind P<'.'lIlUtatiolls in S. whil'h do not correspond to allY lIymmetry of the square, Rnther, we have an exampl(~ of what is ealled It lIuhgroup of the group (S., 0). In the following Homewlmt t.l'('hnieal definition, WE' introduce a special typf' of Iwnllutllt ion c'alll'c11t c'yc'I". sqllllrl~ (ExaIllJlIc~

f! I, n2, . , . , nl: be k distinet perlllutatioll f E S,. is such that

Deflnition 2-17. LI't

If

It

for f( 11k)

-

fen) =

nI ,

<

between 1 and n. -

k,

alld for

It

1 $ i

illtl~gCrs

n fl {nl, n2, ... , nl:},

then f is said to be a k-cycle, or a cycle of lenyth k. Simply put, a cycle replaces n. by n2, n2 by n3, ... , and finally nl: by n'l, while leaving all other d"mcnts fixed. For cycles, a more condensed notation than the usual two-linl' form is to write (nl! n2, , .. ,nt), indicating that eneh integer is to be replac('d by its suecessor on the right and the last integer by the fil'l:lt. III the Kymmetrie group (8 6 , .), for example, wo hlwe

Of course, eaeh integer whil'h is omitted in this cycle notation is presumed to map onto itself under the permutation. A given ('yde may clearly be represented in more than one way, since any of its elements can be put in the first position of the one-line form, all with (2

5 3)

=

(5

3

2)

=

(3

2 5).

2-:l

TWO IMPOHTANT GROUPS

61

A final comment, before we formulate a significant result on the factorization of pC'rmutat,ions, is that olle multiplies ('yell's by multiplying the permutations they reprt·scllt. Thus, ill (Ss, 0) again, (2

5

a)

0

(t

2

4

a)

=

C

2:l45) 4

135

F.V(~ry pennlll.nt.illll f E 8 n call be writ.l.C'n M a commutative product of cycles, no two of which have an element in common.

Theorem 2-15.

Proof. First, consider the set of images of the integer 1 under successive powers of /: {f(1),/2(1),/3(l), ...}. As usual. hy we mcan J of 0 ' " . / , m times. Hincd.he domain of J is fillit,e,ji(t) ,- 1'(1) for SOIlW i < j, whmwe P-i(l) = 1. This in turn implies that there exists a lcaHt positive integer k (I ::; k ::; n) for which /"(1) = 1. Let (1,/(1),/2(1), ... ,/"-1(1» be the first cyele of the permutation J. If the element 2 is not found in t.his eycle, repeat the foregoing arJl:llIllC'lIt. 1.0 ohtnill nnotlll'f c'yd(' (2, J(2), f2(2), ... ,P-I(2», where j is thn KIlI1L1I('sl. pOHitive illtcg(!r KliCh that P(2) = 2. III Ilt llIost n Hlleh steps, this proeedunl must ternlillatc. The order of multiplieation of the resulting eycles is immatclrial, for they clearly have 110 clements in common.

r

To illustrate Theorem 2-15, considered above; in this c!ase,

l~t

us return momentarily to the permutation

since f(I) = fl,

r(I) = /(3)

=

1,

and f(2) = 4,

/2(2)

=

=

f(4)

2.

The simplest of pennut.ations arc undoubtably the 2-cycles, for they just int.erehnnJl:(J t,W() dl'flll'nt.s and Il'nve all ol.llI'rs fixed. It is cust.omary to refer to sl.wh eydeH as Imnxposili()ns. Corollary. Every permutation may be expressed as the product of trans-

positions. Proof. In light. of the preceding result, it suffices to write any k-cycle as a product of transpositions. A direct computation shows that this can be done rather simply in the following manner: (1

2

...

k)

=

(1

k). (1

k - 1)

0

•••

0

(1

2).

62

GHOUP TIn:OUY

As the above trnlll'ipOAitiollS havl' all illt,(~jI;('r in ('om mOil, t.hey do not, ill general, commute. Furtlwl"more, the decomposition is by no means unique. For instance, (1 2 3) = (1 3). (1 2) = (2 1). (2 3).

While n jl;ivl'n pl'rlJlut at ion may hI' fUl:tored into It produet of tmnspo8itions ill a variety of ways, tl\(' IIl1mlwr of \,mllsp08itioml involved will alwaY8 have till' same parit.y. That is, if 0111' fadol"i7.lIt,ion has lin ev('n (odd) number of tmnspositions, then cvrry faI'tOl"i7.:t\.ion mllst have an cvrn (odd) numher of transpositions. The notion of permutation as presented in Definition 2-16 is not. sufficiently general for all ollr pllrpos('s, since it restriets liS to finite Rets only. It would sl'em Illore lIat uml to dl'finp this I'OIWI'P\' for arhitrary Het.H, fillite or infinitl', as follows. Definition 2-18. If G is any nOlll'mpty set, then n permutation of G is a one-to-one function from the Ret G Ollto itself.

We rcprl's('nt tIll' spt of all pNmutatiolls of G hy the symhol, sym G (tlll' reason for this ehoi('e will he clear in a moment). To he sure, if G is a finitl' set having n ('11'1111'11\ s, tlwn sym G = 8,.. I n this genrral fmml'\\'ork, it, is possihle to prove a st.rueture theorem from which Thrort'1II 2- 14 will follow as a speeial ense. Theorem 2-16. For a 1I0011'mpty set G, the pair (sym G, ~ constitutes a /!:roup, ('alled Ihe symmetric IImUJI of G. ['roof. Tlw firs\. thill/!: to til' dlllll' is to show that. is adually a binary operation on I:lylll U. (:ivplI nrhit rary pC'rIJlII\.atiom; I, f/ E sym fl, t.he ('omposition f· f/ if! ohviollsly It fUlII'tioll frolll (J illto a. If t.he e1l'llll'llt C E a, thell t.here exists sOllie /) E (J SIl"'1 tim!. 1(11) .. c; silllilarly, thl'm is lUi dellll1llt a in (J for whieh g(a) = b. HC'Il!:I', (f. y)(a) = I(f/(a») = I(b) = c,

whi('h shows that f (I is nil onto function. By definition, hoth f and g nre one-to-one fundiolls; we would like to conclude t.hat, thl' ('olllposition f. II all«) has thii'! propprt.y. For this purpOf!e, consider arbitrary plements a, bEG with a ¢ b. Since g is one-to-one, the images f/(a) and ,,(b) nrC' ne('cssarily distinet. in G. But thpn, the one-to-one ehnrlwterof I implipsl(y(a») ¢ f(y(b»). In otll('r wonll:l, (f. !/)(a) ¢ (f y)(b), whil'h pstahlisill's tilt' (Ie'sirprl OII1'-i.o-OIl('nI'SS of I.!/. From t.he prece
0

f • io = f = io· I,

2-3

TWO IMPORTANT GROUPS

63

rOl'lLny I E sym (f, showing; i(; to Ill' lUI identity elemcnt for the liystem (sym G, 0). If I E sym (;, the inverse fUlwtion I-I exists, is a one-to-one function, and maps the sl'!. (; Ollto itsdr. l\Iorl'ovPI', I-I is the inverse of I with rClipeet to C~olllp()sitioll. All of whi .. h justifies th .. st.atpmpnt that (sym G, .) is 1\ group. PROBLEMS I. Prove thut if a

==

== cb (moel CIl). x < 15, of the equation

b (mod n), then ea

2. a) Find all solutions x, where 0 ~ 3x == 6 (mod 15). b) Prove that 6" == 6 (mod 10) for any n E Z+. 3. I )esI'ribe the partition of Z eletcrmined by the intl'~er8 modulo 5. 4. Ld I'(x) Iw 11 polynomial in x with inte!(rtll codli .. ients. If 11 is 11 solution of the equation 1'(x) == 0 (mod II), and a == b (mod n), prove thl1t b ill also a solution.

+

5. Show that the pair ({O, 4, 8, 12), 16) is a group. 6. Use the fact that 10 == 1 (mod 9) to prove that an inte!!;l'r is divisible by 9 if and only if the sum of its cliJ!;its is elivisible by 9. Illint: Express the integer in decimal form as a sum of powers of IO.J 7. For any integer n, prove that either n 2

==

U (mod 4) or n 2

==

1 (mod 4).

R. a) Detc'rmine solutions of the cOIlJl;ruen!:c' equations £2 == -.1 (mod 5) and x 2 == -1 (mod Ia). This shows, loosdy speaking, that. the ~quare of an integer in Z" may be negative. h) If thl' I'qul1tiull J.2 ~ a (mucin) has 11 solution .1:), show X:l = n - XI is also a solution. 9. Huppos(' a 2 "" b2 (lIIocl n), whi'rc: a"'" b (mod n) or a "" -b (mod TI).

11

is

prime· nllmhc·r.

It

Prove thll I. I'i t.\wr

sYIIIII\(~trit: group on t.wo symhols, (8 2 , 0), is t:ommut.ative. h) J)('lIIonstmt.n Umt. t.hl' jl;rclUp (,"h, 0), :mcl henel' IUIY lurger symmetric group, ill 1l0n!:omlllutllt ive hy c:on"idering the pcrmll(.llt.ions

10. a) Prove t.he

and 11. Express the following permut.ations as (a) productll of cycles having no elements

in common, (h) prudul'l.!S of transpositions.

2 :J (j

4

4 5 6) (1 1 2

5'

Ii

234567) 7

(j

2

1 3

4

12. Hhow t hat, the e·,\'C'le (1 2 ;~ 4 5) may he written as a product of 3-cycle.
11)-1 = (II 11 1 ... 2 1); in particular deduce t.hnt. C'vl'r,\' tl'lIl1spo"it.ion is it." own inversc'. 14. Thn sYIIIIII('tril's of thl' S(pIII.rll (Exulllple 2-24) muy be interpreted &8 permutalions of t.he. vl'rti(,C>!. HCI'fl'sent (,ach of thCI!IC lIymmctricli by a corresponding permutation.

64

2-4

GROlTP 'fHEOIlY

15. Consider the set G consisting of the four permutations

(11 22 3344), (2I 21 a4 34),

~).

Show that (G, 0) constitutes a commutative group. 16. Form the sct G = {f,j2,f3,r,F',r) , where! is the permutation

! _

(1 2 3 4 5 6), 2

:J

4

a

6

1

and prove that thl' pair (0,0) ill a eommutative group. 2-4 SUBGROUPS

There nrc two Ktnnd:ml t.eehniqul's in at.ta(·king t.lw problem of t.he strueture of a parti('ular j!;roup. One method ('ulls for finding all the subgroups of the group, with the hopl' of gaining information about the parent group through its local structure. The other approach is to determine all homomorphisms from the given group into a more familiar group; the idea here is that the images will reflect some of the algebraic properties of the original group. On dOKl'r Kcrutiny. WI' KhaH KI'C that while HICKe lill('to! (If inveto!tigation aim in difT('rent dirpdions, t.hey nrc not entirely unrelnt(!d, hut rather'll.spects of th(! Kiln\(' probl(·m. For the moment, however, our attcntion ito! focused 011 analyzing It group by means of its subgroups; the question of structure-preserving mappings is a morl' sub tic matter and will be deferred to n lat.er section. From variolls examples nnd exercises, the reader may have noticed that "I'rlnill suhsl't.s of 1.1 ... 1'II'III1'II1s of n grouJI Il'llcl 1.0 III'W I!;rlluJls wlll'lI UlIt! n!Htri(·l.s tlw group olIPratioll to these subsets. It iH this situation in which we shall be primarily interested. Definition 2-19. Let (G, *) he n group and II ~ G be a lIonempty subsef of G. The pair (II,.) is snid to be 1\ subgroup of (0, *) if (II,.) is itself n group.

Eaeh group (0, *) hUH two oLvioUl~ subgroupH. For, if e EGis the idcntity clement of th(! group (0, *), t.hen both ({e}, *) and (0, *) arc subgroups of (G, *). These two sUhgfOUJlIl arc often referred to as the trivial subgroups of (G, *); all suhgroups hdw(,pn thelle two extrernefol are calIed nontrivial subgroups. Any subgrollp different from (0, *) is termcd lIroller.

Z. and Zo dcnote the setH of even and odd integers, rcspe(~­ tively, then (Z., +) is a subgroup of the group (Z, +), while (Zo, +) is not.

Example 2-28. If

Example 2-29. Consid('r (Zo, +6), the group of integers modulo 6. If

II

~

{O, 2, 4},

2-4

SUBGROUPS

65

then (H, +6), whose operation table is given below, is a subgroup of (Zo, +0)'

+0

0 2 4

o

024

2 4

2 4 4 0

0 2

Example 2-30. Let (G, *) he the group of symmetl'ies of the square (Ree ExILlIlP'" 2 2·1), wlll'I'I' a :U uo , H IH ." H 2711 , H:lflfl , II, I', /)It 1)2] IUIII t.11I' OPI'I'ntion * l'onKiKt.!oI of following olle mot.ion by anot.her. Thi!ol group I~ollt,ltin!ol eight. nont.rivial subgroups. We leave it to the reader to V('rify timt the following sets comprise the ei(!ment.s of these subgroups: 0_

{If lilli, /(:11111, II, V}, {R IIlO, Raoo,

[)It

D 2 },

{Raoo, D 2},

{R 180, Ra6u},

{R 360 , H},

{R a60 , V}.

Suppose (H, *) is a subgroup of the group (G, *). Since the identity element of (H, *) satisfies the equation x * x = x, it mURt be the sn.me as the identit.y of UI(' pnnmt. group (a, *), for otherwi!olc we would hlWI~ two idl~mJlotcnt elenll'lIt.R in a, l~olltrlLry to Theorem 2-li. The identit.y demellt of a group thUl:l also serves 3.'> the identity element for any of its subgroups. Moreover, the uniqueness of the inverse elements in a group implies the inverse of nn element hE H in the subgroup (H, *) is the same as its inverse in the whole group (G, *). I'll l'fltnhliHII tlml. n l(ivl'lI !oIlIhH(·I. 1/ flf a, nlnlll( wil,h I,h,' illllul!l'Il IIJlI·rntiml of (a, *), eonHt.it.utl'!oI n subgroup, W(! IIlU!:!t verify tlmt nil the eonditions of Definition 2-11 arc sl\tiRfied. However, the asROdat.ivit.y of the operation * in II is an immediate consequence of its 3.'>sociativity in G, since H !; G. It is neeeRsnry then to show only the following: 1) a, bEll implies a * b E H (closure), 2) e E II, where e iR the identity element of (0, *), 3) a E II implies a-I E II. Needless to say, the saving in not having to eheck the associative law can prove t.o btl cOIl!olidl'l·able. A theorem whieh e!olt.ablishcs a single eonvf'nient criterion for determining subgroups i!ol given below. Theorem 2-17. Let (G, *) be a group and 0 "t; H !; G. Then (H, *) is a subgroup of (a, *) if and only if a, /J E II implies a * II-I E H. Pl'Oof. If (II, *) i!ol a !oIuhgl'oup and a, bEll, tlWII II-I Ell, 1UIIl!olO a * b- I Ell by the dosurl' "oJl(lit ion. COllvl·rMl·ly, suppose 1/ is tL 1101l(!Jllpty suhset of G

66

2-4

GROUP THEORY

which contains the element a * b- 1 whenever a, bE H. Since H contains at least one element h, we may take a = b to sec that b * b- I = e E H. Also, b- ' = e * b- I E H for cvery b in H, applying the hypothesis to the pair e, bE H. Finally, if a and b are any two members of the set H, then by what was just provl'd b- I also belongs to H, so that a * b = a * (b-I)-I E H; in other words, the sct 11 is dosed with resped to the operation *. Because * is 11.11 associative opl'ration in 0, II inherits the associat.ive law as a subset of G. All the group axioms arc satisfied and the sYi'lielll (11, *) is therefore a subgroup of (0, *). Definition 2-20. The center of a group ("'lit.

a=

{c E

a Ic * x

(a, *), dcnoted by cent G, is the = x

* c for all x

Ret.

E G}.

Thus cent. (J ('onsists of those clements which commute with every element of O. For example, ill the group of symmetries of the square, cent

a=

{R 1140, R360}'

The reader may already have deduccU that a group (G, *) is commutative if and only if c('nt. a = O. As ilIustratiolJs of the usc of Theorem 2-17 in determining when a subset of the clements of a group is the set of elements of a subgroup,..we present the following two theorems. Theorem 2-18. The pair (cent 0, *) is a subgroup of each group (G, *).

PmoJ. We finlt ohserve that cent Gis nonempty, for at the very least e E cent G. Now consider any two elements a, bE cent G. By the definition of center, we know that a * x = x * a and II * x = x * b for every clement x of G. Thus, if x E G, (a * b- ' ) * x = a * (b- I * x) = a * (x- 1 * b)-I = a * (b * X-I)-l

* (x * b- 1 ) (a * x) * b- I = (x * a) * b- I = = a

X

* (a * b- I ),

which implies a * b- I E cellt G. According to Theorem 2-17, this is a sufficient ('ondition for «(,pnt a, *) to he a subgroup of (a, *). Theorem 2-19. If (Hi, *) is IUJ arhit.rary indexed collection of subgroups of the group (a, *), then (nil;, *) is alHO It subgroup.

Proof. SilH'e tht' Sl't,; IIi all ('ontain t.Iw identity (·lement. of (a, *), the interscl'tion nil. ~~. 1\('xt" supposc a !\luI II Ilrc any t.wo clcmcnt8 of nH,; thcn

2-4

SUBGROUPS

67

a, bElli, where i rangeM over the index Met. The pair (Hi, *) being a subgroup, it follows that the product a * b- I 11.\:;0 belongs to Hi. As this is true for every index i, a * b- I E nIl.-, which implies (nHi , *) is a subgroup of (G, *). In regard to the group of symmetries of the square, we could take HI = {R 90 , R I!JO, R 270 , R 360 }, H2

= {R lIw • Rallo, Dt, D2}.

The system (HI n H 2 , *) = ({R UIO , R 360 }, *) is obviously a subgroup of this group, for its elements eomprise the eenter of thc grollp. III j!;PIII'r:d, wit.hollt fllrthpl" restric·tioll 011 the Imbp;roupll (1/;, *), it ill not true that the pair (Ulli, *) will 'IlJ.!:Uill he IL sllbg"oUII of (0, *). 0\1(' lIilllply eannot !!;uarallt('(l that. Ulli will c:olltain produets whose faetors come from different Hi. To give a eonerete illustmtion, both ({O, o}, +12) and ({O, 4, s}, +12) are subgroups of (Z 12+ 12), yet on takin!!; the union, ({O, 4, 0, 8}, + 12) fails to be so. The diffieulty in this ease is t.hat the modular sums 4 -h2 6 and () -/-t2 8 do not belong to the set {O, 4, 0, S}. By the way of an aualog to Theorem 2-1H, we have: Theorem 2-20. Let (IIi, *) be an indexed collection of subgroups of the group (G, *). Suppose the family of subsets {Hi} has the property that for any two of its members Hi and Hi there exists a set Hk (depending on i and j) in {H;} sueh that Hi S;;;; Hk and Hi S;;;; H k • Then (UH i, *) is also a subgroup of (G, *). Proof. By now the pattern of proof should be clear. We assume that a and b are arbitrary elemcnt.s of UIIi and Rhow that a * b- I E UHi. If a, bE UHi, then there exist subsets II i, II j cont.aining a and b. rel'lpectivcly. Aecording to our hypothesis, HiS;;;; H k and H j S;;;; Ih for some choice of lIt in {Hi}. Since (H k , *) is a subgroup and both a, bE H k , it follows that the product a * b- I belongs to H k • Accordingly, a * b- I E UHi as was daimed at the beginning. Asa particular case of the foregoing result, (~onsider just two subgroups (H I, *) and (H 2, *). Theorem 2-20 may be interpreted as asserting that (H I U H 2*) will again be a subgroup of (G, *) provided either HIS;;;; H 2 or H2 S;;;; HI' 'What is rather interesting is that thil:! condition is I1eeessary, as well as sufficient. The next theorem gives the details. Theorem 2-21. Let (HI, *) and (H 2 , *) be subgroups of t.hc group (G, *). The pair (H I U H 2,) is also a subgroup if and only if HIS;;;; H 2 or /J 2 S;;;; HI. Proof. I n view of the prececiing remarks, it il'l enough to I'lhow thut if (II, U 11 2 • *) is l~ tiubgl'Oup, then olle of the I'lets H I or 112 must be contained in the other. Suppose to the contrary that this assertion were false: that is, HI g; 112 and 112 g; II,. Then there would exist elementli a E III -- J/ 2 and bEll 2 - Ill'

!i8

2-,1

WUltI!' TIH:OItY

Now, if the procille! a

* IJ were n member of the set Il" wc (~ould infcr that b = a-I

* (a * b)

E Ill,

which it! c·lpurly not. t.rul'. On the other hand, the possibility a * bEll 2 yields the equally false conclut!ion

That it!, the den1t'nts a, b E HI U Il 2, but a * b Ii HI U Il 2. This conclusion is obviously untl'nublC', for it contradicts thc fact (II I U 1/2, *) is a group. Having arrived at n contradic~tion, the proof is compl(~te. The next t.opie of intC'rC'lit c'O!\ecrns eye·lil' subgrollps. To facilitate this disc'lIliliion, WI' fir:-;t int.rodll(·e some spceinlnotution. Definition 2-21. If (U, *) is an arbitrary group and 0 :;o
18 ~ II;

(Il, *) i:;

tL

~

G, then thc

subgroup of (0, *)}.

The set (8) dearly exists, for G itself is a member of thc family appcaring on the right; t.hat is, (G, *) is a (trivial) subgroup of (G, *) and 8 ~ G. In addition, sinee S is ('olltnillcd in eaeh of the sets being intersect&d, we always have thc inclusion S ~ (8). Theorem 2-22. The pair «S), *) is n subgroup of (G, *), known cithcr as tl\(' f'nveilJpiny .~ulJyl'Oul' for 8 or the RUIJI/roup yenerated by the set S. P/'(}of. The proof is !In imnwdiutl' consequl'lwe of Theorem 2-19.

Definition 2-21 implies thnt whenever (Il, *) is a subgroup of the group (G, *) with s ~ II, then (8) ~ II. For this reuson, one speaks informally of «8), *) us being the smallest subgroup which contains thc set 8. Of course, it ma:y well happen that (8) = G, and ill such a situation, the group (G, *) is said to bc (Jenel'ate(i by the subset S. For example, it is easy to sec that the group (Z, +) is generated by Zo, thc set of odd integers. We shall givl' an a\1ernative description of the sublict (8) which is frequently elllii('l' to work wit.h than Definition 2-21. III what follow:;, the symbol 8- 1 is u&-'
(at

* a2 * ... * ani ai, ... ,an E

S

U 8- 1 ;

n

E

Z+}.

Although the' notalioll is self explanatory, it would be helpful to explicitly poillt out I hnl till' :,;pt 011 the right ('()n:oli~tM of all lillit.p product::; whose fl~etors arc' C'itiwr 1'11'1lI1'1It.:oI ill S or inven«'1i of I'IPIlI('lIiM ill S. I..<·t 11M t.mll(lornrily desiglIate this set. of produc:ts by [S].

HIJBWWUI'H

All abhl·pviaj.pd proof of tIll' ass!'rtion ahov!' llIi!1:ht filii as folloWll: TIll' "yst(,11I (181, *) is a subg;roup or Ih(' g;I'OUp (a, *) wilh tlIP prnppl'ty '" ~ [81. A" (8), *) is tll(, slll:dlpst. s\II·h sUbg;l'OlIP, it follows t.hat (8) ~ IS). Tlw rever~(' inl'\usion ill just ifi!'d by tlw fad that any l'HIi>gI'OUP whieh (~ontllins the set 8 Illust J1('(~!'sslll'ily ('olltain nil the l'I!'llI!'nts of [S). A ('asl' of spp(~i:t.I illlportlllH'!' aris!'s \\'11('11 S ('onsists of It singl(· plpllJ('nt a. In I.his sitllatioJl, it is US \I: II to \Hitp (It) im;tpad of (fn:) alld n,f!'1' t.o t.ht' Ilsso(~iatpd sUhg;I'Olljl (a), *) as the cydic SUIJUI'IJIlJl (/('naaled IllI a. The subs('1. (a) is rather !'asy to d!'s('I'il)('; as all ils produds involve the cI('llIent a or its inverse, (a) silllply I'('dw'('s to t he integral powel's of a:

It. is ('ntirdy possihl(' that. the gl'oup (a, *) is <'qllal to OIIP of it.s (·yelie suhgroups, that. is, for Rom!' (·hoiee of a E a, (a) =, a. lInd!'I' t.heRe rircllmstane!'s, the group (G, *) is I'ef('rrl'
Example 2-31. First eonsider the group of int.eg!'1'S (Z, +). In this case, silH'P t.h!' 1I;1'01IP opl'mt iOIl is t.hllt, of addit ion, the ahst.md pro
+

(a) =

{na I n E Z] .

Using this notation, [O} = (0), Z = (1), while Ze = (2). We may thus eondude that both thc groups (Z, -f ) ntHl (Zc, -I ) are (·ydie, with the integers 1 and 2 as their reHpcC't.iv(' gelwratOI'S.

Example 2-32. Another illustration is furnished by (Z 12, +12), the group of integers modulo 12. Not.!' that we now write a +12 a +12·" +12 a for a * a * ..... a. The ('y<,li(' subgroup gellerated hy, say a, is ([0, :l, (i, 9}, +12), for herc, (a) = {an (mod 12) In E Z} = 10, :l, n, !l}.

+

As (1) = Z 12, til(' group (Z 12, 12) is its!'lf cyelie; other possible generators of (ZI2,t 12) are;}, 7ltnd 11. It is lIot difficult, t.o SPC' that., in general, the group of int!'g;!'!'s Illodulo 11 iH eyelie with 1 as a g(·lwmi.or·.

Example 2-33. 'Th!' group of SylllllH't!'il's of tI .. , s
70

2-4

(;Itol''' Tln:OllY

powl'rs of allY of il s 1·11'1II1'1I1H comprisl' lJIP nlemlu'rs of a nontrivial cyclic subgroup. TIll' rol al ion f( 1111, ill part ielllar, generates til(' suhgrouJl whose clcmcnts art' :U IIII , N nUl , /(27111 R:III11:' By I\. finite !/mup, WI' Illmn any group whose underlying set of clements is I\. finite set. The order of 1\ finite group is defined to be the number of its elements. AnalogO\\I!ly, n groUJl wit-h nn infinite number of element.s is said to have infinite order. In til(' following lIworpms, we shall s
«11), .)

is

!L

finite eyclic group of order n, then

Proof. As the sl't (a) is finih', not all the powen; of the generator a are distinct, There must be 801ll{' rl'JlI't.ition at = ai with i < j, On multiplying this equation by a- i = (ai)-I, it follows that a i - i = e, Thus the sct of positive integers k for which ak = e is noncmpty, SUppORe m is the smallest positive integer with this prop<>rty; that is, am = e, while a k ¢ e for 0 < k < m. The sct. 8 = :1', n, a 2, ' , , ,am-I} con8ists of distinct clements of (a). For a' = a', with 0 ~ r < 8 ~ m - 1, implies that a'-' = e, contrary to the minimalit.y of m. To I'olllplete the pl'Oof, it remains to show each member ak of the group «a),.) i;; {'qual to IlIl clem('nt of S. Now, by the division algorit hill, we Illay writ(' k ~ 1J1It ~I l' for SOIllI' integers q !lnd ,. with 0 ~ r < m. II CII(,(, , ak = (am)q. a' = e * a' = a' E S, This means 011' 1-<1'1 (a) C;; S, yielding (a) = sel)uent ellul\lity 111 = n.

["~, a, a:.!, . . , ,am-I)

and the Bub~

If a is an el(,lllent of t.h ... group (G, *), we define the order of a to be the order of the eydic! subgroup «a), *) generntec.l by a, The last result permits an altcrnativ(' vil'wpoint: the orch'r of a is the 1(,!tHt )lositivc integ{!r n, provided it ('xiHtl'l, HIII'1r t Iral (I." ('. Of 1'0111'1'1(', if 110 sud\ integ!'r (~Xilltll, a ill of infinite orll('r. AM 1111 illusl ntl ion, I'ollside'r I II(' gl'OlIp (Z4' 14); Il('re, the elemcnt 0 has order 1, 11m:; orcll'l' .. , :! hal'l 0('(\t'1' :!, whill' a hitS order 4. III certai'l (,ltSI'S (unfol'ftIlIlL(('ly, far 100 f('w) , it. is pOl!sihle to characterize compll'll'ly tIl(' suhgroups of a give'" gl'OUP, To cit.e 0111' instance, ill the aclditivi' group of illll'gl'rs (Z,·I), 1I11' I'IlIhj.!;I·OIlJlH 11I'I' all of the form «n), +) for H()IIII' 110111 tI·j.!;:t t ivl' illll'gl''' II, Adllally, 1he I-
2-4

71

KUBGHOlTPS It ('Y('Ii(~

Theorem 2c-24. EVl'I'y subJl;roup of

group is

(~y('li('.

I'/'III/f. LI't. (a), *) 1)(, n ('y('li(' Jl;roup gl'lIl'rah'lI by Uw 1'11'1IWIlt. a llllll let, (1/, *) be OIW of its liubgl'Oups. If /I '''' :(':, LlII' t.!WOI·1' II , is t.rivially l,rllI' , for (teJ , *J is the l'y(·lie subgroup gl'lIcmtcd by the idcntity e1emcnt. We may tlll.IS supposc the set H ¢ {e}. If am E H, whcre m ¢ 0, then a- m is allio IlII clemcllt of H; hence, II must. contain positive powers of a. Let n he thl' smalll'l'It. llOsit.ive illtl'l!;l'l' sueh that an E /I. We propose to show Il = (a"). To ('stablish till' inl'lusioll Il <;: (a"), h,t a k bl~ all lU'bit.I':tI'Y I'lellll'nt in till' KeL 1/. The divisioll ulJl;orithlll illlplil's 1.llI're I'xist illtegl'J'M q nnd r fill' whieh I,:

1/11

I /',

I)

S;

I'

<

II.

Since both a" and a k arc elcmcnts of 11,

If r > 0, we have a contmdietion to the assumption that an is the minimal posit.ive powl'r of a in 11. Aecordingly, l' = 0 and k = qn. Thus, only JlOwers of an lic ill 1/, illdil'ut.ing II <;: (a"). On the other hand, since the set H is dosed under t.he group operation, any power of an must again be a member of Il. Consequently, (an) <;: H. The two indusions demonstrate that H = (an). Corollary. If (H, *) is a subgroup of (a), *) and H ¢

{e}, then H

=

(an)

where n is the least positive integer such that an E H. We shall Tl'turn to a furt,hm' diseussion of eydie groupl-l at the appropriate place ill t.he sequel. For the mOllll'lIt, though, Id us inlli('lltc another useful method for manufueturing new subgroups from given ones. For this, some speeiaJ terminology is r('quired.

(a, *) be n group ILIHI /I, K hI! IHIIII'JIlpty subsets of The product of /land K, ill thltt order, is the set

Definition 2-22. Let

q.

II

*K =

{h

* k I It E

H, k E K}.

A brief ('Olllml'lIt 011 IIot.at.ion that should Ill' lIuule is that the usual custom is to writ.e Hie produl't II • /I 1I11'1'('ly 11.8 1/ 2 :lnd, if olle of the sets eOllsist.s of a sillgle delllt'llt a, silllplify tal * II to a * II. At first sight, th(~ readl'r JIlight real-lollahly ('ollj('('ture that whenever (H, *) !lud (K, *) an' both subgroups of (0, *), tllI'lI (II • K, *) will also bl' a subgroup. The group of l-Iymmdril's of the square, howev('r, showH that such It simple outeOIlHl is 1101, to bl' exp(,(,t.('d. IIl'n', it is ('Ilough to (~ow'iider UII' subgroups hltvillg I'h·nlt'lIt.s /I = {N: wo , /),: nlld K =- {U: wo , A lJui(~k dw('k ('stahlishes that there is 110 subgrouJl whose melllbers comprise th(' produl'l. set

V:.

1/

*K

=

{H 300

*

Il36o,

Il360

* V,

D\

* R360,

D\

* V} =

[R3Ilo, V,

Db

R 270}.

72

2-4

<1Il0UP TIIEOUY

In fad., Hl(, set II • K isn't even dmled under the group opemtion. One ne('d not b(· dismaYl'd by this state of affairs, for an additional assumption on til\' subsets II and K readily overcomes the difficulty. Theorem 2-25. If (II, *) and (K, *) are subgroups of the group (G, *) such that II * K = K * II, then the pair (II * K, *) is also a subgroup.

Proof. Il1nol!uolIS as the equality /I * K = K * /I appears, it is nonetheless th(! source of sOllie difficulty. This notation docs not mean Cll.!!h element of /I eommutes with ea('h element of K; all it signifiC/!! is Umt whenever hand t.Itrc arhitrary lIIelllhers of 11 and K, then ther(! exist elements h' E 11, k' E K for which It • k = k' * h'. Bearing this in mind, let us proceed with the proof proper. Plainly, HI(' produd. set 11 * K iii noncmpty, for e = e * e E II * K. Now, I('t a and b he allY pair of clements in II • K. Then a = h * k and b = h, * k, for suitable .. hoice of h, h, E II and k, k, E K. As usual, our aim in what follows is to show that the produet a • b-' lies in II * K. This is achieved through first noting a */)-'

=

(h. k). (h, * k.)-' = h * «k * ki')

* hi').

Since K is elOS('
By virtuI:' of the eOllditioll K k2 E K

* /I

=

II * K, there exist elements h2

E

II and

satisfyill~

We lIlay thus 1·lIIwlllll .. t.hat. a * /)-, = It * (1t2

* k 2)

= (It * h 2) * k2 Ell * K,

for the closure of t.he set II insures II * 112 also is a member of it. To complete the proof, it suffices to invoke Theorem 2-17. Corollary. If (II, *) and (K, *) arc Imbgroupli of the (!ommutative group (G, *) UH'1l (II * K, *) is llJl;aill Il :,;ubgrollp. Tlw III ility of Th"l/fI'lIl 2 2!i IiI'S in the flld tlmt it pcrmitH another characterizat ion of the suhgroup Jl;l'rwml.ed by a union of sets. What happens is thi:,;: if th(' pair (II * K, *) forrw:I a suhJ!;roup of (G, *), it mllst in fact be the subgroup gl'nerat('d by II UK; in symbols,

Let us briefly outline the argunwnt involvt'd. To start with, the two inclusions

II

=

ll*c~II*K

and

K=('*K~II*K

SUBOltOUPS

indicate /I U K ~ /I • K. We have ah'eady observed t.hat the subgroup genemted by II U K is til(' smallt-:;t l!ubgroup to cont.ain thiH union. Thus, whenever (H. K, *) is a subgroup, it follows that (H U K) ~ H ... K. On the other hand, thc sct (II UK) by definition must contain all products of the form h • k with It E H, k E K. This I'Clmlts in the reverse inclusion II • K ~ (f/ u K) and the subsequent equality f/ • K = (/l uK). The l!ignilieant point. ill this discussion is tlmt we have gained a great deal of insight into Ow Htrud,ure of the group generated hy t.he union II u K, where (II, *) and (K, *) arc bot.h sUhgrollpH of the group (U, .). To be Hp('c~ifie, in t.he event t.he c'olldilioll /I • K = K • /I holdH, eac~h 1Ill'lllher of UK), .) is expressihle as the produet of all element, of H wit.h all element of K. This stat('m('Jlt obviously nppli('s in the c':\se wlwrc the p:u'pnt gnmp (fl, .) is eOlIlmutative.

«/I

Example 2-34. For purposes of illustrating the above remarks, let us return again to the commutative group (Z 12, 112) and the t.wo subgroups ({O, (j}, +12) and ([0,4, X}, + 12)' To obtain the l!llIalIC'st subgroup which containl! {0,6} and -rO, 4, 8}, it suffiees merely to compute the produc,t of these subsets:

{0,6}

+12

{O, 4, 8}

°

{O +12 0, +12 4, {O, 4, S, 6, 10, 2}.

°+12

8, () +12 0, 6 +124,6 +12 8}

Hence, the subgroup of (Z 12, + 12) gel1emtl'd by the union {0,6} u {O, 4, S} is just ({O, 2, 4, 6, H, 1O), + 12)' PROBLEMS

t. In c'lwh of t.hl' r"lIowinl(

elL~C'S,

c'stahliHh tlULt (//,·) iii

IL

rmhl(fUUp of tho p;roup

«(,', .) : a) II = {I,-I}, (} - {1,-I,i,-iJ, whnrci 2 - -I; b) II = {2n 111 E Z}, G = Q - {O}; e) II = Q -" {OJ, G = n' - {O) ; d)/I = {(1 + 211)/(1 + 2m) 111, /1/ EZ}, G = Q - {O). 2. Prove that ({O, 4, X, I~; ,I-Jn) is It subgroup of (Ztn, -hn), t.he group of integers lIlodulo IIi.

:1. III CIt(, SYllllllc'tri .. f,\rc>UJl (S,.. uJ, ld If tlt'noh, t.he 1:ICL of (l1'rIlIlILlttiuns 1l'ILvinl( the integer n fixed: II = {f E S. I/(n) = 11).

Show that the pair (11,0) is a subgroup of (8 .. ,0). 4. "Prove that if (II, .) is a subgroup of the group (G, .) and (K,.) is a subgroup of (II, .), then (li.,.) is also a subgroup of (0, *). 5. L{,t (II, .) he a suhf,\roup of tIl!' f,\rouJl (G, .). We say that t.wo eJem{'nts a and b of G an' ('ol/grul'lI/modulo II, wrill('11 a == b (lIIocl II), if unci only if a. b- I E II. Establish that congru{'nc'l' mocluJo II is an cquivalC'[H'c rl'lation in G.

i-l

2-4

GHOUl' nn;OHY

Ohsl'rvl' that in tIll' additive group of intl'gl'rs (Z, +), where the subgroups an' of thl' forlll «n), +), 11 a nonnl'gatin' intl'gl'r, Ihis fl'latiun reducl's to eongrul'nl'l' modulo 11.

6. SIIPPOSl' thnt. (0, *) is a p;roup of () whieh ('ollllllute with a: Cia)

11.1111

aE

a.

Let Cia) denol.e the Sl't of all elements

= {x E G I a * x = x * a}.

Prove that. thl' pair (C(a), *) is a Ruhp;roup of (U, *), known /1..'1 the centralizer of a in a. ,\Iso vl'rify the equality, l~cnt G = n.'EG C(a).

7.
(a, *), define

II = {a E

a I ak =

the Sl't II by e for Home k E ZJ.

Detrrmine wlwtht'r t1H' pair (II, *) is a 1mbgroup of (G, *). 9. Let (0, *) hr a p;rollp allti a, bEG. Establish the following facts regarding the order of an ell'ment. a) Thl' I'll'mrnts a, a-I anti b * a * b- I all have the same order. Il) Both tIll' protiul'ts a * band b * a havl' the sam(! order. [Hint: Write a * b = a * (b * a) * a-I anI! use (a).) r) If a is of orr\rr II, thcn at = a i if and only if i == j (mod n). 10. Ol'termine the l'yeli(: subp;roup of (85,0), the symmetric group on five symbols, generate. I by the ryclc (I 3 5 2 4). II. Prove that a group of even order containl'! an element a ~ e such that a 2 = e. [llint: if a ~ a-I fllr all a, thl' group contains an odd number of elements.]

12. Let (II, *) bc a Ruhl!:roup of thc group (G, *) such that II ~ G. Prove that the subgroup genl'rated by the complement 0 - H is the group (G, *) itself. . 13. Suppose (G, *) is a p;roul' and S iii a non empty subset of G. If the elements of S all l'olllmutt', !:!how that the subgroup generated by S, «8), *) is a commutative group. 14. Given a group (G, *) and equivall'nt:

0~

J[ ~

G, verify that the following statements are

a) (II, *) is a l'ubl!;roup of (G, *). h) J[

* H t:;; II

anti 1/-1 t:;; II.

l') 11* 1/-1 t:;; II.

15. If (II, *) is a suhp;rllllp "I' the p;roup (G, *) and implirs K t:;; 1/.

0 ~ K t:;; G,

prove II * K t:;; /I

16. Let (II, *) and (K, *) he suhgroups of the commutative p;roup (G, *) with orders nand 111, TI'sp"div(·iy. A"''1ulllinp; 1/ n K = (eJ, verify that the oreler of the group (11* K, *) is nm.

NORMAL SUDGltOUPS ANI> QUOTn:NT GROUPS

75

17. Consider the group of symmetries of the square. Use Theorem 2-25 to obtain the subgroup J.!;cnprll.ted by II U K, where II = f RplO, R:wo) , K = {Raoo, Dd. IS. Ll't (0, *) hI'

is

Ii

sqIlIL),l'

It

(i.('.,

J.!;rollp of ord('r n, W}IPf(' n is odd. Prove thllt ('Iu:h "Ielnpnt. of G if rEO, thl'1l .x = y2 for ~Olll(, y ill G).

2-5 NORMAL SUBGROUPS AND QUOTIENT GROUPS

Althoul!:h WI' have dl'rivt'd sOllle interesting results eOllcel'lIing suhgroups, this e()(le(~Jlf" if unrcstrieted, is too gelwrul for muny purposes. To obtain (·Nt.ain highl.v dl'sirahlC' (:onehlsion>-l, :ulditiOlml assulllpt.ioJls t.hat p;o IlI'.vond I )I'Iinitioll:! I!I II I 11M I. I", iIlIPOM(·'1. Thus, in the Jll'esent S(,et.ion, we nurrow the field and focus attention on u rt'Ht.ri(,ted ('(ass of :mhgroupl'l which we I'IhnIl refer to ns normal subp;roups. FrollJ IL eon('l'pt.uul point. of view, sueh groups ure "noI'llHLI" ill the liense that t1H'Y make til(' )'t'sull,illg Hl('ory so Illueh ridH'r Ullin would oUlPrwi>-le he th(' ('!lS('. Whill' not. I'vl'ry suhgroup nl'l'd h(' of t.hiH t.YJlI', norlll:ll :.alhgroups O('plll' nOlwt.h"ll'lis with I'onsideruble frequenpy. It. will lioon he('ollle uPJlal'pnt. that, for the lIlajor part of OUI' work, t.he signitit':lnt. usped of this ('lILIiS of :lIIbgl'oups resides in the faet that they permit the ('onstruetion of algebraic structures known as quotient groups, Having already divulged some of the content of this section, let us now proceed to develop these idea.s in detail. As a starting point, we prove a sequence of theorems lmding to the conclusion that eneh subgroup induc:es a decomposition of the dementH of the parl'flt group into disjoint subsets known a.s cosets.

Deftnition 2-23. Let (H, *) be a subgroup of the group (G, *) and let a E G. The set a * II = {a * It I It E II} is called a left coset of ][ in G. The clement a is a representative of a

* H.

In u. similar fashion, we eun define the rip;ht eosets H * a of II. The right cosets of the same subgroup nrc in general different from the left cosets. If the grouJl operat.ion * of (G, *) is commutative, then dearly a * ][ = ][ * a for all a E 0, In HI(! suhsequent diseussiolls, we will generally consider only left cosets of a subgrouJl. It is obvious that, u parallel theory for right cosets may be developed. Before proeeedillg to all exampl£', we lihall make several simple observations. First, if e i:; thl' id!'lItity pll'nH'nt. of (0, *), thl'lI e*

II

=

[e

* It I Ii

so that 1I its('lf is a I<'ft. ('Os('t. of 1I.

E

II} "" {II. I II :\[OI'('OV('J',

E

II}

=

II,

Hill('c e E Il, we have

a=a*eEa*lI,

71i

2-1i

t.hat is, every c1('m('nt a of G beJon~8 to some left coset of H, and more specifically, to tIl!' ('081'1. (l • II. We shnll nmkl' uS(' of t.his fad in 1\ Iitt.J(, while, We nolp further that t.here is It onl'-to-on(' ('orreHpondence between thl' ('1<'111('11 Is of II llIlIl t.hoS(' of any "OHet of II. I n
Example 2-35, Heturning Olll'e again to the group of symmetries of the square, let us sl'icd. the subgroup (8, .), where S = {R360, V}. The tn.sk of computing the I('ft eosets of 8 is .straightforward, sinee we have the operation tahle for this group at. our disposal (HeC Example 2-24). R90 • S = {R90 • fl360, R 90 • V} = IlI!!o· S = {R IHO

s= R 3Ro * s = N 27u

•

H

*S =

V

*s

*

{R 27 I!. {R 360 {H

*

{Roo, D 2 },

R360, Il Iso • V} RaRIh

* Ra6o, H * V}

~~ {V. Nann,

V

* V}

R 270

R360, R 360

* V:

{R ISO, H}, {R27o,

D I},

* V} = {R360, V}, = {H, R ISO},

=

{V, Rann}, {I) .. 1l27nl,

{D2' Roo}. From n quick inspeetion, the render will observe that there are only four distinct (~osets, {Ilon, J)21,

{ /lillO, II},

f R270'

J).},

and

{fl36o, V}

=

S.

Thl'sl' '~OH('tH aI'<' disjoillt alld their union is the un
/). * S

=

[D" R 2701 ¢ {D" R 90}

Theorem 2-26. If (II, *) is if awl only if n EO II.

It

subgroup of till'

=

j!;l'OUp

S*

/)1'

(G, *), then a * H

=

H

Proof. HUPJlos(' lil'sl Ihal a. /I = /I. As W(' havl' just. r(,IlUlrlwd, the fact thnt t h(' id('nt il y /. is n II\I·II.h,·I· (If /I iIllJlli{·s I hili IhI' ('II'IIIt'IIt. a 1lt'lollKS to a * II, alld thus hy hypotlll'sis 10 /I also. Oil till' oth.'!' h:llld, if a E II, t1wn a • /I <;;; II,

2-5

NOHMAI, SUlIGlWUPS ANI> QUOTIENT GROUPS

77

since the set H, being the set of elements of a subgroup, is closed under the gnlllp op{'\"IL\.ion *. The oPpoHite inclusion is ohtained by noting that ench elcment It E If may be w\'itten as h

= a * (a-I * h).

Here, a-I * hEll, since both a, hEll and (II, *) is a subgroup of (0, *). This implics that It E a * If and consequently II ~ a * H. Our next theorem provides a simple criterion for the equality of two left (:osets, whcn It rcpn~lICntativc of each is known. Theorem 2-27. If (//, *) is a Huhgl'Uul' of the gnllll'

(a, *), thCIl

a*H = b*H,

if and only if a-I

* b E H.

Proof. ASRume that a - II = b - II. Then, if a

* hI is an arbitrary element of

a- H, there must exist all h2 E H such that a - hI = b * " 2 , From this we conclude that and, since the product hI - h;1 belongs to H, that a-I - b E H. Conversely, if a-I - b E H, then by Theorem 2-26 we have (,,-I

This implies that nny

elt~lII(mt

* b) * H =

hI E 1/ (:an he

H. cxpn~l!s(!d

as

"2'

for some 112 E II, from which we infer that a - hI = b * Thus each product a * hI in the coset a - II is equal to an element of the form b * h2' and consequently lies in the COIICt b - 11. Sincc this stl\tmllcnt also holds, with a and b interchanged, a-/I = b-Il. Remark. When working with right cOIICts, the requirement a-I - b E H must be rcplaced by a- b- I E II; that is, Ii * a = Ii - b if and only if a * b- I E H.

As an imnwdiatc (~onsequence of the 1I11;t theorem, we tICC that any element al of the ldt em;ct a *11 detcrmincs this eosct. For if al E a - II, then al = a * hI for suitable hI E II. Thus a-I - al E /I, so that by th(! theorem, a * H = al - H. This Illl'tUlH tim\. (,twh pJ('nwnt of 11 (:oset ean be thought of ns It represcntative of that. ('oset. In IL (~ert.ain HI'IlHI! we ure heing prejutiic('(i whenever we denote n ,'oSt'\, hy a * II, for SOI\l('()I\(' !'lSI! lIIi!!:ht dlool!(~ to "all it. ,) - II wlwre b ;t! a, but a-I -" Ell.

78

2-5

G1WUP THf:ORY

We are now in a position t.o prove a fundamental result concerning cosets to the effect that if two left cosets have an clement in common, then they are precisely the same set.

TheoNm 2-28. H (II, *) is It subgroup of the group (0, *) then either t1w cosetH a * II null /) * /I are Jisjoillt or e1l'!e a * H = b * H. II. II and /1 * II ('ont.ain HOIll!' .. (.. nll'lIl. r. ill (~Onlll\llll. c iH ill 1I • 1/, I h(,I'1' ('xi:":; 1111 It, I Eo 1/ Hlwh that c a . It I. Hilllilarly, we have c = b • h',l for SOIlI(' plellwllt 112 E 1/. It follows then that

Proof. HlIJlJl(.HI' HIIlI,

Hin('(~

or RilleI' (II, *) is a slIhl(l'OlIP, t.lw pl'Odud hI * h'2l, alld t.I11IH a-I * b, 1II11tlt lie ill the HI'f 1/. 0111' IIl'l'd ollly apply T(lI'orl'l1l 2 -27 to eOlwhul1'

a

* 1/

= b

* 1/.

We saw earlier thllt eneh clement a EGis a member of /:lome left CQI,lct of H in 0, namely. the C'Ol'lPt a * H; that is, is exhausted by its left ('OsetR. Theorelll 2-2S illdil'llt.('l'I that nn clement can b!'lollg to one and only one left. C'O!wt of 1/. Thlll'l t ht' ~('t a is partit i()J}('d hy 1/ into disjoint sets, each of whi('h has eXlwtly as mllny elements as H. For case of future reference let us summarize these rellJllrks ill tIll' followiul!; theorem.

°

Theorem 2-29. If (II, *) is n ~lIbjl;roup of the ('osetR of II ill (J forlll a partit.ioll of till' set a.

I/;l'Olip

(0, *), the left (right)

Example 2-36. ConsiJer (Z 12, + 12), the group of intcgefl:! moJulo 12. If we take {O, 4, R} for the set H, then ({O, 4, S}, +12) is evidently a subgroup of (ZI2, +12)' The left {,ORets of II in ZI2 arc

0+ 12 II

=

{O, 4, 8}

= 4 +12 H = 8 +12 H,

141',l1l= [1,5,9} =;,-1-!211=9+12H,

2 t 12 1/ .~ {2, ~

(I,

to} = 6 +1211 = 10 +12 H,

I 1211 = [a, 7, Il} = 7 +1211

= 11

+12 H.

In this ('aRp, thp I'oset dl'('omposition of Z12 relative to the subset H is just Z 12

=

[0,4, Rl U {I, 5, 9} U {2, 6, to} U {3, 7, Il} .

RUJlpOHI' 1I0W Ihal (r:. *) is a lillit .. jl;rollP, Hay of order n, nllll (H, *) iR n l'Il\hjl;roup of (0, *) of or«ll'l' k. WI' (·:tI1 thell deeomposc the set (l iuto 1\ union of Il liuitt, 1l1l1lllwr of diHjoillt left ('osets of 11:

a=

(al */1) U (a2

* H) u· ..

U (aT

* H).

2-5

NOltMAI. HUBGHOUI'H ANI> QUOTIBNT GHOI1I'S

79

The number r of dist.inet left cosets appearing in this ueeompositioll is called the index of Tl in G. Since cueh eoset in the above decomposition has k clements, the s('t. G itself must hlLvil 1". Ie dements; hel\(~e n = 1". k or order

a=

(inel(~x

II) • (order 1/).

Theorem 2-30. (l.a!lranl/c). The order and illd('x of any subgroup of a finite group divides the ordpr of the group.

There is

It

("orollary to Theorem 2-30 which is of some int,rinsie interest.

Corollary. If (a, *) iH II I-(I"IIUp of OI"Ile.· n, thl'lI till! orel('r of lLllY dClIlent a c r; is lL fador of It; ill Ilclditioll, an ~ (:. I'f(}o!. L('t the ('(Plllellt a have order k.

By detillitioll, the cyclie subgroup must alRo be of order k. According to the condusioll 9f Lagrange's Theorpm, k is a divisor of n; that is, n = rk for some r E Z+. Hence, . (a),

*) geliPrated by

a

an =

ark .~ (a K)' = c r =

e,

('ompleting the proof of both assertions. From Lngrange'H TheOl·em, w<' are able to condude that any finite group of prime ordpl' hal-i no 1I0ntriviai l-iuhgroupl-i. Actually a Ht.rongm· Htat.enwnt can he mud(~:

Theorem 2:-31. If (U, *) is a finite group of composite order, then (G, *) has nontrivial subgroups. Proof. If the group (G, *) ill not cyclic, any element a E G with a ~ e generates a nontrivial cyclic subgroup (a), *). Thus, it. sufficeH to consider cyclic groups of compoHite order. To t.his end, l-iuppose G = (a) when' the g<'lIera\.or a has order nm (n, 111 ~ J). Tll('n (a")m = e, while (an)m' ~ e for 0 < 171' < m. From t.his, it. iH obviolll-i thut (an), *) is a lIontr·ivial cyclic l-illbgrollJl of (U, *) with order m. Corollary. Every group (U, *) of prime order is eyclil'. Proof. Consider the cydic subgroup (a), *) generated by any a E G, with a ~ e. Now, the order of «a), *) must divide the order of (G, *), It prime; sinee (a) eOlltains more than one elenwnt, order (a) = ord(~r G, whenee (a) = G.

Al-i a fllrt.(lI'r applicat.ion of Lagrange'H TIll'orelll, proof of Theorem 2-7: Theorem 2-7. (Revisited). clements.

W(! ("Illi 1I0W

Any noncommutative group

give

hUH

It

t!implified

at least six

HUOlll'

2-5

'l'II~;Oln'

Proof. A group of prime order, being a cyclic group, is necessarily commutative. nny gToup hnving order 2,3, or 5 will be! commutative. SuppoS(l next I.hnl (fl, *) ill It grullp of oTilc'T 4. My (.ngrILlIgC!'H TIUloTem, eadl elenwllt of (J of order 4 and therdore commuta.tive. On the other hand, a group elU'h of whose clements other than the identity has order 2 mu!!t be ('ommutative by Problem 6, Section 2=-2. This argument establishes that all groups of order less than () are eommutative groups. Ac(~ordillgly,

III('idl'lIlally, till' implic'at.ioll of Lagrange's Theorl'm cannot be reversed; that is to say, a group of order n need not have a subgroup of order k, where k is a divisor of n. To 1)(' more spcc!ific, a group of order 12 exists which has no subgroup!! of order Ii. Thc! particular group we arc referring to happens to be a subgroup of the symmetric group (8 4 , 0) and has as its clements:

C G

2

(

:~

2 :J 4)

C

4

2 3

G G C~

4 3

1 2 3

2

C

2 3 2 4

:3 4 3

C G: G

3

2

2 3 '

3 3

1 2

(:

2 3 3 2

This pemllltntion group c\O<'R, however, Imve suhgroups of orders 2, 3, and 4. We shall introduce next a particularly important class of subgroups which we shall refer to as normal subgroups. Definition 2-24. A subgroup (H, *) of the group (G, *) is said to be normal (or invariant) in (G, *) if and ollly if every left coset of H in G is also a right coset of H in G.

Thus, if (H, *) is normal and a * H is any left coset of H in G, there exists some element bEG such that a*H = H *b. Since a is in the IcCt coset a * H, this means that a is also II. member of the right ('Ol'll't· If * II. The COK<'ts 11 * b I\nd 1/* a have t.he clement a in common; so the Ilnalog of Theorem 2-28 for right cosets implies that

11* b

= 1/* a.

2-5

NOItMAL KUI!G1W\1I'1'! ANI> QUOTJIo:NT GROUI'S

81

In other words, if a • H happens to be a right coset of H, then it must be the rip;ht COtlCt /I * a. 'rhil:! observation u1l0w8 us to reformulate Definition 2-24 1\.1:1 followH. Definition 2-25. A subgroup (1/, *) is normal in the group (G,.) if and only if a * 1/ = H • a for every a E G. For a normal subgroup (11, *), we may thus speak simply of the cosets of II in G without spl.'t:ifying right or left. The triviallmhgroups arc obviously normal l\Iore generally, every subgroulJ of a commutative group is a normal subgroup. We will HOmetimes speak of a simple group (in the technical sense), meaning thereby tlmt it 1111.8 no normal subgroupl:I other than the two trivial ones. For instance, the finite cyclic groups of prime order are simple groups. Definition 2-2."i indicates that normality of a subgroup (H, *) guarantees a w('ak fOl"lll of commutativity f('lative to II. For, if hEll, while it cannot in geneml he cOllcluded that a • It = h • a for any a E G, we do know that there exists an clement h' E J/ such that

a. h

=

h'

* a.

It would be gratifying to have a less cumbersome procedure than to compute cosets for determining whether a given subgroup is in fact a nonnal subgroup. Just such a criterion is given in the next theorem, and we shall have frequent occasion to make use of it. Theorem 2-32. The subgroup (II, *) is a normal subgroup of the group «(], *) if and only if for each clement a E G,

a.H. a-I

~H.

Proof. First, assume that a * H • a-I ~ H for every a E G. We must prove that in this case a * H = 11 • a. Let a • It be an arbitrary element of a • II. Since a • II * a-I ~ II, a. h • a-I =-- hI for some hI E H. Thus

a. h

=

(a. h • a-I) • a

= hI •

a.

The product hI • a lies in the right coset II • a, so we conclude that

We obtain the opposite inclusion, 1/ • a ~ a *11, by a similar argument upon observing that our hypothesis also implies a-I. II

*a =

a-I. II

* (a-I)-l

~

II.

Conversely, suppose a. f{ = II. a for each a E G. Let a • hI • a-I be any element in a. H • a-. I • Then, since a .11 = H • a, there exists an element

82

2-5

(lIWITI' TlIJo:Olty

h2 E /I such that Consequently,

which iml>lies a *1/ * a-I s;;; 1/. To demonstrate the convenience of this result, we now prove the following assertion: (rent G, *) is a normal subgroup of each group (G, *). In terms of elements, it must be shown that if c E cent G and a is arbitrary in G, then a * c * a-I E cent G. But this is fairly obvious, since from the definition of the center of a group, a * c = c * a. It follows at once that a

* c * a-I =

c ... a'" a-I

=

c ... e =

C

E

cent G.

Examp'e 2-37, Let us return to the noneommutntive group (G, 0) of order 6 presentnd in Example 2-22. The reader may f
.- fl f2

fa i4. is f6 fl It 12 f3 f4 hi f6

f2 f3 f. f5 fa

f2 It f4 f4 f3 fro f6 f6 fs

fa

f6 fs f2 f6 f4 f3

It fs

f4 fa f6 f5

12 It It 12 !2 It fa 14

If we take as H t.he subset {fl.J• .J6}, thcn it is easily verified that the pair (H,o) is a normal suhgroup of (G, 0). The I:osct breakdown for the subgroup in question is

fk fk

0

H

0

II

= {!t.hf6} = = {h,Ja,fs} =

of,.

for k

II·f,.

for k

H

= =

1, 4, 6, 2, 3, 5.

On the other hand, the subgroup ({It,!2},.) is not normal; a short computation indi{'atcs why the criterion of Theorem 2-32 fails to be satisfied:

f. 012 of. 1= f4 ·f2 ·f6 = f" e {!t.12}· The Rignifil'nJlc(' of normal subgroups-indeed, our main purpose for introducing them-is that they enable us to define new groups which are associated

2-5

NOItMAJ, RlJlIOItOtTl'H A Nil

~~lJOT"';NT

(mOUI'S

83

in a natural way with the original group. More specifically, we shall show that the set of eoscts of a nonnal subgroup is itself the set of clements of a group. If (H, *) is a lIornmll:!ubgroup of the group (G, *), then we shall denote the collectioll of distinct (!oscts of H in G by GIH: Gill = {a*ll/aE(l}.

Thesc are also right coscts, since the definition of a normal subgroup guarantees that a * /I = /I * a for every a E a. A rule of (!ompositioll ® may be defined on (JIll by the formula

Since this definition is stated in terms of coset representatives, we must first show that the multiplication of cosets under ® is unambiguously defined, independent of the arbitrary choice of representatives from these sets. That is, it mwst be shown t.hat if a*H

=

al *H

and

t.hen also

According to Theorem 2-27, it is enough merely to prove that the product (a

~ b)-I

* (al * bl )

is a member of H. Now, a * H = at * H and boll H = b l * H imply both * at. b- l * b1 E H. Silll.:e (H, oil) is normal in (a, oil), we know that

a-I

for every x E G. In particular,

From this we conclude b- 1

* (a- t * al) * b E

H and, since H is closed, that

The above argument shows ® to be a well-defined binary operation on GIH in the scnsc that the product of two coscts depends only on the cosets involved and in no wny 011 the representative elements chosen from them; any other choice would have yielded the same product. Having thus prepared the way, we now state and prove the principal result of this sedion.

84

2-5

GROUI' TIU;OIty

Theorem 2-33. If (11, *) iR a normal Rubgroup of the group (G, *), then the system (a/II, ®) formR a group, known as the quotient group of G by II.

Prool. First, lelt 11M ohHCrve t,hat, UIC! ulIIIO!lintivity or the Opl!ratioll ® is a direct consequence of the associativity of * in G:

[(a * /I) ® (b * II)] ® (c *11)

= =

«a * b) * II) ® (c * 11) «a * b) * c) *11

= (a*(b*c»*11

=

(a * II) ® «b * c) * II)

= (a

* 1/)

® [(b

* II)

® (c. 11)].

The coset If = e.11 is the identity clement for the operation ®, since

* /I = a *11 = (e * a) * If = (e * JI) ® (a * If).

(a • 1/) ® (e. II) = (a. e)

.

It is equally easy to sec thllt the inverse of the coset a * If is a-I * II, where a-I denotE's the inverse of a in (0, *). This is evident from the computation (a * II) ® (a- I * II)

= (a * a-I) * II = e */1 = (a- I * a) * II = (a- I * II) ® (a * H).

/

Hence all the group postula,t.es are fulfilled and the proof is complete. Example 2-38. Once n.p;ain we fnll bl~ck on th(l group of symmetries of the. square for an illustration. Here, the subgroup

(S, *)

=

({Rilla, R 36o}, *)

is normal, being the center of the group. Its distinct cosets, that is, the elements of G/S, are

I

G/S

= I {R I8o ,

R 36o }, {R go, Rna}, {V,N}, {Dh ~2}1.

A typical coset multiplieation proceeds as follows: ID" D 2 } ® {R go, R270}

= (D I • S) ® (Rgo • = (D t • R go) • S = II*S = {V, II}.

S)

NOHMAI, :-;U/lGlUIlJI'S ANIl tll)UTn:NT (JIWUI'M

85

To lIluJt.iply two (:Ol'!ds under ®, nil we really need to do is select all arbitrary rnprcflCmtnl,ive from ('/wlt 1'()l'!HI" Illultiply tlwlic c1emcnts under tho group uJlI'/·ILI.ioli ... ILlIII tIHt.l'rlllillC! to whit·h eOIi(·I. the rt·liIlIt.illg prothwt. bolollgll. The operation table for the quotient group (G/S, ®) il:l shown in Table 2-4. Table 2-4

®

{Ii pm, Raoo}

{Ruo, R27u}

{V, II}

{Rpm, Rauu} {RIIO, R27o} {V, II}

{RIBO, Rauo} {Ruo, R27o}

{V,ll} {{h, /)2}

l/JI, /J:!:

(/)" /J2}

{Ruu, R27o) {R18o, RallO} {J)I, ])2} t V, II}

.. _ - - -

-------_.

{V,II:

{/)I, /)2}

_._-------

{Ii PUI, 1l:IHII} l Uuu, U27U}

{Dl, D2} {V,lI} {lluu, R27(J} {lll8u, /lauo}

Example 2-39. A simple, hut. lIscful example to kl'l'p ill milld when working wit.h IJllot.i(!lIt. grouJlIi iii furnislu·tJ hy till' atltJit.iv(! gJ't)up of integers (Z, +). 11. hu.s heell prl'viollsly ('stnhlislll'd that tlw (IIOl'mal) suhgroups of (Z, +) are the eydie suhgroupli «n), I), n a lIollllcgatiVl' illll'ger. The (~osets of (n) ill Z t.ake t.he form a + (n) = {a + kn IkE Z} = [a].

In other words, the cosets of (n) are merely the congruence classes modulo n. Coset multiplication in Z/(n), moreover, is given by

(a + (n» ® (b

+ (n»

=

a + b + (n),

or, with a judicious change of notation, [a] ® [b] = [a

. Wc thus (!I·duce that. the quot.ient ~roup of int,I'I!;('1'lI modulo n,

~roup

(Z/(n), ®)

=

+- b] .

of Z

by

(n) is none ot.her than the

(Zn +n).

Am()n~

ot.hl'r t.hings, t.ltis illllieates that had we so desired, t.he study of the integcr!:! modulo n eouid havc been subsumed under t.he more general theory of quotient groups. It is a simple matter to sec that any quotient group of a commutative group is ne(:('sl'iarily eommutativc. A natural qucstion is whether a noncommutative group ('1U1 poss('ss eommlltnt.iv(' qllot,i('nt grollps ILnd, morc pointedly, what eondit.iow; (if allY at. all) would insure their exishmee. As a concluding topic in this seetion, we investigate t.hil'i particular liituatioll. Our analysis begins wit.h a basic definition. Definition 2-26. <:ivcn a group (U,"') ILnd c1cments a, bE 0, the comlIlutator of a ILnd b is defined to be the product a * b'" a-I ... b- I .

86

2-5

GHOUl' TlU:OItY

To simplify matters, Hw symbol [a, b) will be used to represent the commutator of two element.s a and h; any other symbol would do as well, but this notation is standard. Inasmueh as [a, h] satisfies the identity a

*b =

[a,

bl * b * a,

one may view the commutator of a and h as a measure of the extent to which a * b differs from b * a. Indeed, the elements a and b commute if and only if [a,b) = e. In general, the commutators do not by themselves form the elements of a subgroup, !linee thC'y fail to be closed under multiplieation. The mmal procedure for bypassing this difficulty is to work imltead with the suhgroup generated by all the commutators [a, h), a, bEG. The resulting subgroup is known either as the dcril'cd s1iligroup or commutator subgroup of (G, *) and may be denoted simply by (lG, G), *). Now, the illver~c of a eommutator is again a commutator: [a, bl- I = [b, al. There if! no nC'ccl'l'ity then of explicitly conKidering inverses in the definition of the set [a, G]; ilK clements consist merely of products of finitely many commutators of G. That is,

IG

J

G) =

{ilIa;, bi ] I ai, bi

E G},

whcre t he ~Ylllhol il l'hould be construed as representing a finite product with one or mor!! fadors. With HlI's!' prl'paratory remarks out of the way, we pro(~eed to establish some of the Kp<'('ial properties of the commutator subgroup. Theorem 2-34. The group

(fa, aI,

*) iR a normal subgroup of (G, *).

Proof. The proof procemls along the usual line. Namely, it must be shown that for c E [G, G) and a in G, a * c * a-I lies in [G, G]. But, a

The element [a, c) to [G, G].

* c * a-I = * c is a

(a

* c * a-I * c- I ) * c =

[a, c)* c.

finite product of eommutators and accordingly belongs

The quotient group (G/IG, G). ®), which exists by virtue of Theorem 2-34, is called the commutatol' quotient (/!'Oup or abelianized (J!'OUp. The motivation for this In\.t,er (·hoi(·(, of t.errnillology will only h(,(~()IIJ(l llppnrl'nt nH(lr the next. r(!Huit. Theorem 2-35. Lc-1. (II, *) he n normal subgroup of t.he group (G, *). Thl' quotient. group (G/lI, ®) is cOlllnlllt,nl.iv(! if anel only if rG, G) ~ II.

Proof. Suppose a * /I and b * /I arc two arhitrary elc-mcnts ill G/H. Sinee the coset II = e * 1/ is the identity clement of (G/II, ®), the group operation

2-5

NOItMAI, SUIlGHOUI'S AND QUOTmNT GItOUPS

87

® will he (:ommutative if and only if

H= [a*II,b*lIJ= (a*H) ® (b

* II)

® (a*H)-1 ® (b*H)-I,

or, whitt amounts to the sllme thing,

But, Theorem 2-26 tells us a necessary and sufficient condition for the last equality to hold is that

la, bJ

=

a * b * a-I * b- I

E

H.

In other words, commutativity of the quotient group (GIH, ®) is equivalent to requiring that the subgroup (H, *) contain all the commutators of G. As ([0, GJ, *) is by definition the smallest subgroup with this property, the latter eOlldition may be replaced by [G, GJ S;;; II. A special case, but itself of interest, occurs on taking H

=

[G, 0):

Corollary. For any group (G, *), the commutator quotient group (Gil G, 0], ®)

is commutative. The foregoing theorem says, in effect, that the commutator group is the smallest (again, in the sense of inclusion) normal subgroup whose associated quot.ient group is commutative. The transition from a group to its commutator. quotient group is mfermd to II-S the abelizalion of the group and provides a cOllvenient melUUI of mallufacturing commutative groupt! from noncommutative ones.

PROBLEMS

J. If 11 = {O,6, 12, Ig}, show that (II, -t2.) is a cyclic subgroup of (Z24, +24). Also, lillt t.he elementl! of eaeh coset of 1/ in Z24.

2. In the symmetric group (84,0), let the set 1/ consist of the four permutations ( 1 2 3 4), I 2 3 4 LiK!,

(21 2I 34 4), 3

(1 2 3 4). 432

1

Ow c!c'IllI'" til or I'lli'll ('OHI't or 1/ in 84.

3. AK.~III1l(l (II, *) iH a

HllhJ,!;TOIlP

or (0,

.J.

a) :Show that ('very left ('oset of /I ha.~ the Harne numher of elements as every ri/!:ht coset.. h) Prove that (c * a) * /I = (c * II) * 1/ implit's a • II = b * II. (.) Show that. tht'rtl cxistH a IIne-to-one (,orreIlJlon(\ence betwl'('11 the left cosets of 1/ in G and the right cosets of II in G. [Hint: a * II -+ II * a-I.)

88

2-5

G1WUP Tln:OItY

4. Det('rmin(' till' l('ft ('OH('t dc('omposition of ttl(' jl;roup of !:!ymmetriefl of the squarc with rcsl'ed to the slih/l:roup ([ Raun, /)I!, *).

5. In the group of symmetries of the elillilateral trian/l:lc, fin(l: a) all Rllhjl;roU(lR, b) all normal IIU hgrou pR, I') the "('1\ t('r of t.hl' /l:flllIl'. 6. Hhow that if the cyelie jl;roup «a), *) is infinite, then a and a-I are its only generators, and all subgroups except ({e}, *) are infinite. 7. Lilt (II, *) he a Ruh/l:fOllP of index 2 in t1l1ll(rOllp (0, *). Prove that (II, *) i!i a normal Sllhl(fOUp. [lfint: /I U (a * /I) = 0 = /I U (II * a) for any a E G - /1.1 ~.

Given t.hat (III, *) Imd (11 2 , *) af(~ hoth normal HlIhl(fOlIPH of the prove t.hat, tlH' SlIhl(rflllp (1/1 n 112, *) is also normal.

1(ff1IlP

(G, *),

9. Let (1/, *) be a Huhl!:rollp of t.he I(roup (G, *) and the set N(l/) be defined by

N(ll)

=

I * II * a-I = ll}.

{a E G a

a) Prove that the pair (N(ll), *) is a subgroup of (G, *), called the normalizer of IT in G. b) Prove that (1/, *) ill normal if and only if NUl) - G. 10. Rllpposc that (II, *) an,1 (K, *) are normal Rllhl(roUpR of the group (G, *), with /I n K = ':('1. By ('onsidl'rinl!: l'il'lIH'nt.s of thl' form" * k * II-I * k- I , !lhow that. h * k = k * It for all It E /I, k E K. II. Given (If, *) and (K, *) arc subgroupR of the group (G, *) and one of these subgroups is normal, prove that the pair (ll * K, *) is a subgroup of (G, *) i when both are normal subgroups, show the group (11* K, *) is also normal. 12. Find an example of a I(rollp (G, *) having a subgroup (//, *) for which the produet of two left CORllts of // in 0 need not be a left coset of ll. ' 13. Describe the quotillnt group of a) (Z.,+) in (Z,+), c) (Z,rt-) in (Q,+),

b) ({O,2, 4, 6, 8},+1O) in (ZIO,+lO), d) ({I, -I},·) in ({I, -1, i, -i}, .).

14. Let (G, *) be a cyclic group with generator a and (II, *) be any subgroup of (G, *). Prove that the quotient group (G/II, ®) is also cyclic with the coset a * II as a generator. 15. Given (II, *) is a normal subgroup of the group (G, *), prove that the quotient group (G/II, ®) is I'ommutative whenever (G, *) is commutative. 16. For any group (G, *), describe the quotient grouJlS of the trivial normal subgroups ({e},*) and (G,*). 17. Consider the eyclie I!;roup «a), *) of or(ler 15 and the subgroup «a 3 ), *). List the elements of eaeh coset of (a 3 ) and construe!. the multiplicat.ion table for the quotient grouJl «a)/(a 3 ), ®). 18. In the commutat,ive group (G, *), let the set II consist of all elements of G with finite order. Prove that a) (11, *) is a normal subgroup of (G, *), called the torsion 8ubgroup, b) the quot.ient group (G/II, ®) is torsion-free; that is, none of its elements other than the identity are of finitl~ order.

2-{i

1I0MOMORl'HI8MS

8U

19. Rhow that a group (G, *) is commut.at.ive if ancl only if (G, GJ = {e}. 20. For any group (a, *), h-t (II, *) he the l"Iuhgroupgcnerated hy the Kct of Ilquares of clmnenl.!I of a. Jo:stnhlish t.he following: a) (II, *) is a normalllllhgfoup of (G, *). h) The qllotil'nt p;rollp (G/II,0) is commutative. (llint: [G, GJ ~ II, since [a, II) = (a

21.

* II)".! *

(/}-I

* a-I * /})2 * /}-2.)

u-t

(IIi. *) he a ('olll-I,tion of nontrivial norllIal ~ubgroup>! of the group (G. *) such that G = U/h Assume further that II, n IIi = {e} for i ;>C j, Prove that the parl-nt group (G, *) ill n"""I'C j, w.,e Prohlem 10. In (~a.~e i = j. choose any clement, c E G - II;. Then c aud c * a "OIlIllIUIl- wit,h IWI~ry ('lllment of IIi; in partkular, e = [e * a. b * al =[a, bl.]

22. Prove that if th!' qUoti"nt group (G/cent G, 0) is mutative group,

I~y('lil'.

then (G, *) ill a I:om-

2-6 HOMOMORPHISMS

Up t.o thiH point, in the t(~xt, w(~ hll.vI~ not eonHidcred ml\(lpinll;f1 from one group to Ilnothel'; in(l<~('d. any knowledge of funetions was irr<,l(lvallt to most of the t.opi(·s (~onsid('r('(1. TllI'Y now ent.er in an eHslmt,illl way. for we wish to inl.rodu<:e It ('0111'<'(11. IIII' idm of algl'iH'uienlly indistinguishable sYHI,ems-whieh will be of fundamental importance throughout thc remainder of the book and whieh ill, in fat't. one of t.he most important notions in mathemlltics, This section begilll'l, however, with an analysis of a class of functions which preserve alj!;ehmic Rtrueture. Deftnition 2-27. Let (G, *) and (G', .) be two groups and f a function from G into a'. f: a -+ G', Tlwn J is said to be a homomorphism (or operationpreserving fundion) from (G, *) into (G', .) if and only if f(a

* b) = f(a)

0

feb)

for every pair of clements a. bEG. A few remarks are in order before considering any examples. First, notice that on the left.-hand side of the above equation, the product a * b is computed in G, while on the right side the product f(a) feb) is that of elements of G'. The fundions indicated in this definition have the charaeteristic: property of earryillg productll into products, A common way of expressing the situation is to llay that t.he image of n produet under J is equal to the product of the images, 0

«((a), ( ( b l l - - - - . . . : - - - -.... ((a)

0

((b)

~

f(a • b)

!}()

2--1;

WlOtTP T1n:OHY

Anotlwr vi('wpoint is perhaps 1)(,IH'ficial. The requirement. that f(a * b) = f(a) f(b) for eVl'ry pair of clements a, bEG is sometimeH des(~rihed by saying thnt till' dinJ.(mlll of mappings on page 8!J is ('ollllllutal.iv(', For this eCllulit.ioll aHsertl'! thaI. if WI' I'!tal't with dl'nwlltH a, b of a and move them to 0' by eit.IH'1' of the two rouh'H indieateu by the arrowli -hy fil'lit forming the product a * /, and t.lWII applying f t.o it or hy firlit. oht.aining UII' imagl'li f(a) and f(b) and then taking I heir produl'l. --the rCHult will be the Hllme, It may niSI) st.rike t.he reauer that the langlmge ill whil'h Definition 2-2i i" coudwd is 01)('11 to I~ritieilim. To speak of IL hOlllolllorphiHm from a group (G, *) illt.o It group' (0', .) iii S
Example 2-40. FOI' lin arhitmry group (G, *), d('filll' the funetion f: G -+ (/ hy taking f = io , the ident.it.y map on G. It is It trivilLlit.y to check that f is !l homomorphililll from t.he group (G, *) into itself, a8 f(a

* b)

= a

*b =

f(a)

* f(b).

Example 2-41. Suppose that (G, *) and (G', 0) are two groups with identity elements (' and p', r('sp('divl'ly. Thl' fun('tion f: G -+ G' given hy f(a) = c' for eH('h a E r; is a homomorphiHm: f(a

* b)

=

e' = e' e' = f(a) of(b). 0

This part.i('ulal' mapping (the so-ealled trivial homomorphism) is the ollly constant. function whieh satisfies Definition 2-27. Example 2-42. Considpr the two groups (R', +) and (R' - {OJ, .), where Ill'! usual, lind· d('llOtt' ordinary addition and multiplicntion. For a E R', dcfilw t.he funl'liollf hy f(a) = 2". To show that the mappingf is operation-preserving, we mllst ('stahlish whether f(a b) = f(a) , f(b). This is readily verified,

+

+

f(a

+ b)

= 2"+b

= 2"· 2b =

f(a) . feb).

Example 2-43. L(,t (Z, +) be the group of intcgcl'!! under addition and (Zn, +n) be thl' group of integers modulo n. Define J: Z -+ Z,. by f(a) = [a]; that is, Illap eal'h intcgl'r into the congruence dass containing it. That f is a homomorphism follows dirl'ctIy from the definition of modulnr addition: f(a -I I) =

For fulure

liSP, W('

la -I

b] = [a] f,. [b] = f(a) +,.f(b).

shall label the set of all hOlllolllorphiHtns from th(, group

(0, *) inlo ilsdf (thp so-calh'd ('1u/olllurphis1ns) hy t.he Hymbol hom Gj a fr('-

!(llI'ntly IIs!'ll altl'rnal iv(' notal ion iH to write end G. Both notations hllv(' a el'rlain HUJl;gestive pow('r, and it rctIuces to 11 matter of pel'l:lOnal preference.

2-6

IIOMOMOltPlIlHMS

IlIterCl:it ill)!;ly ('lIoll/!:h, the set hom stmdur!' :

a

91

cnll be enuowl,d with an nlgebraic

Theorem 2-36. Tilt' pail' (hOIlI 0, 0), wh('l"(' 0 dl'lIot.ml flllwtionnl tiOIl, forms a semigroup wit.h idcnt.ity.

compOl~i­

PJ"()of. For' III!' proof, whidl is quite dCllwnt.nry, it, IIlUHt fif/lt. he Hhown that t.h(l ('olllposil ion f· (I of two flllwLionH f, (I E hom G Itl(uin pr('HerVCS tho group operation. ThiH is pltsily Itcl'ompli!lhed by noting that whenever a, bEG,

(f

0

(f)(a

* b)

= f(y(a

* 1))

= f(r/(a) * f/(/I») = f(a(a»)

* f(I/(II»)

= (f

0

g)(a)

* (f

0

g)(b).

As we havp s('('n in an ('arli('r !l(lction, ('o!llposition of funetions is a.'!sociative. Filially, Example 2-40 in!iil'nt.es that. the identity mapping io (the identity dement for (~orllposit.ioll) is it.Helf opl'rat.ion-preservillg. A r!'Hsonabltl l'luhjed. of curiosity would 1)(' th(' question of whether or not For inverses to exist, on!' must plainly single out the one-to-one functions. i\loreover, in order that the domain of f-I be the set G, consideration should be further restricted to Ihos(' fllnd ions whil'h lIIap onto G. Thus a natural lIndl'rtaking is to invest.igate I hI' (~()II('etion of all one-to-one homomorphisnu; from the group (G, *) onto itself; as a matter of notation, we shall designate this set of mappings by UI(' symbol A (G), for auloTnoJ'plii81n. The clementI:! of A(G) arc now restricted to the extent that (A(G), 0) does indeed have the agreeable property of being a group. Let us give some details. t her(' !'xistll a subset S ~ hom G such that (8, 0) is a group.

Theorem 2-37. The system (A (G), 0) is a subgroup of t.hc symmetric group (sYIll a,.). Proof. For funetions f, (J E A(G), we already know the composition of fog is in sym G. In conjunction with the lal:!t result, this showsf (J belongs to A(G), aH does the ident.ity map i a. It remainH ollly to verify here that whenever a funetion f E A (G), it.s inver!le f- I (whieh clearly is a member of sym G) is a homomorphi!llll. If if, 1) E G, the onto dmmeter of f implies if = f(a), lj = feb) for some choice of a, b in G. Therefore, 0

f- I (if

* b)

= f-I (f(a)

* f(ll»

= f-I(J(a * b) = a*b =

f-I(if)

* f-I(b),

and t II(' proof is ('olllplete. There are many important and inlN('sl illg fad.s ('olwerning homomorphic mappings. I n I h(' :-;u('(~eedin)!; theorellls, we shall examine some of tJwse results in detail.

92

2-6

G1WUP THEOHY

Theorem 2-38. If J iH n homomorphiHm from the group (G, *) into the group «(;', .), then 1) f mapll the identity clement e of (G, *) onto the identity clement e' of (G' , ·):f(e) = e' , 2) f maps the inverse of an element a E G onto the inverse of f(a) in (G' , 0): f(a- I ) = f(a)-l for each a E G.

Proof. To prove the first assertion, it is enough to observe that under the hypoUw8is of the t.lworl'lll, J(a) • e'

= J(a) = f(a * e) = J(a)

• J(e),

whenever a E G. By the cnrwl'Untion hlW in (G', .), we then have

=

j(e)

e' .

In ttw second part of the t.1worem, it ill first necessary to show that

We can then conclude from the uniqueness of the inverse of f(a) in (G' , .) that = j(a- I ). To obtain this result, we make use of part (1) t.o get

f(a)-I

f(a)

0

f(a- 1 )

=

f(a

* a-I) =

Similarly, j(a- I )

•

f(a)

=

f(e)

=

e' •

e/.

Example 2-44. As an immroiate application of these ideas, we propose to establish that for each real number r ~ 0 there is exactly one homomorphism f from the group (Z, +) into the group (R' - {O},·) for whichf(l) = r. The ('XiHteIW(, of Btwh n function is trivial, for we need only (!onKider the mapping f(n) = r", n eZ. / To prove there can be at most one function satisfying the indicated conditions provides a more chall(mging problem. The basic idea is simple enough: assume there are two fmwtions, f and g, having the required properties, and show that they are actually the same. Now, each positive integer n may be written as (n summands). n= 1+1+···+1

The operation-preserving character of J and g thus implies f(lI) = f(1)"

=

r"

=

g(l)"

=

g(n) ,

On the othC'r hand, if n is a nonzt'ro negative integer, -n E Z+. Hence, f(n) = f(-(-1I»)

=

f(-n)-I

=

y(_n)-l

=

g(-(-n»)

=

g(n).

2-6

HOMOMORPHISMS

93

The erlJ(~inl st.ep, f(-n) =-: (I( -n), is justified by til(! fact f lmd g arc already known t.o agree on the positive intl'ge,'s, By the firnt. plLrt of Theorem 2-38, f(O) = 1 = y(O), so t.hat fen) = yen) for every integer n; therefore, f = y, The next result indicates the algebraic nature of direct and inverse images of subgroups under homomorphisms, Among other things, we shall see that if f is a homomorphism from the group (G, *) into the group (G', .), then (f(G), 0) forms a subgroup of (G', 0), The complete story is told below: Theorem 2-39. Let f be a homomorphism from the group (G, *) into the group (G', 0), Then 1) for each subgroup (H, *) of (G, *), the pllir (j(II),.) is a subgroup of (0',0), 2) for each subgroup (H',o) of (G', 0), the pair (I-I(H'),.) is a subgroup of (G, *), Proof. To obtain the first part of the theorem, recall the definition of the image set f(H) : f(H) = {f(lt) I h e H},

Now, suppose f(h) and f(k) are arbitrary clements of f(H) , Then both hand k belong to the set H, as does the product h * k- 1 , Hence, f(h) of(k)-l

=

f(h) of(k- 1)

=

f(h

* k- 1)

ef(H),

Our argument shows that whellever f(h),f(k) ef(H), thenf(h) of(k)-l lies in feB); this is a sufficient condition for (j(H), 0) to be a subgroup of (G', 0). The proof of the second statement proceeds in a similar manner. rememher thut j-I(H')

=

First,

{aeGlf(a)eH'},

Thus, if a, b ef-'(H'), t.he images f(a] and feb) must be elements of H', It follows at once that f(a

* b-')

= f(a)

0

j(b-I) = f(a)

0

f(b)-t

e

H'.

This means a * b- t ej-I(ll'), from whi('h we ('OIwhule (J-'(Il'), *) is a subgroup of (G, *), L(·ft. ullrl'solved, as yet, is the mat.t('r of replacing t.he term "subgroup" in Throrem 2-39 hy "normnl suhgroup." It. is IIot pIll'ti('ularly diffil'ult to show that. part (2) of the theorem remains true under sueh IL subst.itution, Preeisely speaking, if (/1',0) is a normal subgroup of (G', 0), the subgroup (I-'(H'), *) is normal in (G, .), In establishing this fal·t., we will ut.ilize both implications of Th('or!'1II 2-:t!. Huppose \lOW II ef-'(lI'), M t.hat f(lt) ell', nnd let a be

94

2-(S

GROUP THEORY

an arbitrary dement of f(a

In other words, a * h

a. Then,

* h * a-I) =

* a-I

f(a) of(h) of(a)-I E H'.

Ef-1(H'), or in terms of sets,

According to Theorem 2-32, this inclusion is enough to make (j-I(H'), *) a normal subgroup of (G, *). Without further restriction, it cannot be inferred that the image subgroup (f(H), 0) will be normal in (a', 0) whenever (H, *) is itself a normal subgroup of (a, *). One would need to know that a' ·f(lt)

0

(a,)-I Ef(H)

for all a' E G' and It E H. In general, there is no way of replacing the element a' by someJ(a) ill order to (!xploit the normality of (H, *). A slight strengthening of the hypot.hellis overcomes this difficulty; simply tak!l f to be a.n onto ma.pping. (Rc<:all that the word "onto" r<-'quircs every member of (J' to be the image of at. I<'l\~t one clement of G.) Summarizing tlH'sC r('marks, we mny now state: Corollary. I) For !lll.dl normal lmhgroup (H',o) of (G',o), the subgroup (f-I(II'), *) is norlllal in (0, *).

2) If f(G) --=- a', Uwn for ca<:h normal Imbgroup (ll, *) of(G, *), the group (j(l/), 0) it! normal in (a', 0).

HUh-

In much of our subsequent work, the object of interest will be the kernel of a homomorphism. Definition 2-28. Let J be a homomorphism from the group (G, *) into the. group (G', 0) and let e' be the idenhity element of (G', 0). The kernel of f, denot,ed by ker (f), is the set

ker (f)

=

{a E G I/(a)

=

e'}.

Thus ker (f) consists of those clements in G which are mapped by f onto the identity eicment of the group (G', 0). Theorem 2-38 indicates that ker (f) Is a nonempty subset of a, since e E ker (f). It may well happen, as Example 2-41 shows, that kl'r (f) = G. Except for the trivial function indicated there, the kernel ill IlIW:1YII a proper subset of G. Our definition of It homomorphism did 1I0t require that it be a one-to-one function, and iIHl('cd, we hllve preRCntcd several examples where it failed to be so. There ill, howcV!'r, 11 simple charudcrization of a on<-'-to-one homomorphic mapping in tenns of the kenlel.

2-(i

95

HOMOMOUPHISMS

Theorem 2-40. l.,(~t f he a homomorphi8m from the group (G, *) into the group (G', .). ThenJ is one-to-one if and only if ker (f) = {e}.

Proof. SuppoS(! the function f is one-to-one. We already know that e E ker (/). Our aim is to show that this is the only element in the kernel. If there existed another clement a E ker (f), a ¢ e, then we would havef(a) = e' = I(e). That is, f(a) = fee) but a ¢ e. This would contradict the hypothesis that I is oneto-Gne. On the oUler hand, !lUppOH(! thaI, ker (f) = {e}. Let a, bEG and f(a) = feb). To prove f is one-to-one, we must 8how t1mt a = b. But if f(a) = feb), then f(a * b- I ) = lea) of(b-I) = f(a) of(b)-I ~-= lea) f(a) - I '" e', 0

which implieR a a = b.

* b- I

E

ker (/). But, ker (J)

= {e}. Therefore a * b- I

=

e or

The next theorem will establish the algebraic character of the pair (ker (f), *). Theorem 2-41. If I is a homomorphism from tho group (G, *) into the group «(/', 0), then t.Il1l pair (kllr (f), *) is u nornll~IHubgroup of (G, *).

Proof. We have already indieated that t.hc trivial subgroup ({e'), .) is a normal subgroup of «(t, 0). Hin/~e k/~r (f) = f-"(c'), I.he /:onduHion follows from the general result tltl1u!<1 ill pnrl. (1) of the lust /~orollnry. Example 2-45. As IL Him pie two grollI'M (Z, I ) and (Il' -

illu~trntion

of the! ILbovn Uworems, cOIIHi n' -- {OJ defin<-'fi

{O},')' Thll nmpping f: Z

by

fen) = {

I -1

if n E Z., if n E Zo

is a homomorphism, us the reader may vllrify by checking the various cases that could arise. III the situation considered, ker (f)

=

{n E Z Ifen)

=

I}

=

z.,

while the direct image

feZ) = {I, -I}. It is not pnrl.i/·lIlnrly diffieult 1.0 show that (Z., +) i!l a normal subgroup of (Z, -!) ancl t.hat ({I, -I},,) is a subgroup of (n' - IO}, .). We IlILvc jUHt. H(~(m that every homolllorphiHm determincs It normul Huhgroup by mcanfl of its k(!rncl. On I.he other hand, thc following theorem will show that every normal flllbgroup givefl rifle to a hom()morphi(~ mapping, the socalk-d nutunl1 llJapping. Simply put, the problems of IIIIIJlIIg homomorphisms and normal subgroups are inscparable.

96

2-6

GHOUl' THEORY

Theorem 2-42. Let (H, *) be a normal subgroup of the group (G, *). Then the mapping natH: G -+ G/ H defined by

is II. homomorphism from (0, *) onto the quotient group (G/H, ®); the kernel of natll is prc("iscly the set H. Proof. Tlw fal"!, t.hat. t.he mapping natH is homolllorphie follows directly from till' mlUlIIl'r in whi!"h 1I\11lt,iplic~llt.iClII i>l clC'linml ill thc~ ,,1IC1t,icmt KrOllI':

nllt.,,(11 • I,)

=

(a. II)

=

(a

* /I

* H)

® (b

* H) =

nutH(a) ® natJl(b).

To show thnt, nat/( is an onto funet ion is almost trivial, sinee every element of G/If is Il c'ost't a * H wh('\'(· a E G find Ilntll(a) = a * H. IlIIuml\lc·h lUI the ("0>1('1, /I serV!'s IlS the identity clement for (O/H, ®), we must have ker (natll) = {a E G I natH(a) = H} {a E G I a

* /I

=

/I} = II.

Th(' InRt !'CllIlllit.y was ac'hievec! by t1w lise of TheorNIl 2-2fi,

It is pO>lsihl(', IUlcl sonwt.illll'H {!onvl'lIient, to phru.sc TIl(lorem 2-42 so that no rdl'rc'nc'C' is lIIaclc' to t.Il1' notion of CJllotil'nt p;roup: Theorem 2-43. Ll't (1/, *) be a nOl'mal Imbp;roup of the group (G, *). Then there exists a group (G', 0), and 11 homomorphism f from (G, *) onto (G', 0) su{~h that ker (f) = H. Of COUl'R(', we tak(' (G', .) to be the quotient group (G/II, ®) and

f = natH,

The usual custom is to refl'r to the function nntll as the natural or canonical Provided there is no danger of eOllfusion, we shall frequently omit the sub8cript II in writing this funl,tion. As a rl'bt.t'd rc'mark, it might be emphasized that the natural mapping is not generally one-to-Olw. For if a, b E (J arc clements such thaI, the product a-I * b is in H, tl1<'11 by Theorem 2-