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 2-27, a * 1/ = b * II, and consequently nutH(a) = nat.[{(b). Lt,t us pausl' for a 1lI0nlC'lIt to inil'rpr('t Theorl'lll 2-42 ill the ease of the additivc group of illt('~t'rs (Z,!~). We already know that its normal subgroups are the cyclie grouplS «n), +), where n is a nonnegative integer. i\Ioreover, the' qlloti!'nt group C'OI'I'('sponding to !lny fixed n E Z+ is simply (Z,,, +n), the group of int('~ers modulo II; that. is, Z/(n) = Zn. It ilS fairly evident from this that. the nlltuml mapping na .. : Z -+ Zit docs nothing morc than send each integer into its eongnwnce ('hls8 modulo n: nat (a) = [a). mappinfl of G onto G / II.

2-(3

HOMOMORPHISMS

97

Definition 2-29. Two groups (G, *) and (G',.) are said to be i8omorphic, denoted (G, *) ~ (G', .), if there exi~ts a one-to-one homomorphism f of (G, .) onto (G', .), that is, f (G) = G'. Such a homomorphism f is called lUI i8()mOTp"i.~m, or isomorphic mapping, of (G, *) onto (G', .).

Any property of (U, *) whieh clLn be expres!:!ed in terms of the operation * is prc~rvcd under f and consequently be(~ome!:! a property of (G', .) as w'lJ1I. The uJlHhot is that t.he nmpJlinj!; f haH the effect of I ransferring the algebmic Htrtll'l.ure of the group (a, *) to the group (G', .). I!:!()llIorphic groups are thus incliHt.ingllislmhl .. fmlll t.\J(' ILhslral't point of vi(~w, (WI''' though t\u'y limy lIiffc'r ill t.hl' not.al ion for arid lIal Ul'(' of UU'ir "Iements mill ()p(~flLtiom~. Two slleh groups, whill' not. ill W'lIl'ml fOl'lnally iucntil:al, arc the HIUIIe for all pl·act.ieal purposes. Actually, the concept of isomorphism is applicable to all types of mathematical systems, for it seems reasonable to treat two systems as essentially <'qlUlI when they hnve exactly the Hallie properties. The eR.'!Cnec of the notion is that we can alwnys find a one-to-onc mapping betwccn the clements of the two systems whil'h preservcs whatever structure we are int.erested in studying. The ohservant. reader probahly noticed that Definition 2-29 is unsymmetric in t.hat it IImk('s nH'ntion of a fllll(·t.ioll f!'Om one pllrt.iellhlr group t.o Ilnother. Howl'vPr, if f: (J -+ (]' is a one-to-one, onto, op<'ration-preserving mapping, the fundion f- I : G' -+ G also Jut~ these propertil's. We may therefore ignore this illitillllaek of symmet.ry and merely Hpeak of two groups (G, *) and (G', .) us heing iiiOll\orphie to ea(~h other; to indieate t.his, it. Imffiees to write either (fl, *)

~

(a',.) fir (a',.) ~ (f~, *).

Before procecuing further, 1\ knowleuge of several specific examples will provide some basis for an understanding of the general idea of isomorphism. Example 2-46. Consider the two groups (Z., +4) and (G, .), where G

and i 2

=

= {I, -1, i,

-i}

-1. The operation tables for these two systems arc

+. 0 1 2 3

0

I

1

-1

1 -1 i

-1

i

-1

-1

-I

-~

1 -i i

2 3

0 2 3 1 2 3 0 2 :3 0 1 :3 0 2

1

-1

-i

-1

1

1

-1

Wc wish to provc that the groups (Z., +.) and (G, .) are abstractly "equal." To do so, we must produce a one-to-one homomorphism f from Z. onto G. Since the preservat.ion of identity clcments is !\ general feature of any homomorphism, f llIust be such that f(O) = 1. Let us suppose for the moment that we were to define f( 1) = -1. The image of an inverse clement must equal

98

2-6

GROUP THEORY

the invel'HC of the image. We would then have f(3)

=

f(I- I )

=

f(I)-1

=

=

(_1)-1

-I,

or f(3) = f(l). This, however, would prevent f from being one-to-one. A more appropriate choice, in the sense that it avoids the above difficulty, is to iake f(l) = i. The condition on inverses then implies f(3) = -i. Since f is further required to preserve modular addition, f(2)

=

1(1 +.1)

=

1(1) ·f(l)

=

i· i

=

-1.

Wo are thus lNI in a llatuMLl way to eonl!ider Uw f\llletion defined by 1(0)

=

1,

f(l) = i,

1(2) = 1,

I(a)

=

-i.

Clearly this fundion is 1\ ono-t.()-()no mappinll; of tho HCt Z. onto the set G. Furthermore, 1 uetuully pn~HCrve8 tho operations of the groups. Merely to verify one instanco, we ObH(~rv(l that f(1

+. 2) = 1(3) =

-i

=

i· -1

f(I)· f(2).

±

Consequently, we have (Z., +.) ~ (G, .). Loosely speaking, two finite groups are isomorphic if it is possible to obtain each mUltiplication table from the other by merely renaming the elements. The nature of the function 1 suggests the appropriate rearrangement of the table for (G, .) is 1 i -1 - l i -1

-i

i -1 -i

i -1 -i 1

-1 -i 1 i

-i

1 i

-1

Apart from the particular symbols used, this group table is identical to that of (Z .. , +.. ), for corr<'sponding elements appear at the same place in each table. Both groups are simply disguises for the same abstract system. In passing, we might note that (Z .. , +.) is also isomorphic to (G,.) under the function g, whereby g(O) = 1, g(l) = -i, g(2) = -1, g(3) = i. Example 2-47. Let G = {e, a, b, c} and the operation * be defined by the table at the right. The reader may verify that the pair (G, *) is IL group, known as Klein's fourgroup. The two groups (Z., and (0, *) are not abstractly equal, however, for every one-to-one function I from the S('t. Z. onto G fails to be operation-preserving.

+.. )

*

e a b c

e e a a a e b b c c c b

b

c

c b e a a e

2-6

HOMOMORPHISMS

99

From t.his we conclude t.hat. there arl' at least. two distinct algebraic structures for groups with fOllr clements. To illustrate this point, we shall check several possibilities for the function f. Consider the mapping defined on the set Z4 by f(O)

=

e,

1(1

-I-.

3) = f(O) = e ~ b = a. r.

f(l) = a,

f(2)

=

b,

1(3)

=

c.

Then

whidl Ilhows that. Uw proposp.<1 for f might, bn f(O)

=

e,

= f(l)

• 1(3),

f is not a homomorphism. Another posllibility

f(l)

=

f(2) = c,

h,

1(3)

=

a.

Note that wo mUlti, ulwaYIt mlLp idt!lltity n1mmmtll to it.ioutify clements. This ehoico of f also faill'! to pr(!S('fve th(' operations, Ilineo

1(1

-h 1)

=

f(2)

=

r ~ e

=

b·1J =- 1(1) • f(I).

We shall leave the test of the remaining possibilities as an exercise. A standard procedure for showing that two groups are not isomorphic is to find some property of one, not possessed by the other, which by its nature would necessarily be shared if these groups were actually isomorphic. In the prellCnt case, the group (Z4, +.) and the four-group are differentiated by the fact the former is a cyclic group whereas the latter is not. Example 2-48. The two groups (Z, +) and (Q - {O},·) arc not isomorphic. To sec thiN, IlUppotIC t.hat t.here exists a one-to-one onto functionf: Z -+ Q - {O} with the property I(a

+ b) =

f(a) . I(b)

for all a, bE Z. If x donotes the clement of Z such that f(x) f(2x) =f(x+x) =f(x)·f(x)

=

(-1)· (-1)

=

=

-1, then

1.

According to Theorem 2-40, the identity element of (Z, +) is the unique clement of Z corresponding t.o the identity of (Q - [O},·), so that 2x = 0 or x = O. Consequently, both f(O) = 1, and f(O) = -1, contradicting the fact that the functionf is one-to-one. This argument shows that (Z, +) cannot be isomorphic to (Q - {O},·), for no fUlH'tion satisfying Definition 2-29 can exist. Example 2-49. For an instructive example in connection with the additive group of integers ('ollsidcr the following ll..'!SCrtio:l: the only functions under which (Z, +) is isomorphie to it.'lClf are the idcntit.y mapping and its negative. A fairly suceiru:t dl'seription of all this is that A(Z) = {iz, -iz}.

100

2-6

Gnoup THEOny

Perhaps the quiekest proof of til(! above WI.'!crtion consists of ahowing that if J E A (Z), then the eY('lie subgroup «(j(l)), +) generated by J(I) is the group (Z, +) itself. Since the illdusion (j(l» ~ Z trivially holdR, our aim would bc achieved by establishing Z ~ (/(1». But this is a l:!traightforward matter. If n is an arbitrary integer, n = J(m) for some mE Z-recall J is a. ma.pping onto Z-so that n = J(1 1 1) (m summands)

+ + ... +

= J(I) + J(I) + ... + J(I) = mJ(I), whence n E (J(l». Since land -1 are the only generators of (Z, +), either J(I) = lor J(I) = -1. However, the preceding computation indicates fen) = nf(l) for each n E Z; fromthis,itisclearthatJ= izorJ= -iz according asf(I) = 10rJ(I) =-1. Let us return to general considerations by showing that the groups (Z", +,,) and (Z, +) are the prototypes of all finite and infinite cyclic groups, respectively. Theorem 2-44. Every finite cyclic group of order n is isomorphic to (Z", +,,) and every infinite cyclic group is isomorphic to (Z, +).

Proof. First, suppose the cyclic group «a),.) is of finite ordeF- n. case, we know from Theorem 2-23 that

In this

It scems nutural tht'll t,o illvel:!tigate thc mapping J: (a) --. Z,. givcn by thc rule f(ak) = [k], 0 ::; k < n. This function plainly carries (a) onto the sct Z,.. Next observe that f is Ollc-to-one: if J(a k) = f(a i ), then k == j (mod n) so that a k = a i . Finally, for any clemcnts a\ a i in (a), we have

This shows t.hc function J prescrves the respective group operations and eomplctes the proof of the iRomorphism C(a), .) ~ (Z,., +.. ). For t.he seeond purt of the theorem, thc cyelic group C(a),.) is assumed to he of infinite order. HI're, the choice of a mapping I betwccn (a) and Z is obvious: simply take I(a fr ) = k. It is immediate that J, so defined, is an onto mapping. Further, this function is one-to-one, for all the powers of the generator must be distinct; if two different powers of a were equal, the argument of Theorem 2-23 could be employed to obtain the contradiction that (a) is a finite set. The rest is routine:

Hence, as we wished to establish, C(a),.) is isomorphic to the group (Z,

+).

2-6

HOMOMORPHISMS

101

Corollary. Any t,wo "Yl'lic grouP!! of the same order arc isomorphic.

Trivially, llny group (0, *) is isomorphi(~ to itself under the identity mapping i(}. A reasonlLhl" query is whether (0, *) is isomorphic to any group other than itself. The concluding th(!orem in this HCetion, a classical result due to Cayley, answers the question in the affirmative. We begin, however, by reenlling HOIlt(! definitions and notation. For an arbitrary clement a in 0, the left-multiplication function fa: G -+ G was defined by taking fa (x) = a * x for every x E G. The collection of all functions obtained ill this way is labeled by Fo: Fa = {fa I a E G}. Example 2-25 established the stnlctural nature of the pair (Fa, )-in the present context indicates the operation of funetional eompoHition-when this system was shown to be a group. Our tn.sk now is to prove the isomorphism of (G, *) and (Fa, .). 0

0

Theorem 2-45. (Cayle'lJ). If (G, *) is an arbitrary group, then (0, *)

~

(Fa, .).

Proof. Define the mapping f: G -+ F G by the rule f(a) = fa for each a E G. That the functioll J is onto Fa is ohvi()uH. If f(a) = J(II), so fa = fb, thell a * x = b * x for all clements x of a. In particular,

a

= a *e = b *e =

b,

which shows that f is one-to-one. We complete the proof by establishing that f is a homomorphism: I(ll

* IJ) =

f ... b

=

fa fb 0

=

f(a)

0

f(l,).

As all iJluHtration of this theorem, (~onsider the group (Il', +). Corresponding to an element a E R' is the left-multiplication function fa, defined by

f .. (x) =

a

+ x,

x E R'.

That iR, the function fa merely has the effect of translating or shifting elements by lln llmount a. Cayley's Theorem MHerts that the group (R', +) and the I/:rollP (Fa, .) of transl:tt.iollR of t.he real line nre indiRtinguishablc a8 far as their nlgl'hmie propl'rt.i(·s an! (!OIw(!rlled.

PROBLEMS

I. In the following situations. (Ietcrmine whether the indicated function J is a homomorphifoull from the first group into the sc{'ond group.

a)J(a) = -a, (R'.+). (R',+) h} J(al = la!. IN' :0: •. ), (R'+,·) c) J(a) = a + I. (Z,+), (Z,+)

102

2-6

GHOUl' TIU;OHY

d) I(a) = a 2 , (R' - {O}, .), (R't, .) e) I(a) = a/q (q a. fixed nonzero integer), (Z,+), (Q,+) f) I(a) = na (n a fixed int.eger), (Z, I ), (Z, I ) 2. l'IUJlIHll!1l I iii a IWlIlulllurphitim trolll t.he group «(},.) into the group (G', 0): a) If e d(!!!ignu.tel! the identity eiemont of (a, .), show tha.t the kernel of I may be dMcribed by ker (I) - I -I (f(e». b) Provided the group (G',o) is commutative, establish the inclusion [G, GJ S;;;; ker (f). 3. Let (ZK, I K) he the group of intogertl modulo Ii and «a),.) be any finite (lydi(l group of order 12. AJ;l!ume further that. the maJlPing I: ZS -- (a) is defined all follow!!: 1(0)

1(2)

= 1(4) = Il, = 1(6) = a6 ,

1(1) =/(5)

1(3) = 1(7)

= aa, = all.

a) Prove that. t.he function I, 110 defined, is a homomorphillm. b) Dell('rib(' the ~UhgroUPN (kl,r (f), -I II) anel (f(ZK) , .). c) If II = {e, aft), show the pair (f-I(II), +11) ill a Iluhgruup of (ZI!,-t-s). 4. Consider the two groups (Z, -1-) and ({ J, - J , i, -i},'), where i 2 = - J. Hhow that the mapping defincd by I(n) = i" for n E Z is a. homomorphism from (Z, +) onto ({ I, -- J , i, - i J, .), and determine itH kernel. 5. Let I be a. homomorphism from the group (G, .) into itselr and let II denote the Aet of elements of a whit-h are left fixed by I: II

=

{a E G I/(a)

= a).

Prove that (ll,.) is a subgroup of (G, .). 6. Let (G,.) he 11 gruul' and tho (,)omont a E a be (ixlle\. Provo that (G,.) ill isomorphic to itllclf--that is, (0, .) ~ (G, . ) - under the mapping I defined by

xEG. What is the kernel of thi!! function? 7. Prove that if the group (G,.) is commutative (cyclic) and (G, .) the group (G', 0) ill also commutative (cyclic). 8. Let the St,t

~

(G',o) then

a = z x Z and the binary operation • on G be given by the rule + r, b + d). It ill easily verified that the pair (G,.) is a

(a, b) • (c, d) = (a

commutative grollp. a) Show that the mapping I: a -> Z defined by I[(a, b)) = a is a homomorphism from (G,.) onto the group (Z, +). b) Det,C'rmin(' the kernel of this mapping. c) If II = {(a, a) I a E Z}, prove that (II,.) is a subgroup of (G, .), whieh is isomorphiC' to (Z, +) under the function I. 9. Show that the two J!;roups (R', +) and (R' - {O},·) arc not isomorphic. 10. Prove that all finite groups of order two are isomorphic.

'2--7

TU),; FUNOAMENTAL THEOUEMS

103

II. If

wlll're i~ (G, .)

~

=

.

I, thl'lI the pl1ir (0,') rorlll~ 11 group.

(Za, -l-:i).

]2. Let land g be two homomorphisms from the group (G, *) into the group (G', 0). Define tlw fUlldioll h: G -+ (/' by h(a)

= 1(11)

0

g(a).

Rhow that. if t.l1I1 group «(/',0) is ,'ommutative, then h ill also a homomorphism. 13.

COIl~idcr

the following two groups: (GI, *), where GI = {RUIO, Raoo, II, V},

and the operation * "on~iHtli of following onllllyrnmetry of the IIlluare by another; (02. 0). wlll'rl' (/2 I'ow.;ists of t Iw four fUlwtiollS on R' - {O:,

/1 (.c) and

0

= .1',

12(.1') = -.1',

/:I(x)

=

I/x,

dl!llOt!'s fund-ional I,ompollitioll. Verify that (aI, *)

14 (x) ~

=

-I/x,

(G2, 0).

14. (liven I is a homomorphil:!m from a Aimple group (G, *) onto a group (G', 0). show that, eith,'r (0, *) ~ (0',0) or e1lSe that I mUlSt be the trivial homomorphism.

15. Let (G, *) be an arbitrary group and i be the mapping of the set G onto itself defined by ita) = a-I. for aK a E G. a) Prove that the function i is a homomorphism if and only if (G, *) is comlllut,n.I,ivr.. b) Generalize thr. result of Examplo 2-49 to the following: if (G. *) is an infinite cyeli() group, then A (G) = {io, i}. 16. Prove that any group (G, *) is isomorphic to some subgroup of its symmetric group (Kym G, 0). rUint: Use Theurem 2-45.] 17. Obtain the group of left-multiplication functions eorresponding to the group (Zli, +8) j set up the homomorphism which results in the isomorphism of these groups. 2-7 THE FUNDAMENTAL THEOREMS

In thill flectioll. we shall clifleuSS a number of lIignificant results having to do wit.h the relationship bet-w('1'1l homomorphisms and quotient groupli. Of thesc. Theorem 2-47, genl'rally known as the Fundamental Homomorphism Theorem for Groups, iK perhaps the most crueia!' The importance of this result would be difficult to overcmphnsizei in !1 scnse. all which follows thereafter may be vicwed as an enumeration of its spccial cases and implications.

Throuylwut this lIection, / denotes a IwmomoTllhism from the group (a, *) onto the group (G'.o). t.hat is. f(G) = a'. In order to simplify the statements of

104

2-7

OIlOUP TIIEOIlY

various tl\('or('m~, WI' shall frequently 1I0t troll!>l!! to spedfy this familiar opening phm&·. Ae('ol'dingly, unl('S!; Uwre is ('I('ar indicat.ion to the eontmry, any rI·f(~rel\(·e to til!' fUIH'tionj i~ undel'Mtood implieitly t.o involve the aforementioned hypoth('~ill.

W(~ h(·gin wil II It proof of the Fnctor Theorem, stated here in a form beij!. Huitl'd to 0111' immedinte needs.

Theorem 2-46. (Far/or Theorem). LC't (//,.) be a normal subgroup of the group (0, .) l>uch that H ~ ker (f). Then there exiijts a unique homomorphism7: Gill -> G' with the property

1= 7· nntl/. Prool. Before becoming explicit in the details of the proof, let us remind the r('ad('r thnt the symbol nat/l designates the natural mapping of G onto GIH; that is, natu: G -+ Gill with natH(a) = a. II. To start. with, we ddine a function 1: GIH -+ G', called the induced mapping, by taking aE G. l(a. II) = I(a),

The first question to be raised concerns whether or not 1 is aetual1y well-defined. In other words, it must be established that this funetion depends only on the eoSC'ts of H and in no way on the partieular representative used. -To see that thil> is so, SUppOIiC a • H = b • II. As the elemcnf,ij a and b belong to the same ('oset, the product a-I. b E H ~ ker (f). This means that f(b) = j(a • a-I • b)

=

f(a) • f(a- I

•

b)

=

f(a) • e'

=

f(a),

I\nd, by the mllnner in whieh 1 is defined, that l(a • H)

= l(b • II).

Hem'e, the function 1 is const.ant on the cosets of H, as we wished to demonstrate. A routine computntioll, involving the definition of multip\i('lltion in (G I H, ®), shows 1 to be t\ homomorphism: f«a.H) ® (b.H» =f«a.b) .H) = f(a. b)

= f(a)

• feb) = l(a • II) ·l(b • H).

Xext, we ohserve that for et\('h elenwnt a E G, f(a) = l(a .11) = l(nntu(a»

=

a· natll)(a),

whl'nce the equality f = 7. lIatll. The proof is complet.l'd upon showing that thil> fact.orization is uniqul'. Thms, l>UPPO:-;C also that f = Y lIat" for IIOmc other 0

2-7

105

fUlletioll (/: 0/11

-+

a' .

But t.JWII,

7(a • /I) = f(a)

= «(/

0

nll.tll )(a)

=

g(a • ll)

for nil a ill fl, Ji()] 1/. '1'111' irllitweci nlllppillg ] iK tllI·r..ror() 1i('(~11 to hI' thl! Ollly functioll from a/ /I to (J' lSutilSfying the equation f = ] natH' 0

Corollary. The function] is one-to-one if and only if ker

U) s; ll.

Proof. What is required here is an explidt description of the kernel of 7:

ker (/)

=

{a. H I](a. H) {a. H If(a)

=

=

e/}

e'l.

where, of CO\lI'8(', e' denotes the identity of the group (G' , 0). Another way of saying the same thing is ker (])

= {a. H I a E ker (J)} =

natll(ker (f».

Now, from Theorem 2-40, a necessary and sufficient condition for 7 to be a one-to-one mapping is that ker (]) = e .11 = H. In the present situation, this eondition reduces to requiring that natll (ker U»

=

H,

H. . In view of the equality f = 7 natll, the conclusion of Theorem 2-46 is

whieh is equivalent to the inclusion ker (f)

S;

0

often described by saying that the function f can be factored through the quotient, group (G/H, ®) or, alternatively, that f can he factored by natH. The followillg dillgmm muy help to elurify the relations umong the various function!! :

G/H

What. we have just proved, ill effect, is that there' ('xi!lts one and only one function 7 whieh makcs thilS triangle of maps commutative. Example 2-50. Lct us eit.(~ a SI)(!cific instance of the Factor Theorem. Our content.ion is that whenever the group (G', 0) is commutative, the function f can nhvays he fadored t.hrough the commutator quotient group (G/[G, G), ®); ot.herwise stated, f = 7· natlo,o).

lOG

(;)(WP

2-i

TIIEOHY

For this, f'ollsidl'r allY ('Ollllllutator fa, 11J = a * b As a ('OIlS('q\l('IH'1' of till' ('OIltllllltativity of (0', 0),

* a-I * /,-1,

with a, II E G.

f([a, b)) = [f(a),/(b)] = e',

so that [a, b] E k('r (f). 1'\ow, the ('omnlllt.at.or subgroup «(0, G], *) is, so to speak, the smallest subgroup of (G, *) to contain :\11 the commutators. Thus we must ('OIl1'lude that [0, OJ C;;; ker (f). Having t.his inclusion, it is only necessary to apply Theor!'1ll 2-·1(i to obtain the dl'sired factorization. Although

WI'

did Ilot I'xprl'ssly IWluire the information in {hp proof of Theorem

2-·W, it Illight II(' poilltl'd out thaI til(' ilulw'l'd nlappinll:7 ('lllTil'M (;/11 ont~) till' set (I'. 'fhi,.; is I'mdily ohtainable 1'1'0111 til(' defillition of 7 and t.hc fact f itself is un onto fundion. To hp more Sl)('('ific, if a' E (;', thcn a' = f(a) for some c1ellll'llt a in 0; h('III'I', a' = f(a) = 7(a * II). A rather simple obs('rvat iOIl, with far-reaehing implieations, is that whenever II = k(')' (fJ, so t hat bot h t he FIl(~tor Tlworem and its eorollary are applicable, f inclu('('s a nlllppinI!;JIIIUI('r ",hi('h (G/II, ®) and (G', 0) arc isomorphic groups. Thp cI iseu8sion of the for('l!;oinl!; paragraph may be ('onvcnicntly summarize(1 in the followillJ!; thN)f'I'Ill, a I'l'slIlt whi('h will hI' invoked rcpcnt.eUJy. Theorem 2-47. (Fundamental Theorem). If the group (G, *) Ollto thp group (G', 0), t1K'n

f

is

It

homomorphism from

(ajk('r (f), ®) ~ (G', 0). Remark. If in the statl'nl<'nt of the th!'Or('m, t.he word "onto" is replaced by "into," the e'OIwlusion takl's t.!u' form (a/kcr (f), ®) ~ U(G), 0).

Theof('1ll 2-47 is :ulmitt('dly rather t.Pehnical in nature and therefore perhaps n hril'f I'xplallat ion of its signifi('atH'(, is in ordl'r. Ruppm;p t.hnt. (G, *) is nn ullfalllilinr grollP WIIlISI' IIlw,bmi(' pl'Opl'l·t.il'; W(~ wiHIr t.o (1t,t.('r11lill('. CI('nriy, if (G, *) ('oHld hI' slroWII to hl' ilSolllorplri(' to SOIllI' well-known group (G' , 0), then OUl' prohlplll is soh'l'd; for (G, *), h('inj!; a rl'plica of (0 ' , 0), would POSReISS the SUIIlP alg('lmlil' Ht ruf'lurl'. Anofiwr approaeh, whieh usually givl's a lcss completc picture of (G, *) is t.o ('xamill(, itH illlngl's Ulldl'l' hOlllolllorphilillls. The (litrieult.y h('n~, of eOUrRI!, is that wlH'1I t.IrI'SI' flllld iOlls fail to 1)(' onl'-to-one, not all t.hc alj!;(·hraie properties of the imag:<'s an' 1'('f\I'I'I('tJ in t h(' original group. For instance, it is quite possihle for the ('()fIlIllH tilt ivl' law to hold ill :III illlag(~ group wit.hout (a, *) itself being (:olllmutat iv('. TIII'or('1ll 2---17 assl'rts t.hat the images of (G, *) ullder hornornorphisllIs (':111 II(' dllpli('atl'
2-7

Tilt<: FUNUAMENTAL THEOREMS

107

Example 2-51. A simph', hut illuminating, example of f.he Fundamental Theor(,1lI is flll"llislwci hy UII' I!;I'OIlJlS (Z, ~I ), «( 1, -I}, '), and thll homomorphism J; Z -> :1, I}, where fen)

={

1 if n

-1

E Ze,

n EZ o ,

if

Hen', Ihe kern!'1 of J is I.IIC' sci. Z" so t.hat Z /kcr(f)

=

Z /Z.

=

{Z., Zo}'

The FlIIuianu·nl.al '1'1\('01'('111 l.ill'n I!;lIarnntel~s thnt ( (Z., Zo}, ®) and ( {I, -I}, ,) Itrl! il'«lIIll1rphi(! Jl:rllllps illlll'l'ci, t.hiH is flLirly I'viill'lIt fnull 1m inspection of their multipliclLtioll tables; 1

Z. Zo

Z. Zo

1 -1

Zo Z.

-1

1-1 -1 1

7

Further, the proof of the theorem indicates that the function which actually establil:lhm; thil:! iHOmorphil:lm (the induced mappinp;) is given by

+ Ze) = Z(1 + Z.)

7(Ze) = 7(0

= f(O)

7(Zo)

=

f(l)

= =

1,

-1.

Example 2-52. For n more Jl('netrntinp; exampll~, consider an arbitrary group (0, *) and It fixed dl·nwllt. a E (1, Deline the nmppinp; f; Z -> G by the rule fen) = an, nEZ, It ill not difficult to chC(~k that f, so defined, is a homomorphie mapping from til!' :uldil ivl' group of intcgcrs (Z, +) onto the cyclic subgroup «a), *), Hence, by virtue of Theorem 2-47,

wherc, in t.he sitllation at hand, ker (f)

~

(n E Z I an

= e},

Now, t.WO pIIssihilit.i(·s Ilril'«' IU'l'orclilll!; 10 t.Iw mnl!;nit.Ilc1(· of t.he kernel, tho first hl'ing I hat kl'r (f) = :0;; ill other wordfol, an = e implies n = 0, Under the ('ireulllstall(,!'s, (Z/kcr (f), ®) is just t.he group (Z, +) itself. On the other hand, if kC'I' (f) ~ (0], tlH'I'C I'xistH SOI1W least posilive iute-ger n for which aft ~ f', 0111' ('lUI ('asily d('d\l('(~ fl'OIll this that k('I' (f) = (n), HO W(~ IlIlIst have (Z/k('r (f), ®) = (Z, .. -f .. ), III Sll 11111 lal'y , til(' pr('('('dillg disellHsion reveals t.hat (I) if the 1!;(,lwmtor a ill of infinite order, then «(a), *) ~ (Z, +), and (2) if a is of finite order n, then

108

2-7

GROUP THEORY

«a), *) ~ (Z .. , -III)' Tlwsc fads arc already familiar, of (~ourse, but thc argument involved furnishes an alternative approach to Theorem 2-44. The next theorem not only provides further evidence of the power of the Fundamental Theorem, but is of independent interest sinee it gives additional insight into the stmeturc of the quotient group (G/cent G, ®). To prepare the wny, it is 1\(!(~(,SS:U'y t,o
for every x in G. I.('t, us obtain a few of the special properties of this (unction. Finlt, CT" turns out to he a homomorphiHm: if XI, X2 E 0, then CT,.(.fl

* .1'2) = a * (XI * X2) * a-I = (a * XI * a-I) * (a * Xl * a-I) =

CTa(XI)

* CTa(X2)'

The next. t.hing to noti('c is that CT a maps the set G onto itself; specifically, for any e1<'n1<'lIt. ;r E G, CT "(a-I * x * a) = x. Filially, it can be proved that CT a is actually a one-t.o-one function. For this purpose, assume CTa(XI) = CT 4(X2), so that a * XI * a-I = a * X2 * a-I. The superfluous el('ments may be removed through the e:tnc('lIation law, allowing us then t.o conclude that XI = X2. All of these observations may be eonvenientIy summarized by saying-that CT" E A(G).

Funetiow; of t.lw form CT tI, with a E G, are usually (:aIlcd inner automorphism8 of the groUJl «(1, *); to he more preeise, CT a is the inner automorphism induced by the elelllent a. For hrevity, we label the set of fum:tions arising in this way by I(G): I(a) = fCT" I a E G}. In the ease of a commutative group, I(G) reduces to just the identity mapping io = CT.. Thus, it is only when (G, *) is noneommutative that the notion becomes meaningful. Lemma. The pair (I(G),.) constitutes a group, known as the group of inner aulomo/'phisms of (G, *); in fad., (I(G),.) is a normal subgroup of

(A(G), .).

Proof. The proof that, (I(G),.) is a group presents no difficulties; it consists of nothinl!: more than noticing that whellever CT a, CTb E I(G),

In addition to disposinJ!; of the closure condition, thili relation also indicates CT. is the iden! ity elemellt for the system and CT;I = CTa-l.

2-7·

THE FUNDAMENTAL THEOREMS

109

Regarding the second assertion, it suffices to show that if J E A(G), the product J (I' a J- I is an inner automorphism. The argument proceeds as follows: for each clement x in G, 0

0

(f

0

(1'"

0

rl)(x) = f((I'a(r1(X») = f(a. rl(x) • a-I) = f(a) • I(r\t» • f(a-I)

= f(a)

• x • f(a)-I

=

UI(a)(X).

The reasons for earh of these steps are reasonably KClf-cvident, and the reader should mnke Hure he ullderstlUuls them. What is Hignifimmt is that

f

0

U"

0

I-I

=

(l'/(n)

E leG),

as was to be proved. We are now in a position to obtain the result which has been our goal. Namely, that the groUpH (G/I!ent G, ®) and (I(G), .) refICmblll Clwh other in all essential aspects. Theorem 2-41. For each group (G, *), (G/cent G, ®) ~ (I(G), .).

= u o • That this function maps G onto the set /(G) is quite plain. It is equally easy to see that 9 is a homomorphism, since

Proof. To begin, consider the mapping 0: G -+ /(G) whereby g(a)

g(a • b)

=

(l'a.b

=

.

(l'a

0

Ub

=

yea) • g(b),

a,bE G.

The crudul IUipect of the proof is now to IdlOW that ker (g) = cent Gj once this is done, the desired conclusion will follow immediately from the Fundamental Theorem. III f,he clll;e at hand, t.he inlier automorphiHm U c serves as the identity clement for the group (I(G), .). A('cordingly, the kernel of the function 9 is defined by ker (0)

=

{a E G I o(a)

=

{aEGlu a

= u.}

=

ia}.

RI'ferring t.o the ddinition of p<)lInJif,y of funetions, we eonclu
llD

2-7

GROUP THEOUY

cvery subgroup (II, *) of the group (G, *) determines a subgroup (f(II), 0) of the group (a',o). It. goes without saying that group theory would be eonsidernbly simplified if the subgroups of (a, *) were in a one-to-one eorre:-;pondelwe with t.hm«' of (a', 0) in thill mamwr. Unfortunately, this need not be the CIlHl'. The !)ituation is refll'eted in the fact that if (II, *) and (K, *) arc two subgroups of (C, *) with 11 ~ K ~ II * kcr (f), then (f(II), *) = (f(K), *). The quickest way to see this is to note that f(lI) ~f(K) ~f(1I

* ker (f)) = f(lI)

of(ker (I»)

=

f(II),

from ",hii'll Wl' infpr Ihat. all thc inclusions are actuuIly equalities. In essence, we llre observing that diRt.inet RubRet.s of G may have the same image set in G'. TIl(' diffieult.y in thc last paragraph could be remedied by either requiring that kcr (I) = {e} or clse by narrowing our view to considcr only subgroups (II, *) with ker (f) ~ II. In eith('r event., it followH I,hat

II

~

K

~

11

* ker (f)

~

II,

yielding the subsequent equality II = K. The first of these aforementioned ('onditions has the eff'eet of making the function f onc-to-one, in which case (a, *) llnd (G', 0) are isomorphic groups. The seeond possibility is the subject of the next theorem. We pallsc to establish a preliminary lcmma which will provide the kcy to later success. Lemma. If II is llny suhset of G such that ker (I) ~ II, then II

=

f-I (f(II»).

Proof. Suppose the cl!'ll1!'nt. a is ill f-'(j(II»), so that f(a) Ef(II). Then f(a) = f(h) for HlHne choi('(~ of II. E 11. As the equation f(a) = f(lt) is equivalent to f(a * It-I) = e', we have a * It-I E ker (f) ~ II. This implies a also belongs to the set II ulld yield:-; the in('lu:-;ion f-I (f(II») ~ II. The opposite illdusion' always holds (Theorem 1-7), whenl'e II = f-I(j(II». Tim n'latinllHhip "I'IW('('II t.11I~ HII"I(I'OIl(l1l of (0, *) lun! Un~ suhgrou(l of (0', 0) may he stall·1I "" follow ... : Theorem 2-49. (Corrcl!pllluience Theorem). There is a one-to-one correspOndell('e between t,ho:-<e subgroups (H, *) of the group (G, *) sueh that k!'r (f) ~ II and the HI'! of all subgroups (H', 0) of thc group (G', 0); spccifi('ally, /I' i:-< j!;iV!'n by Il' = f(H).

Proof. Let us first dJ('('k t hal the indj(·at.ed correspondencc is onto. In other words, if (II', .,) is any :-
2-7

III

a subgroup of (a, Oo) and, sincc e' Ell',

l\Ioreover, t.hl' fund-ioll f hcing Itll OlltO IIlltppillg, I,he corollary to Thcorem 1-6 indicatcs thatf(j-l(II'») = Il'. Next, we vcrify that this corrcspondence is also one-to-one. To this end, supposc (Ill, Oo) and (H2' *) are both subgroups of (G, Oo) with ker (I) £;;; H., kcr (I) £;;; H 2 and such that f(1I 1 ) = f(1I 2 ). According to the preceding lemma, we then must. have

It follows that thc eorrespondence (H, *) completing the proof.

<->

(J(II), 0) is one-to-one, thereby

Tlw t.Il1'orl'lll llpplil!s, ill pllrt.il!ular, to I hp I·as(' ill whidl we st.art with It normal :;uhgrollp (//, *) of (a, *) alii I lake f \'0 be the lIatural mapping natu: G -> a/II of U onto a/ Il. Hinee ker (Ilntu) = Il, the conclusion is modified slightly. Corollary. Let (II, *) be a normal subgroup of the group (G, Oo). There is a onc-to-one eorrC'spondenee b('t\V('CIl those subgJ"Oups (K, *) of (G, Oo) Hu(~h that. II ~ K Ilnd thc set of all Hubgroups of the quotimlt group (a/II, ®).

Before proceeding, it should be remarked that the Corf('spondence Theorem remains valid if WI' replacc the teml ":subgroup" throughout by "normal Hubgroup." That is to say, there is also a one-to-one correspondence between those normal subgroups of (a, *) which contain kcr (f) and the set of all lIormal suhgroups of (a', 0). The addit.ional nrgunwnt needed to establish this fact is left for the reader to supply. Example 2-53. As nn application of these ideas, consider the following statement: if (a, *) iH a finite I!ydie p;roup of nrlll'r n, tl\()n (0, *) hn.s cxaCltly ()III! suhgroup of ordl'!· 1r/. for I'lldl positivI' c1iviHllr 1/1. of n !LIlli no oUlI'r propl'r suhgroups. Alt.hough t.\lI'rl' is nothing vt'ry HllrpriHing about LhiH assertion, our aim is to demonstrate it.s validity hy IIHing the corollary to the Corresponuenec Theorem. First observe that, sincc the groups (a, *) and (Zn, +n) are isomorphic, there is np losH in generalit.y in working with (Zn, +n). Furthermore, as we have poirill'd out 011 :several occasions, (Zn. -In) = (Z/(n), ®). By the corollary, we learn there is a one-to-one eorrespondelwe between those subgroups of the group (Z, I) whi("h eontain the spt (n) and IIII' subgroups of (Zn, -hi). Bul., the suhgroups of (Z, +) are jm;t the ('ydie subgroups «m), +), where m is It lIonnl'galive intpger. Comhining these rpsult,s, we arrive at the COIlc1u:sion then' is a one-to-one eorrcsponucnce bctween the subgroups of (Zn' +n)

112

2-7

GROUP THEORY

and those subgroups «m), +) of (Z, +) such that (m) 2 (n). clusion occurs if and only if m divides n.

This last in-

Tlw two ':olwluiling tlwOr'!IllH of thiH Kedion arc KOmewhat dl.'cper results than U811111 IUIII rl''1l1ire the full forI".' of ollr necllmlllutcd machinery; thl!y eOlllpriSl.' what un' oftl'n <:ulled the FirKt allli Second lliomorphilim Theorelllli for Groups and are of great importanee in the study of group structure. Theorem 2-50. If (II, *) is allY normal subgroup of the group (G, *) such that ker (I) ~ II, then (G/II, ®) ~ (G'/f(ll), ®/). Proof. As a prefatory remark, we might point out that the corollary to Theorem 2-30 implil's the pair (J(II), 0) is a normal subgroup of (G',o), so thnt it. is I'l'rt.llinly J1l'rmiHMihlc to form f he quotient group (0' /f(II) , ®'). Let us now define thl' function 7: (; --> 0' /f(J1) by

1=

nat/Ol) f, 0

where natf(ll,: 0 --> O'/f(lI) deRiglllLtes the naturnl mapping. diagram helps to visualize the situation:

The following

G'/I{H)

Observe that 1 nwr(·ly ILssigns to (·ach element a EO the coSet f(a) of(H) of 0' /f(II). Since both the functions f and nat/(II) are onto and operation-preserving, the composition of these gives us a homomorphic mapping from the group (G, *) onto the group (G'/f(ll), ®'). The main line of the argument is to show that ker 0) = H, for then the desired reRu!t. would be a simple eonl!l'quence of the Fundamental Theorem. Now till! ililmtity dlmll'nt of (a' /f(lI) , ®') iM jUHt the (:OKl.'t f(lI) = e' • f(II). This means the kernel of 1 ('onsists of those members of G which are mapped by ] onto f(lI); that is, ker

m=:a E G I](a) = f(lI)} fa E

=

G If(a) of(H)

= f(lI)}

:a EO I f(a) Ef(H)}

=

f-I (J(ll».

As we Ilrt· givI'n f.Imt, kl'f (f) ~ II, the lelllllll\ prm'eding Theorem 2-49 may b(' invoked t.o (,OIl1'lud(' II = f-I(J(ll». Hence, k('r (]) = H, which completes the proof. In applicat.ions of this theorem, we frmluently Htart with an arbitrary noml1l1 subgroup of (a', .) lind utilize inVl'rsc images rather than direct images.

2-7

1I:J

Corollary. If (II',.) is any normal suhgroup of the group (G', .), then (G//- 1 (1I'), ®) ~ (G'/H', ®'). I'mo/. By the ('llrollul'y to Tlwllrmll 2--:m, nUl pair (f-I(lI'), *) iK I~ Ilornml f:!lIhl(rIllip of (a, *). :'Itofl'llVl'r, kl'r (j) r;;/-I(II'), SIl 1.111' hYIlOUlC'siK of th(, UIC'()J'('III is ('()/IIJ1h,tdy sut isfil'd. This leads to till' iSOIllOl'phiKIIl

Since / is a mnpping onto

a', 1/' = /(j-I (1/'»,

and we arc done.

Our final theorem, a rather technical result, will be crucial to the proof of the ,Jor(lnll-Htildl'r Theorem. Theorem 2-51. If (//, *) lUlU (K, *) an! KuhgroliPH of 1.1)(' KrouP (a, *) with (K, *) llorllllLl,l.hen (II/II n K, ®) ~ (11* K/K, ®').

Proof. NeedlesH to Hay, it should be che(,ked Uu~t the quotient groupII appearing in the stat.emf'llt of the theorem nrc actually definc'd. We leave to t.he reuder the routine t.usk of verifying thltt (11 n K, *) ill U Ilormal Huugroup of (II, *), I.hnt.-(II * K, *) is 1\ group, lUui that (K, *) h~ llorlllal ill (1/ * K, *). Our proof is putt.erneu on t.hat of Theorem 2-flO. H(!I't~, the problem is to construct a homomorphism (I from the group (Il, *) ont.o t.he quotient group (/1* K/K, ®') for whieh ker «(I) = II n K, To n(,hil've this, cOllsili('r t.he function (I(h) = It * K, It E /I. Note, II = II * e ~ /I * K, so that a can be obtained by composing the inclusion map ill: /I .-. II * K with the natural mapping natK: H * K .-./1 * ~/K. In other words,

or, in diagrammatic language,

H oK/K

The foregoing factorization implies a is a hOlllomorphismund a(l/) = /I * K/ K. WI' next proceed to estahlish that. the kerlwl of a is precisely t.he sct II n K. FirHt., obHl'rw I hut. tI\(' cosl'!. K = e * K K(,\'V('S as till' id(,lltit.y ('1(,llwllt of I,h(! quotient gi'ouJI (1/ * K/K, ®'). This llIeans ker (a) = [It E III a(h) = Kl = Ut E /lIlt [It E /I lit E K} = II n K. The required

i~olllorJlhism

*K =

KJ

is /lOW cvili('nt from t.he Fundn/lwlltal Theorem.

114

2-7

GROUP THEORY

Example 2-54. As an illustration of this last result, let us return again to the group of integers (Z, +) anrl consider the cyclic subgroups «3), +) and «4), +). Both these Imbgroups are normal, since (Z, +) is a commutative group. Morc!Ovcr, it is fllirly obviolls tlmt (:J)

n

(4)

Th<,'orem 2-;;] then tells

liS

=

(a) -I- (4) = Z.

unu

(12)

that

«:J)/(]2), 0) ~ (Z/(4), 0'), where 0 and 0' ucsignate the resp<.'ctivc quoticnt group operations. TIm 1101 ntioll t"'IlIls to nhscnln,' t,IIP sintplil.ity (If ollr l'olldIlHion. For the r('Iull'I' will duull!.ll'sl'l I'(,I'nll !.Imt (Z/(1J, ®') il'l jill'll, til" grollp (If illl,egt.f/'I modulo 4. A dOl'l(!r c'xlunillnt,ioll of I,hn C'Ol'l(,'tl'l ullci OpC!rlltioll in the sYHtmn «:1)/(12), ®) n,'v('nls t.his to be not.hing more t.han the group ({O, 3, 6, 9}, +12). Whtlt we ac:tunlly ImvIJ is n disguiseu versioll of the isomorphism,

PROBLEMS 1. Let (G, *) he the group of symmetries of the square and (G', oLbe the Klein

four-grouJl (see Examples 2-24 and 2-47). homomorphism from (G, *) onto (G', 0):

I(RIKo) 1(11)

=

I(R36o)

=

The following mapping defines a

I(Roo)

e,

= I( V) = b,

1(1)1)

=

I(R27o)

=

= 1(1)2) = c.

a,

/

a) Est.ablish the iHomorphiRm (G/cent 0, 0) ~ (0',0). b) Write Ollt the indul:ClI mapping l: G/cent G --+ 0' which leads to this i.
/

G,

[Uint:

I,(a)

{·St·

""arly the same argument a.~ for Theorem 2-46; that is, for any element

E (/" !ldint'] fly

lU,(a»

=

h(a).1

2-7

115

3. Decitl('l' the Fador Theorem from the result of Problem 2. II. I!;roup (r., *), establish the followinl!; fads rl'l!;arciing the inner automorphisms uf U.

4. Given

II) \Vh"lIl'vc'" U:, *) is 1I11111·Ollllllutut.ivc', 1«(:) ~ (ill). h) If (/1, *) is I~ suhl!;rollp of (a, *), tJ1C'1I tJI<' plLir (u,,(//), *) is also

II. subgroup for every a EO. <") A slIhl!;rollp (//, *) is nurmal in (G, *) if anci only if the set II is mapped into itsl'lf hy ,'ul'h illlll'r autolllorphislll of G. tI) If till' Pi"lIIelit J' E a is stwit thut u .. (x) = x fur c'very (t E 0, tl\('11 the eyclil' subv;roup «x), *) is normal in (G, *).

5. Let. (G, *) bl' a v;rollp anI! t.he elements x, 11 E G. writtl'lI .r - !I, if (Lilli ollly if u .. (J') = !I for it< 1111 "'1l1ivlll"II1'1' n·lllt ill" ill fl.

1<01111' (/

E

We Ray x is conjugate t.o II, a. I'rovl' t.hnt, 1'lllljUl(lItillll

II. Oht.uill UII' illomorpldsm (S:I, 0) ....... (1(8 a ), .. ). 7. ASHlIme! is a homomorphism from t.he I!;rollP (0, *) into it.self having the propert,y of c,ommllt illl!; wit.h cwnry inner automorphism of 0; that. is, u" o! = ! every (t E a. If t.he set. /I is cldinecl by

0

u. for

11= {xE GI/(x) = x}, prove that a) the pair (II, *) is a norma.l subgroup of (G, *), b) the quotient group (G/II, ®) is commutative. [/lint: [G, G1 ~ II.]

8. Let (//, *) be a normal suhl!;roup of the I!;roU[l (G, *). Further, let (KI, *) and (K2, *) be 8ubl!;roups of (r., *).such that II ~ KI, II ~ K2. Prove that Kl ~ K2 if and only if nat II K I ~ na.tH Ii: 2. n. (:iVI'/\ (0, *) is lhe I!;rollP of symlllet.ries uf t,llll sqllare. Exhihit thl' ('orrespolltl"11"1' hd.wenll t,hoSI' Imhv;rollps (II, *) of (a, *) wit.h I'I'nt. a ~ II allol UIIl HulIgroups of (O/l'.Imt G, ®), In I'ruhlc'lIIl< 10 throu~h 14, I clc'lIotl''' a homomorphism from th(' group th" I!;I'Ollp (r.', 0).

(a, *) onto

10. Hhow that till' Flwtor Thl'on'lII implil's the flllH"tion I I,an hc' c'xPrl'ssl'd (nontrivilllly) as t.11I' I'ompositioll of IlII ollfo flllH"tion Ilnd n Olll'-to-ow' flllll'tioll.

II. If (II, *) is 11 Hllhjl;fOllP of (G, *) for whil'h II [/lint: I-I (/(/1) = 11* knr (/).1

= /-1(/(1/», vl'rify that. ker

12. For 11 proof of Thl'orcllI 2·-50 t.hat cloeH not depl'lId defille thl' fundion g: 0/11--. a'/IUI) hy takinl!; g(a

* II) =

I(a) 01(11).

al Show that, g i" WI'U-oh'filll'd, ClIw-tn-OIlI', opl'ration-pn'sprving, ancl onto G'//(II); 11('1\('1', (0/11, ®) ~ (O'//UI) , ®'). II) Estllhlish that g is thc' IIl1iqll(, mappinl!; whi"h nUlk,'s till' ,!ingram at till' right l'OllllHlltlltivl'.

G

notH

011

(f) ~ II.

t.lw Funclalllentlli Theorem,

-----.!..----.. G' =

fiG)

noff/H)

9 G/H-----=~----•• G'/f(HI

116

2-7

GROUP THEORY

13. If ker (J) k [G, G), prove that (G/[G, G), 0) ~ (G'/[G', G'), 0'). 14. Suppose (/I, *) and (K, *) are normallluhgroups of the group (G, *) with II k K. Prove that

a) (II, *) is a normal suhl{roup of (K, *), b) (K/Il, 0) is a normal subgroup of the quotient group (G/II,0), e) (G/II / K/II, 0') ~ (G/K, 0). [llint: Apply Theorem 2-50 with natll and (G/ 11,0) replacing f and (G',o).) 15. Givcn (//,.) anti (K, *) are subwoups of the group (G, *) with (K, *) normal. AsslIllling II n K = {eJ, derive the isolllorphism (II, *) ~ (1/ * K/K, 0). 16. In the Kymrnetric~ IotrOUp·(S4, 0), Int the Ket K conHist of the four permutations

(I 2 4), !l

123 4

( I 2 3 4), 2

I

4 3

(I 2 3 4) 3

4

1 2 '

(I 2 3 4). 432

a) Show that the pair (K, 0) is a normal subgroup of (84,0). b) Establi!!h that (S4/K,0) and (8:1, 0) are hlOmorphic groups. 1/ = {fE 841/(4) = 4} and use Problem 15.)

1

[I/int: Define

17. Let (G, *) and (G',.) be two groups with identity £'lements e and e', respectively. Recall that th£'ir direct product is the group (G X G', .), with the operation' definE"d on the ('artesian product G X G' in t.he natural way: (a, a') . (b, b') = (a

for all (a, a'), (b, II') E

aX

* b, a' • b')

G'. Prove the following statements:

a) If // = G X c' anti II' = eX G', 'then (II,·) an(1 (II',·) are both normal Huhlotroll(lH of (0 X 0', .). h) (1/,.) '"" (0, *) Ilnd (1/',.) ' " W', 0). e) Every 1'\I'IIII'nt of /I eOlllllllltl'K wit.h I'very e\ellllmt of II'. d) )':adl IIwmlJl"r of (/ X 0' ('un hi) ulliqul'ly CX(lrl'SKCd B.'< I.IH' product of an element of /I hy an (·\emnut. of 1/'. e) (G X 0'/11, 0) ~ (II',·) and (G X G'/II', 0) ~ (II, .). Derive these isomorphisms in two wayR: use the Fundamental 1hcorem, and then Theorem 2-51.

18. IlIwltrate the varioml parts of Probll'm 17 hy considering the direct product of till, IotroU(lH lib, I :\) 1111.1 (Z4,·1 4). In addition, Mhow that

(llint: (Z:l X Z4, .) ill a cyclic group of order 12.) 19. HU(I)!o,,!, (II, *) aud (/\, *) art' :mbgroU)!H of thn I{roup (G, *) such that

I) every c\1'1II1'nt of 1/ (·ommuteli wilh I'vl'ry e1mnent of K, 0 can he IIllilfll!>\Y expfI.·sliCd &'5 the product of an element of /I hy an {,\l"llIl'llt of /\.

2) nvery mClIlhcr of

Pro\"(' that (0, *1 -.., (II X K, .).

2-8

THE JORnAN-HOLnER THEOREM

117

20. Illustrate the result of Problem 19 wit.h (Zu, -h), the group of integers modulo 6, and the two subgroups ({O, 3}, +6) and ({O, 2, 4}, +6). 21. Prove that the Klein four-grouJl is isomorphic to (Zz X Z2, .). 22. Hhow that if 711 and n are relatively prime, then t.he direct product (Z .. X Z., .) is a cyclic group of order mn. 2-8 THE JORDAN-HOLDER THEOREM

AM the !.it.!!! ill(lil~atn!!, the main PIlIlIOH(~ of thi!! brief Hedion ill to estn.hlish the Higllifil'l1l1t. I1nt! hil>wrie .Jordan-Holder Theorem. For the sake of simplicity, we Hhall lilllit Olll'lielV!'H tll 1,111' I'II..'«~ of finite gJ'()\IJlH; 1.111' inl.l~J'(~Ht.ed fI~n.der ill referred to n lIIore gmwrtll lreutnwllt in [WJ. BI!('nuSt~ the theormn is ruthel' involved, it will be eOllvellicllt. to begin by introdueing t!Ollle special terminology. Definition 2-30. By n. dlain for a group (G, *) il> 1II('l\nt any finite sequence

of SUhsetH of

a,

(proper inclusions), descending from G to {e} with the property that all the pairs (Hi, *) are subgroups of (0, *). The integer n is called the length of the chain. What we are really interested in, and henceforth IIhall confine our attention to, are the so-called normal chains. These are chains in which each group (Hi, *) is a normal subgroup of its immediate predecessor (Hi_\, *). For grouP!! (a, *) with two or more clemellts there is always one normal chain, namely the trivial e1uLin a:J (e) ; however, this may v(~ry W('lI be the only such chain. Example 2-55. In UII' group (ZI2, ure normal dmillH:

+12)

of intl'gl'l'li modllio 12, the following

('hnilU~

Z12:::> (Ii) :::> (O},

z 12 :J (:J) :J (6) :::>

Z.2:::>(2):::>(4):::> (O},

(O},

ZI2:::> (2) :J (6) :J (O}.

All subgroups are automatieally normal, since (ZI2' +12) is a commutative group. Among other t.hingR, this pnrtil:ular exnmp!e indinnt.efll t.hat II. p;ivcn chain be lellgthened 01' refined by the ill&!rtioll of admissable lIubl>Ct.!!. In technical terms, a second chain

lIIay

0= Ko:::> Kl :::> ... :J K m- 1 :J K". = {e} is said to

h(~ It

l'efinement of the (·hnin

a=

Hu:::>ll. J'" :::>Hn- I :::>lln

=

[e}

provided there exists a oJl(_~to-one function] from :0, I, ... ,n} into :0,1, ... , m)

118

2-8

mwul' THEORY

I'IlIl'h that If i ~~ K/(i) for all i. Whut w(~ :lrI~ rl'Cjllirillg, ill elT(!!'t., iH thltt every If i ('oilH:idl' with OIl(' of the Kj. The lellgths of the foregoillg ehains mll!;t ('\I'urly l
Definition 2-31. In the gl'Oup (G, *), the descending sequence of sets

G = lin J III J' , . J 11 .. _ 1 J lin fornls

II (,(}1/I11II.~ilillll

I'hain for

1) (II;, *) is a l
(a,

(a,

=

{e}

*) provided

*),

:!) (IIi, *) is a lIornml suhgroup of (IIi_I, *), a) the indllsion Ili_1 ;2 K ;2 IIi, where (K, *) is a normal subgroup of (IIi_I. *), impli('1'1 either K = Ili_1 or K = IIi.

Bl'forl' prol'l'l~d ing t.o dl'VI'lop SOITlI! of till' properties of composition chainN, I('t us pausl' to I'oll:;id('r l<('v{'ral examples.

Example 2-56. In tIl!' jl;roup (Z24' +24), the norlllal ehnin Z24

J (:!) J (12) J (O}

is IIOt. It l'Olllpo8ition I'hain, sinee it mlly be further refined by inserting either of th(' s('h, (4) or «i). On the other hand, Z24

J (2) J (4) J (~) J (O}

and

Z24 J (:1) J «(i) J (12) J {OJ arl' hot It (·lIlllpllsit.ion 1'lminN for (Z24, I 24)' One way of verifying this is to dll'l'k t Iw ordefH of the l'Iuhgroups eOIl('crncd, For instalw(!, to inscrt, a tmbset b('t.w('t'n (2) and (4) 111('1'(' would have to exil'lt a suhgroup of (Z24' +24) of order n, Ij < n < I:!, s\II'h that n divides 12 and is it.self divisihle by (i; clearly no slIeh l
Example 2-57. For':l ~('('OI\(1 iIIust.ration, let (G, *) he the group of symmetries of till' sIl"an'. III fhil'l ('nl
J : I( I so. I(alll), fl, V} J : Hallllt Il} J .[ H:IIH1}.

It is worth lIutill/( that tIll' slIhjl;l'OllP (:/(:1110. II:, *) is no\. P;l'OllJl «(1, *), bllt only ill its illulH'diaLt' PI'('III'('(·I'ISiIl'.

11111'1111\1

ill till' pnrl'llt,

2-8

'rIlE JOnDAN-I10LD~;lt THEOHEM

lin

A normal ('IUlin for (G, *) which fails 1.0 Ill' a ('oIllJlosit.ioll ehain is

This ('huill admils allY olle of Ht'vt'ral I,t'finemellts; among others, tlwre is

Example 2-58. To see that not every group possesses u composition chain, merely ('onsider the additive group of integers (Z, +), We have previously observed that til(' normal subgroups of (Z, +) are the c:yclic :subgroups (n), +), II It nOlllll'gal iv(' intl'gPl', Rinf't' til(' i!lf'lUHiCIII (kn) <;;;; (n) holdH for nil k E Z+, th('n' always ('xisls It PI'OJl('I' suhgroup of any J!:ivell J!:roup. Aecordillgly, each dillin for (Z, I) Illay 1)(, rdilled ilult'finitt'ly, Condilion (3) of Definition 2-:n merits e\OS('f attcntion, Looscly speaking, it prohibits liS from sql\('ezing U normal suhl/:roup hl'tween (l/j_l, *) and (H j , *), Th<'l'c is a sl)('cial tC'rminoloJ!:y for this situat ion.

Definition 2-32. A norIllal suhgroup (1/, *) is cl1lled It maximal nurmal SUbyJ'OllJl of tl\(' group (G, .) if H ~ G and ti1l'rc exists no normal subgroup

(K, *) of (G, .) such that II eKe G, III terms of I.his IWW notion, a chain

is a composition chain for (G, *) if each subgroup (Hi, *) is a maximal normal subgroup of (lI i - h *), We :should point out that. Definition 2-:t~ doe8 not, imply (H, *) is maximal in the sense of being the 8ubgroup of (G, *) with largest order, A given group may well have many distinct maximal normal IlUbgroups of various orders. Indeed, ill the group (Z24' +24), the eydie subgrouJls (2), +24) and (a), -h4) Ilrc hoth maxirnal/lorllllli with orfi('rs 1~ ami x, rps(lpdiv('ly. The II('Xt. th('orelll dev('lops the eOIlIlI'('tioll ht'twl'ell It maximal nOl'mal subgrouJl and its Ils,'>o('iat.ed quotit'lIl. group, For this, r('('all that 11 group is said to h(' Hilllple if !.IH' ollly lIorlllal subgroups Ilr(' t he two trivial OI1(,S,

Theorem 2-52. A normal subgrouJl (H, *) of the group (G, *) is maximal if and only if the quotient (G/lI, ®) iH simpl('. Proof. This result follows immediately from the COI'I'(,Hpondence Theorem, We saw that to elU'h norlllal subgl'oup (K, *) of (G, *) with /l <;;;; K there (~orre­ s(lollds a lIorllllll subgroup of t.h(' quoti('nl group (G//I, ®) and I.his ('OI'I'('spondI'IH'(' is 011('-10-011(', 111'11('(', (//, *) is III:txilllal lIorl11al ill (a, *) if awl (lilly if Ih('f'(' al'(' ('xndly two 1I0\'llwi Huhgl'Oups in (0/11, ®), Ihp ('/llin' v;rnup Ilnd the idl'llt ily subJ,(l'Oup,

2-8

120

C('rtainly all till' quotient groUpH (Hi_.! 1/ i, ®) of a composition chain are simpl('. Conven,l'ly, any normal chain who!!C quotient groups are all simple groups is not suhj('d 10 furt·her rpfinemcnt and is therefore a composition chain. Not evpry j!;rouJl ('ontains a muxinml normal suhgroup, so we eannot always find a ('ol11position chain for a given group. However, in the case of finite groups, th(')'(' iH the following result. Theorem 2-53. Ev('ry finite p;roup (0, *) with more than one clement hUH

a cOlllpoHit ion chain.

Proof. If (a, *) is a simpl!' group, then till' tI'ivial chain O::J {e} iH a composit ion I"!tain. HUJlPos!' (a, *) iH not simple, so t.hat. there exists a normal suhgroup (II, *) 1111"1'(' may he s(~veral choil:l'H for (/I, *). If (/I, *) it! maximal in U;, *) nnd (:1': , *) is maximal in (II, *), t.hplI

is n composition ('hnin, and we are through. When (H, *) iSllot already maximal normal, a Illrger normal subgroup (K, *) with 1I eKe G exists. Now, if (H, *) is maximal in (K, *) and (K, *) is maximal in (G, *), a::J K::J H ::J {e] is the
1

::J lin = {e}

and (} = Ko::J KI ::J ... ::J K m_ 1 ::J Km =

are t('rmcd cquil'aicnt if th£'y II/we the same length (n permutation f E Sn Hueh t.Imt

{e},

=

m) and there exists

1\

i = 1,2, ... ,no In oth('r wOrill', two ('huills ILre equivl1l('nt if t1wir assoeiat!'d quot.ient groups are isolJlorphi(~ in Snlll!! order. We are now in a posit.ion, hnvinj!; I1ssemhled the n('C(!HHury machinery, to att ack the .J()fIlan-IIoldl'r Theon'lll. To simplify our prcsentation, part of the proof iH scparated I1H u preliminary lemma. Lemma. If (1/, *) alld (K, *) art' distilwt. maximal normal subgroups of the p;rollP (a, *), thl'\I

1) t.lU' pair (II (K, *),

n K,

*) iH II nmxinl:ll 1I()J"llIul ~lIh~r()lIp of bo\.h

2) (a/II, ®) '" (K/II

n K, ®) and (O/K, ®)

~

(II, *) ami

(ll/ll n K, ®).

2-8

THE JOIU):\:\I-1I0LUER THEOREM

121

J>1"OIIf. Since the suhgroups (11, *) and (K, *) arc both normal, the pair (II * K, *) is ilHl'lf a normal suhgroup of (G, *) [Problem 11, Section 2-5]. Plainly, we have the ill('lusion II ~ H * K ~ G. By hypothesis, the subgroup (II, *) is maximal norlllal in (G, *), which implies either H * K = H or H * K = G. Our eontelltion is that H * K = G. Were this not the case, the equality }[ * K = H would mean K ~ H (with K ~ II), contradicting the m:tximlllity of (K, *); therefore, the only possibility left us is that G = H * K. A stmightforwunl applieation of Theorem 2-51 now establishes the isomorphislll (G/K, 0) ~ (II/II n K, 0). But the quotient group (G/K, 0) is asilllpll' gmup (ThporPIIl 2-!i2),!lO the same must also he true of (H/H n K, 0). Fmlll I,his, w(~ condudp the group (/1 n K, *) is maximal normal in (H, *), again ut.ilizillg Tlwol'l'lJI 2--!i2. If the ldlns /I and K arc interc1umged, the symmetry of the hypothesis yields the r('/uuining parts of the lemma.

Theorem 2-54. (Jordan-Holder). In a finite group (G, *) with more than one element, any two composition chains arc equivalent. Proof. The proof wiII proceed by induction on the order of the group. If (G, *) is simple, the only pOllsible composition chain is the trivial chain G::) {e}, so the t1wol1'm is cert,ninly true. This estahlishes the result for groups of order 2, the I:!mullest admissible order. III general, let

G

= Ho::) HJ ::) ... ::) H .. _ 1 ::) Hn = {e}

(1)

Ko:> Kl :> ... ::) K m_ 1 ::) Km = {e}

(2)

and (J =

be any two composition (~hnins for the group (G, *). Assume further that the theorem holds for all groups having order less than that of (G, *). We distinguil:!h two easel:!: ('A~m I.

III

= K I. In this ('asc, if the

Sl't.

G is deleted from (1) and (2), the

rcsuItin/l: ehllins

anel

III :>K 2 :>··· ::)K m_ 1 ::)Km

= {e}

hoth r('pre~ent (~()/IIJ1()silioll ehains fOI' the I:!uh/l:roup (Il" *). As the theorem is as!';lIllll'd true for (// t, *), whose order is l(~ss than that of (G, *), these two chains are necessarily equivalent. But (G/llt, 0) = (G/K l , 0)j hence, the given I'Omposit.ion ehains (1) and (2) lIlust, also be equivnlent. (' AHE 2. II. ~ K I' Eilhpr II,

n KI

=

k!,

0"

hy Th('Orp/ll 2-53, t.here exists

a ('ompositioll chain

Il, n K, ::) 1-, :> ... ::) 1-r _, :> l"r

= {e}

122

2-S

mlOt1P TIII';()(lY

for the tmbgroup (Ill nK 1 , . ) . Now, a(~(~rding to the lemma just established, (H InK 10 .) is a maximal normal subgroup of both (H 10 .) and (K 10 .). It then follows that the two chains .

nJlli (4)

are actually composition chains for (G, .). ApJX'aling to the lemma once more, we see that and Hence, the eomposition quotient groups ohtained from (:l) are isomorphie in pail"!! to those ohlllilwd frolll Uw e1min (4); let Ul! ngrcll to ubbrcviate tlii" situation by writing (;) ~ (4). Next, consider the following two composition ehains for the group (G, .): and

Since these ehains have their first two terms in common, we can concludp from case 1 that they must be equivalent. In a like manner,

and

are equivalent dmins for (G, .). Combining our results, we observe that

(1)

~ (:l) ~

(4)

~

(2),

from whieh it follows that the original chains (1) and (2) are equivalent. Let us quickly review what has just been learned. First, the preceding theorem implies that the composition chains of a given finite group (G,.) must all be of the same length. !\[oreover, by means of the quotient groups of any sueh chain, we are able to assoeiatl' with (G, .) a finite sequence of simple groups. Up to isomorphism, these simple groups depend solely on (G,.) and are independent of thl' partil'ular eomposition ehain from which they were originally obtained. The important. point is that the composition quotient groups will to some dl'grce mirror the properties of the given group and provide a hint to its algebraic strueturc.

2-8

123

Example 2-59. To illustrate the ,Tordall-lIolder Theorem, eOlll:!ider the group (Zoo, -ho) of int('gerl:! modulo 60 Ilnd t.he two composition chains: Zuo J (:l) J «j) J (12) J {O}, 7,'1111-)

(:!) -) (Ii)-)

(an)

~

:n}.

After suitable rearrangement, thc il:!omorphic pail'l:! of composition quotient groups are (Zao/(a), ®) ~ «2)/(f», ®), «a)/(f», ®) ~ (Z6o/(2), ®), «(j)/(12), ®) ~ «30)/{0}, ®), «12)/ {O}, ®) ~ «6)/(30), ®). All th('s(' flHOt.icllt, groUpH nrc simpl!', heing nYI'lie' groups of prime order: the first pair of iHolllor"hil' groupl:! iH of ol'llel' a, the HI~eOlI(I pair of Imler 5, while the remaining groupl:! are all of order 2. Note that the product of all these orden'! is HO. We 1I0W take a brief look at a wide c\al:!s of groups which contains, among others, all eommutative groups.

Definition 2-34. A group (q, *) il:! solvable if it has a normal chain (possibly of length 1) G=HOJHIJ"'JHn_lJHn= {e},

in which every quotient group (Hi_dH i , ®) [i = 1,2, ... , n] is commutative. AI:! a matter of language, we shall call such a chain a solvable chain for (G, *). It should be clear at onec that all commutative groups (G, *) with more than one element arc Holvltble, sinee in this ('/tRe, the trivial chain 0 J {e} is It solvable ehain. A:; IllI ('xum"le of It lloncomJl1utative solvable group, one need only eOIlHidl'r the symllletri(' group Oil three' :;ymbols, (Sa, 0); here, a solvable chain is

Sa

~ {C ~

:), G: D' G~

~)} ~ {(~

: :)} .

Reccntly, W. Fcit and ,J. Thompson succeedcd in proving the long-standing eonjecture that all fillite groups of odd order arc solvable. Thus, if we are interestcd in nonsolvable groups, t1wy will be found among the noncommutative groups of evell order. SOllie bal:!ic properties of solvable groupl:! arc given in our next theorem.

Theorem 2-55. If (0, *) is It solvablc group, then every subgroup of (G, *) is I:!olvahlc and evcry homomorphic image of (0, *) is solvable.

124

2-8

catoUl' TlU:OItY

Proof. Let

G = Ho:::> HI:::>' .. :::> H n_ 1 :::> Hn

=

{e}

be a fixed solvable chain for (G, .). Given a subgroup (K, .) of (G, .), define Ki = K n Hi Ii = 0, 1, ... ,nJ. We intend to prove that the chain

K = Ko:::> K 1 :::> ••• :::> K n_ 1 :::> Kn = {e}

is a Rolvahl{' chain for th~ subgroup (K, .). First, observe that the pair (K i , is Illlorlllul ~lIbJ(I'OIlP of (K i _ h .); in fact., for each a E K;-h (f.

Ki

•

a-

I

= (a. K i

•

a-I)

n

K

IIi. a-I) n K

S;;; (a.

.)

~

IIi n K

=

Kj.

Also, K; = K

n IIi

= K

n l1 i_ 1 n IIi = K'_ 1 n Hi,

so Umt

According to Theorem 2-51, we have the isomorphism

But the quotient group (K i - 1 • Hi/Hi, 0) is commutative, being a subgroup of the eomlllutlltive group (l1i_dIli' 0). This implim! all the quotient groups (Ki_dK j , 0) are commut.ative, as required. As for tlH' :-;"(""1(1 pnrt. of thl~ t1wormn, let f he a homomorphism from thl! group (0, .) Ollto t.he group (G', 0). Setting Il~ = f(lI i ) [i = 0,1, ...• nJ. we obtain a chain for (G', 0):

G'

=

H'o:::> Ht:::>···:::> H~-l:::> H~

=

{e}.

By the corollary to Theorem 2-39, the subgroup (H~, .) is certainly normal in (HLI, .). What little diffi(:ulty there is arises in showing the corresponding quotient group (HLdH~, 0') to be commutative. For this, we define mappings k Hi_dHi --+ lI';-dm hy taking fiCa • Hi)

=

f(a) • H~,

(i

a E Hi-l

=

It is easily seen that f; is well-defined; indeed, if

1,2, ... , n).

a. Hi = b· Hi, then

a-I. b E Hi, hence

f(a)-' • feb)

=

f(a-' • b) E f(H i)

=

H~,

which, in t.um, implies f;(a· IIi)

=

f(a) • Ht

=

feb)

0

H;

=

f;(b. Hi).

2-8

TilE JOIU)AN-HOI.ln;H TIlEOREM

125

Next, note that each/; is itself a homomorphism; for if a, b E Hi-I, f;«a*H i) 0 (b*H i» =j;(a*b*Hi) =f(a*b)oH~ = f(a) feb) m = (I(a) • m) 0' (f(b) • Hn. 0

0

Finally, because fell;) = 1I~, the function /; maps onto the set H~_dH~. From th('Hll faetH, W(l ILre ahle to eondu
nnd, Hitu!I' (II j . ./11 j, 0) iH n 1·()llllllut.nt.iv(' gnlllJl, HlI niHIl iH t.l1Il (Iuot.il'llt. group (II~-I/lli, 0').

(a, *) ill Hotvabte and (H, *) is a proper normal subgroup of (G, *), then the quotient. group (G/II, 0) is solvable.

Corollary. If

In the way of a convcl'flC to this throrcm, we prove

Theorem 2-56. )Alt. (II, *) hll a lion trivial normal suhgrouJl of (a, *). If hoth (II, *) atlll (a/II, 0) arll solvabl
G/H

=

Kl,:::> K1:::>"':::> K~_I:::> K~

=

{e

* H}

be a solvable chain for the quotient group (G/H, 0). Using the Correspondence Theorem, we obtain a ,sequence of sets G = Ko:::> K 1 :::>···:::> K"_I:::> K,.

=

H

dL'Scllnding from (J to II with the prop(lrty that (K;, *) is a normal subgroup of (K i - 1 , *); in fact, Ki = nati?(Ki). Furthermore, by Theorem 2-50, (Ki-tlK;, 0)

~ (K~_dK~,

0'),

so that the quotient group (K i _tlKi ,0) is commutative. solva},)(!, it haH a solvabl(! ehain running from H to {e}, say

Since (H, *) is

11 = H o :::>H 1 :::>··· :::>H"'_1 :::>Hm = {e}.

By stringing t.heHll Hl'qlll!necll of setl! togcther, we can eonstruct a solvable ehain for (G, *): G:::>K 1 :::>·,· :::>K"_1 :::>H:::>Hl:::>'" :::>Hm_ 1 :::> {e}, Therefore, (G, *) is a solvable group. In conjunction, Theorems 2-51> and 2-.')6 tell us that if (11, *) is a nontrivial normal Hubgroup of (G, *), then (G, *) is a solvable group if and only if both (II, *) and (a/II, 0) art' HlIlvahlll gt'OllpH. The d"dHion r('garding Holvability is oft.en facilitated by this criterion.

126

2-8

WIOI'I' I'll EOH\"

TIl(' following I Iwort'llI giv('s fUl'tiwl' illsight. illlo t.he ('hains for 11 fillile solvahle group.

lIatuJ'(~

of t.hc c'omposit.ion

Theorem 2-57. Ld (0, *) he a finite solvable group. Then the quotiellt. groups of nny (·omposition c·hnin for (a, *) nrc ('yc·lie groups of p"ime or<1c'l'. Proof. \Vc pro('cl'cI by indue·tion on the order of (G, *). If order G is a prime, there is nothing to provc), sinee G ::::> :e~ is the only c~omposit.ion ('hain. (This remark takes carC' of I he basis for the incluetion when order (] = 2,) In the contrary <'a~C, (G, *) has a nontrivial, normal ~ubgroup (II, *), [Why?] By thl' induct.ion as!-mmption, (II, *) ancl (0/11, 0), being solvable, have composition chains whose quot.iPllt groups are of prime orciN: say, the dmins

If=lIo::::>lI'::::>'''::::>lIn_I::::>lIn= {e} ancI

0/11 = K;)::::> K; ::::> ..• ::::> K:"_l ::::> K:" Taking Ki

=

=

{e

* ll}.

nat H' (Ki) [i = 0, 1, ... , m], it follows from Theorem 2-49 that

G = Ko::::> KI ::::>.,'::::> K"'_l::::> Km = H, where (K j , *) is a normal subgroup of (K i _ 1 , *), As before, Theorem 2-[>0 implies (Ki_dKi, 0) ~ (K~_dK~, 0'), when!'e (K; __ I//(;, 0) iH (c'yC'lic') of prillI(' order, Hooking theRe sequnllceH togethl'r, we ohl aill (aftc~r r(,IIIClvillg the H('c'cII\(1 oeeurrellce of II), a composil,ioll c,hain for (0, *), all of whosc~ quotient groUpH are c'yelie groUpH of prime order. By the ,Tordl\lI-l WIder Theor('m I he same is true of every eomposition chain. ilc·fore eOlH'luclilll!; Uw \lI'(,:';1'1I1. H('C'tiOll, \ct Ull :.;t.nt.1' n fcw I'('!mlts, omitt.illg t1wir proofs. WI' hav{' :.;eclI that ev{'ry pC'rmutation in Sn (n ~ 2) can be written III' a product of tmllspositiolls. While this expl'<'ssion is not unique, the number of transpositions o(·(·urrin/!: in the various representations for a given permutation is always (,VC'II 01' IIlwllYS odd. AC'l'ordingly, we define an Cl'en (odd) permutation ns olle whic'h ('lUI hI' (>xprel"llcclas thc product of an even (odd) number of transposil ions. If A,. c\pnoll's t hI' I' {e}, We are thus led 10 ('onduc\e that for 11 ~ Ii (A,,, a) is a solvllhle group if Illld only if (An' 0) is eommutative. f)ince (1,2,3)

0

(1,2, 4)

~

(1,2,4)

0

(1, 2,3),

2-8

THE JOIWAN-HOLUJo;R THEOREM

127

the alterrmtilljl; grouJl (An, 0) fails to be commutative for n ~ 4 and hence is illKolvable for n ~ 5. The Kignificallce of thill obllervation lies in the fact that, by Theorem 2-!l5, the Kymmctrin group (8", 0) cannot be solvable when n ~ 5. Tho foregoing remarks may be stated as a theorem. Theorem 2-58. If n

~

5, then (8",0) and (An' 0) are not solvable groups.

PROBLEMS

1. Check that the following chains represent composition chains for the indicated group. a) For (Zafi, +:m), the group of intcgerH modulo 36: Zao:) (a) J (9) J (18) J (o}.

b) For (G, *), the group of symmetries of the square:

c) For

«a), *), a cyclie group of order 30:

d) For (83,0), the symmetric group on 3 symbols:

2. Find a composition chain for the symmetric group (84,0). [Hint: Consider the Imt of pcrmut,ations on p. 80.) 3. Show that no infinite cyclic group possesses a composition chain. 4. Prove that if (G, *) is a finite cyclic group of order 2", then there is exactly one ('om position ('hain for (G, *). 5. D!'rive the result: if the quotient group (G/ll, ®) is of prime order, then (H,.) is a maximal normal subgroup of (G, *). [Hint: Use the Correspondence Theorem.) 6. Prove that the <:Y('!ic :mbgroup «n), +) is a maximal normal subgroup of (Z, +) if and (lnly if n hi a prime number. 7. Establish that the followinl!; two composition ehains for (Zu, +24) are equivalent: Z24

J (3) J (6) J ~12) J {O},

Z2.' :) (2) :) (4) J (12) J {O}. 8. Find all COmI)()Rition chains for (a) the group (Zao, +:10) of integers modulo 30, (h) tIll' group of llymml'tril'" of the llquare, and verify t.he Jordan-Hold!'r Theorem in both (:ases. [/lint: There are four composition chains for each group.)

128

2-9

GROUP Tln:OIlY

9. Apply the .Jordan-Holder Theorem to the group of integerR modulo n to prove that a positive integer n may be factored uniquely (apart from the order of the factors) into. positive primes. 10. Ll't (n, .) be t.he dirl'd prodlll"t nf three nont,rivial groups (nk , *k) Ik = 1,2,3); that ill to Hlty, G = GI X G z X (fa wit.h t1w biliary oIK,mt.ioll . (Idillcd by (a,

h, c)· (a', b', e')

= (a *1 a',

b *2 h', C *3 e').

a) Verify that

G =>

f, X G2 X Ga => el X e2 X Ga => el X e2 X ea

if! a normltl ('hain for (0, .). h) If thc oril!;illal II.rollJlK «(h, *k) arc Kimple, prove that the forcl!;oing chain mUllt be a composition chain for (G, '). c) State and prove an analogous result for n groups. II. Suppose (II, *) is a proper normal slIhgroup of the group (G, *) and

is a composition chain for (G, *). Prove that G/ II

=> K Ii II:; ... =>

KrlH

=>

{II}

reprl'sen\.s n l"olllposition rhnin for the quotient group (G/II, ®). 12. Show that the symmetric grouJl (84,0) is solvable. 13. Prove that if (G, *) and (G',o) are solvable groups, then their direct product (G X G',') is also a solvable II;roll)l. [llint: Problem 17, Hedion 2-7 and Theorem 2-56.)

14. Let (II, *) and (K, *) be solvahle suhgroups of the group (G, *), with (II, *) normal in (0, *). Verify that· the subgroup (11* K, *) ill alHo Holvahlc. 15. Prove that a Himple grollp with more than one clement j,,; solvable if and only if it is of prime order. 16. Establish the following facts concerning the alternating group (.1 .. ,0), n ~ 3: a) (..tn, 0) is generated by the set of aU3-cycles. b) Order An = n!/2. c) (.4 n ,0) is the commutator subgroup of the symmetric group (8n , 0). 2-9 SYLOW THEOREMS

As noted earlier, if m divides n, we eannot he certain that a group of order n will possess at leuHt one suhgroup of order m. To be sure, under special eircumstanees (for example, given a finite eyc:lie group) it is true that such subgroups always do exist. In general, the problem of finding subgroups of a preserihed order in lin arbitrary finite group is one of eonsiderable difficulty and constitutl's the subjeet. matter of this, our final section on group theory. We begin with a quick survey of a numher of results, some previously encountered in the exereises, whieh will playa prominant role in the sequel.

2-!I

120

If (H, *) is a subgroup of the finite group (G, *), the illdex of H in G was to be the number of distinet left cosets of H (ill light of Problem 3(e), Section 2-1), the numher of right eoscts of H in G is the same as the number of Icft C'Oscls, so I hc'r!' is no 1I('('d to dist inJl;uish b!'tw('cn It right IUlclleft index). To silllplify t.1I" Wl'il illl!; lall'l' 011, WI' sllllll IlI'rI'llfll'l' n~pl'l'l'Il'lIl. til" illdex ur II in a hy [a: HI. Sin!!!! the ord(!r of the group (G, *) may be interpreted as the index of the trivial subgroup (re~, *), one might, ill keeping with the foregoing notation, write order 0 = [G: e). We shall not employ this convention, however, hut instead conform to the praetiee of denoting the order of (G, *) by the symbol 0(0). For any I'IlIhJ!:l'OuJI (IT, *) of tIll' /l;roU(I (0, *) IlllclllllY ('IC'lIl1'lIl, a E a, consider the followilll!; I'IlIhsl'l. of (,: d~fined

a

* II * a-I

=

fa

* It * a-I I hEll}.

If x alld 11 arC' memh/'l'H of a * II * a-I, there exist. * hi * a-I,!I -.-. a * h 2 • a-I. Thlm,

x = a

"It

h2 E H for which,

whprc', sin!!(' (IT, .) is a subgroup, the p ro a * H * a-I by taking J(h) = a * h * a-I; then J is all i::;olllorphism. Now, if (K, *) is any other subgroup of (G, *), we may cert.ainly c~onsider those element::; of K under which (IT, .) is invariant; in ot.IH'r words, the spt

N K(I/) = {k E K

I k * H * k- I =

l/}.

The rt'adpr Hhould provp t.he theorem below. Theorem 2-60. If (II, *) and (K, *) al'e two ::;Ubgl'Oll(lS or t.he group (G, *), then the I}air (N K(l!), *) is also a subgroup, known as the normalizer of H in K.

1:30

onoup

2-!1

THEOHY

When K = G, one customarily writc13 N (H) for N a(H) and refers to th(· suhgroup (N(H), *) simply as the normalizer of H. Although the subgroup (H, *) i:-< not necessarily normal in (a, *), a simple argument shows it to 1)(' normal in it s normalizer; in fact, (N (H), *) is the largest subgroup of (G, ,.) in whieh (1/, *) is nomml. Wit.h t.1t is not.ation ill rn illd, W(~ can now "(lOU!lt" til(! eOlljugate subgroups (a" 1/ * a-I, *) of (II, *). Theorem 2-61. The number of distinct conjugates of a subgroup (H, *) of (G, *) induced by the elements of a subgroup (K, *) is equal to [K: N K(H) J. the index of N K(ll) in K.

"'2

E K, WP haV<' kl * 1/ * k,1 -= * /I * k;-I if and only if /I (I> I I .. k~) * 11 * (k i I * I.''.!) -I, whidl lIol
l'I'ol/!, For "'1, 1>2

Corollary. "Pl. (II, *) and (K, *) he t.wo subgroups of t.he group (G, *), If (II, ,.) is illvariallt ullder n e1ellJ('nts of K, then (1/, *) has o(K)jn conjugat(~ suhgroups by elements of K.

PI'oof. TIIP hypot.hesis asserts o(N K(lI») = n. But, by Lagrange's Theorem tI(K) = [K: N K(H)] o(N K(lI»),

from whi('h t.I11' ('orollnry foHoWR. I.('t us I'ont inuI' our

dis('w~sion

hy introducing

It

llew

elMS

of groups. /

Definition 2-35. A group (a, *) is said to hI! 1\ p-group if the order of ('Iwh 1'1('IlII'IIt. of (l is Jl powl'r of It fixed prinw p. Example 2-60. Any group of order pn (11 It prime) is It 11-grouP, since till' (mh'r of I'lli'll 1'11'111('111, IJIlISt. divide t.he or
2-9

SYLOW THEOREMS

131

identit.y has order 7); this take::! care of the basis for the in
0(C)

=

[G: (a)]o«a»

=

o(G/(a»n

and till' fnd. 1/ tn, it, follows tlmt. o( a/(al) is divisihle hy fl. lTsinJl; our induction hypoUH'sis, U/(a) JIlUHt the equation

t11('1'('fol'(~

1/'

* (a)

=

eOlltain l'!onw dl'nwllt. Ii

(b * (a»P

* (Il)

IIf llI'd,,!' p. But

= c * (a)

implies bP E (a), 80 t.hat (bn)P = (bP)n = c. Were I)n = Il, then (b * (a»n = e * (a) and we would eOlwllld" 1/ I 1t, It I'ontmdi(~t.ion. Thill'! the dement. bn has order p.

Corollary. Let (G, *) be a finite eOJlllllutative group and 0(G). Then (G, *) hItS IL subgroup of order p.

]I It

prime dividing

Our next ohjl'et.ive iH to ('siabliHh tlJ(' preceding lelllllla without. the hypothesis of commutativity. We puu::!e, hOW(lVer, to develop tllll lIe(lCSKary mllthemt~tical tools.

Definition 2-36. The centralizer oj an element a of the group (a, *), denoted hy C(a), is t.Iw s('t. of nil ,,1!·IIWII!.t1 in (/ whi/·h (·Ollllllu1.n wit.h a: C(a) =-

Ix E a I .I: * a

= a

* .r: .

Note, in"idcntally, that. C(a) = (J if and only if a E "pnt. a. For allY eydie subgroup «a), *), we alwaYH have C(a) ~ N«a». The proof of the result below is routine and is left as un exereise.

Theorem 2-62. For any e1!'llwnt. a E G, the pair (C(a), *) cOllstitutes suhj./;l'OlIp of (a, *). .

It

Bdow (·IIt.l'l'inj./; illto a fUl'tlll'l' dis('lIssioll of till' (·(·nt.l'Illiz(·I·, lUI additional eOlwept. IlII1Ht. be intl·otiul'l'd.

Definition 2-37. L(,t, (a, *) hI! a j./;1·01l1> alld u., II E (r'. The dl'lIl1'ut. b il'! Haid t.o be It ('{Inju(!a/I! of a (ill (l) if j.lwre exi~ts ~()IIW £ E (J for which b == x * a * X-I. The reader shoukl now prove th(· following unalog of Thporelll 2-(il which clarities the rel:ttiollship hetween the last two delillitiollH.

2-9

GHOUP TJn:OIty

Theorem 2-63. If a E G, the number of conjugate clements of a is equal to [G:C(a)], the index of the centralizer of a in G.

Given a, bEG, we may define an equivalence relation,.., in G by requiring that a ,.., b if and only if b is a conjugate of a. The relation,.., induces a partition of the Ht't G illto equivalenee dMSCR, IlRually referred to as the conjugacy cla88e.~ of (a, *). EIC'llll'lItH in thl! Hltm/! I~onjllgu.ey dlt,HM ILrH (:ull(~d clmjugates of 01111 nllot.lwr. By 1.111; prC'I'I~dill/l: 1.lwon~m, t.Iwl'e an! [a: C(a)] ('Iellwlltl! in the COIIjUII;IWY ('IUHH ('0111 uinillg It. Certaill of III!'HI! dlLHHeH will (:ollsiHt of eXlLdly OIW mmllber; for example, if a E ('I'llt. 0, tI"'1I ,r ... a ... X-I = a for every x E 0, HO that the eOlljugaey daSH ('(H1t.aillillg a r..dll(' ..H to !a]. It. Hilould he d(!lLr, Oil the other halld, that a E (:ellt (J whell tltl' ('olljugaey claHs of a is {a]. From this, we infer that a group is commut.ative if and only if each of its conjugaey classes is a singleton. Example 2-62. Tlwre are t.hree eonjugaey dasscH of (8 a,o), the symmetrie group on a Hymbols. We leave to the reader the tll.l!k of verifying that these classcl:! ar(' given by

As pach elonwnt of G lies ill one and only one ('onjugaey class, we may determine the lIumber of (·lements in G by eounting t.hem class by class. To do so, first observe that. there arc exactly o(cent G) dasses which contain one element. If we add to O(I:OIlt. G) the number of clements in the nonsingleton conjugacy eiasl:!es, the c1emcnts of G will have all been eoullted. This leads to the important (conjugacy) clas8 equation:

neG)

=

o(cent G)

+ ~ [G: C(a»),

where the sUllllllat.ion rum; over a set of represcntlLtives for the distiuet COI1jUJ!;acy claHH('H having more t.han one memher; that is to Ray, we sum over a sct of noncolljugate elements not in eent G. In the group of 8ymnU'tries of the 8qunre, for inl:!tnnce, the conjugacy classes are {H, V}, so I hal III(' ('Im;:-I 0(0)

I'qll:\ I ion

would rend

= o(cent G)

+ [G: C(R 90») + [0: C(D 1)] + [0: C(H)].

2-!) Using the dass equation of a group, we are able to ('x tend the last lemma on commutative groups to a more general setting.

Theorem 2-64. (Cauchy). If (G, *) is a finite group whose order is divisible hy a prim£' 71, Own 0 ('onl.nills an element of order p. JlrtltlJ. I1t>!C' illdlldioll 011 IIII' III'dl'r of (U, .), If tI(U) :.!, IIII'll (G,.) is It "Ollllllllinliv,' ).(I'III1P nltd II", 11"'01'('111 I'..dlll'('t>! t.o IhC' 1"III11I1L, HIIJlJlIII«!, illdlldivC'ly, 1.11111. till' ,111'01'1'111 holds 11'111' rill' /1.11 ~I'OIIJlt>! of III'dl'l' 11't>!t>! Umn 0(0). Now ,'ollsid"r t.he dnss "'Illation 1'01' (U, .),

where the summatioll, as noted previously, is made hy ehoot>!ing one represent!\,tive from C'll,eh ('onjuJI;!ley ('In.'!s wit.h morn t.han onn df'm£'llt,. If (J =- ('nnl. a, t.11I! (a, .) is ('Ollllllut.al.iv(! and we IlPl'd only apJl(!:d t.o the lemma to cOlllplete t.h(' proof; in the cont.rary ('asp, t.hem is some element a E 0 with a fl ('('111. O. For such an a, [a; e'(a)) > 1, so that. by Lagrange's TIH!orem, 0(0) == rO;C(a»)o(O(a» > 1I(e'(n». Hilll'C' t.Ite t.1,,'or,'1lI it>! IlSSllIIlI't! t.o hold for t.hl! HuhJ,!;nlllp (('(It), *), if Jllo(e(a», then e'(a) has all element of order p, and wc are donc. On tltp ot.her hand, if 1d o(C(a» for evpry a fl eentdl, 11 must divide [a; C(a»); for JI is prime and divides 0(0) = [G; C(a)) o(C(a). But then, in the dass equation, 11 divides each term of t.he summation alld also divides o(G), so that p I o(cent (J). As (cent G, *) is n comlllutat.ive subgroup of (G, *), it follows (again, from the II!mom) t.hat. ,~ellt. (J ('ollt.aiIlH :lII 1'1('IIII'II!. of ordl'r fl.

Corollary. (0, *) is:t finit.1' p-J,!;roup if

:tll(i

only if o( a) = 11k, for some k

> o.

Proof. Rupposc (U, *) is a /I-group, but. q I o(G) for some prime q ¢ p. Then by Cauchy's Theorl'lll, 0 has lUi element of order q, I!ont.radil!tin)!; the fact that (0, *) is a {I-group. Thus, JI is the only prime divisor of o«(J), which implies o(G) = 11k (k > 0). Conversdy, if o(a) = pk, then from Lagrange's Theorem each clement of G has a power of 11 as its order.

As all appli'·at.ioll of C:ulI'hy's Tll<'ol'l'l1I, it. lIlight. Ill! of intl'rl'st. t.o mention

t.he r('sult. hl'lo\\'. Theorem 2-65. If (:ellt

a

¢

(a,

*)

is It finit.!! p-grollJl wit.h

11101'1'

Ihall

OIW

elcl1I('nt., t.hen

f/·:.

Proof. As a start in)!; point, consider the class equat.ioll for (G, *), o( (1) =, o«'('nt G) t

~

rG; C(a)],

the SIIIIlIlWt iOIl Iwilll!; ov!'r ('('1'1:1 ill 1'1"11I(,1I1s a for \\'hi('h [0; C(a) I ¢ 1. For a fll'('II(. 0. (C(a), *) is a nOlllrivial subgroup of (U, *); ill fnd, (C(a),.) is itself a JI-gl'OUp (any subgroup of It 1I-group is It 11-grollp !). By the preceding

IOUOI'I' Tln:OUy

2-!J

('orolhll'Y, it folio",:; that o(C(a»

= pk for sollie k

>

O. Theil

[G:C(a») = o(U)/o(C(a» llIu:;t 1)(' divi:.:ihl(~ hy p. ~ow, /I divide:; clwh term of the summation, as well a:; divicll':; o«(l), :;0 that. JlI n(c·cnt. G). Thi:; implies o(ecnt G) > 1. WI'

II0\\'

:;hifl our I'IlIJ1ha:;i:; frolll nrhit l'ary JI-lI;ronps t.o Sylow p-subgroups.

Definition 2-38. LC'I (G,.) be n fi II if,(' group IUld p It prime. A subgroup (1', .) of (0, .) is said to be a Sylow p-subymup if (P, .) is a p-group and is not properly ('()J\jaill!'d ill nny olh!'r p-subgroup of (G, .) for the same prime p. Example 2-63. Thp :;Ylllllwl.ric· group (.'01:1,.) hILS thrpc (eolljuglltc) Sylow ~-snhgl'llup:;,

Thu:;

It

slll·pili(·ally, t.he Mnhgrnup:.: who:.:c! sf't.s of dmmmts nrc!

Aylo", /I-:;uh!!:l'Oup of n giv('n group Iwed not be unique.

Theorem 2-66. For !'lll'h prim!' p,

th(~

finite group (0,.) has a Sylow

11-suhgroup. Proof. If o«(J) = I or ]I t 0(0), (re~,.) i:;, in a t.riviul I:!CIll~e, the required Sylow p-:;uhgroup. 011 til!' 01 her halld, wlumever 11 I n( G), Cmwhy's Thcorcl)l gllarlllli!'!':': til<' ('XiKh'IH'(' of lit. 11'1I"'\. OIW lIuhgroup (11, *) of (Jrt\('r p. If (/I, *) i:; 1I0t aln·ady a Sylow /I-Kullgl'OUP, Ihl'lI sillcl' a iH fillit(!, we arc led (aft(!r a lillit!' 1I1II1I1l!'r of ill:;p!'ct i()IIH) to a Hylow l'-Muhgrollp containillg II.

HiJl('c HII' nrllt'r of all ('lelll!'lIt. iH prcservNI under conjugation, it is clear that :lIIy ('onjllgat!' of a p-group i:; again a II-grouP. Even more is true. The conjugatl'x of a Hylow II-subgroup (P,.) arc themselves Sylow p-subgroups (aR an exC'r('isc', t 1)(' rmd!'l· Hhollld v!'rify I his fRet); of (~oursc, HOm!! or all of theRe (·o"jugat.c :;lIhgl'OUPS Illay b(' idclltic'al with (P, .). III th(' ('ollling tlwor!'llls, our goal iH to show that. if (P, .) is II. fixed Sylow p-subj!;l'OuJl of th!' fillil!' group (G, *), Ih('1\ allY ot.her Sylow p-RlIhgrollP of (G, *) Illust ill fad hI' a (·()"jugal!' of (/', *). We will also wltnt to determinl', so far ax Jl0i<sihl<', Ih!' 1ll1l11lll'r of dist.inC'l Rylow p-Ruhgrollps of (G, .). If, for ('xalllpl(" tlwrl' is a ulliqlll' Sylow /I-subgroup ('orr!'xponding to Il parti(mlar primp fI, tllI'lI t hi:; Kuhgroup IlIUl,olt 1)(' norlllal, Hill('(' it is (,Olljllg!lt(~ to itsl'lL

2-9

I'IYLOW TIIF.0I0;M8

1:15

Utilizing the Hylow Tlworl'ms, we c!tn obt!tin some rat.her general results conIIII' ('xil'<tplH'p of I'
('('rnill~

Lemma. Let (II, *) h(' a normall'ubgroup of (G, *). If (H, *) and (GIH, ®) nrc bot h p-groupH, 1.111'11 (a, *) it.self iH It l'-group.

Proof. For I'llph a ill a, the ('Ol'
I'
Lemma. Let. (P, *) he

e\enwllt. WhOHI' (lI'Ill'l' is

Hylow p-Imbgroup of (a, *) and a E (J he Imy power of 11. If a * P * a-I = P, thl'lI a E P.

!L

It

Proof. Since the condition a * P * a-I = l' means a E N(P), the problem is one of showilll!; there ('nn be no clement in N(P) - P of p-power order. Suppose sudl an elellll'nt. a Hl'1ually did exist. Now, (1', *) is normal in its normalizer, so we may ('onsider the quotient group (N (P) I P, ®) and the coset a * P. The order of a ('oset as an elenient in the quotient. group divides the order of any of its represent.atives, whence o(a * P) is a power of p. This implies the eyclic subgroup * P), ®) of (N(P)IP, ®) hM p-power order and eonRcqu!'nt.ly is a II-group (eoroIlnry to Cmwhy's Theorem). By the CorreRpondcnee Theorem, (N(J'), *) thus has a subgroup (K, *) with K;2 P and KIP = (a * P). As a G!: K properly eont.u.ins P. Furthermore, from the prcviolls II'mnm, (K, *) Illust be a p-gI"OUp, sillce hoth (P, *) and * P), ®) are sUl'h. But this dpllrly eOlltradil'ts the ftwt (P, *) is a maximal p-tmbgroup. AeC'Cmlillp;ly, t1l1'r(' ('1111 1)(' 110 1'I1'I1II'nl. of N(I') -- I' WhOHI' ordl'r iK IL \lower of fl.

«a

I"

«a

WI' an' now I'(·ady to statp the fil'sf, of I.he

so-(:ItIl(~J

Hylow TheOl'cIII S.

LeI (a, *) lw a rillit.c group and p a primc. Theil thc lIullIlJI'r of dislillet Sylow p-subgroups of (G, *) is eOllgrucnt t.o 1 modulo II and i:; a divi:;or of o(U).

Theorem 2-67. (8yI0l{').

1'1'110/. Let. (P, *) h(' a fix!'d Sylow JI-sllb~roIlP of (a, *). WI' havl' already obsl'rved t hat allY ('onjugate subgroup (a * I' * a-I, *) of (I', *) is also a Sylow 71-suhgl'OlIp. Thus, It natural start.illg point is to ('011111. the lIumber of dist.inct (·olljul!:at.cs of (I', *). If (P, *) il'< a 1I01'111al suh~rouJl (that. is, sdf-f'olljugnte), t.his numhl'r i" I. Othel'Wi,,(', *) has It ('olljugatl' subgroup (PI, *), with PI ~ P. Hill('(' allY ('olljllgat!' of (1'1, *) is, at. thc SIllIU' tinll', a ('olljugate of (P, *), we lirst. ('ollsidl'l' (·onjugat.!'s of (1'1, *) illlhll'ed hy dl'nwllt.s fl'lllll P. Aeeording

(I"

I :~{j

(;I{()l'i'

Tln;o!!y

2-9

till! last, 1('111111:1, Ih" 1'1"lIlent.H of I' 1I1ulel' whidl (1'1'.) prpeisl'iy t.hos(' of 1'1 n I':

10

:11

r 1'1

/'1 •

(1 •

(L

I

I' I,I

IS

illvariant an'

1'1 n I',

Hilll'(' (/'1 n /',.) is II p-sIlIoJ(l'lll/p of (f:, .l, it. follows t.hat. 11(1'1 n I') is a powl'r of p, ~ow, from Theof'(,111 :!-lil, the number of diHtinct eonjugn\,cs of (/' 1, *) by d(!llIl'lItlol of P is (!quul to

ThiH "IIII\\';; Ihali/': N1,(I'll] nll/st. III' >«11111' (low('r of p, "Ily plr.", Furtill'rlllOre, Itl > 0, fof' if Itl = 0, Ih('n 0(1') = 0(1'1 n I'), HO l' = PI n 1', or rather l' ~ 1'\. Bllt I hI' fae! that (1', .) iH a maximal p-subgroup would imply l' = PI, whieh iH impol'l'ihl('. l'\ote also that (1', .) will not be found among the conjlll!;at(' SlIhl!;l'OIlJlH of (1'1, .) by ell'n1l'nls of P; weI'(' P = a. PI • a-I for some a E P, the'n 1'1 = a-I * p. a ~ P, again h':u\inl!; to a eontradietion, l\OW, if t.hl' ('OIijlll!;ah' ~iUhl!;roups of (1', .) are no\' exhausted by (I', .) and the pk, eonjugates of (1'1,.) indul'et\ hy nl('mbers of 1', choose ano\'her conjugate (1'2, .) distini't frolll any yet ('numerat.cd. As above, we can obtain the pk2, > 0, di"linl'l eonjugat.(,H of (1'2,.) indll('ct\ by P. No ('onjugate of (1'2, .) hy an ,,11'1lI1'1I1 of I' will alHo b(, 11I1 (~onjllgal.(! of (1) ... ) by IlII clement of 1', for if a, iJ E P,

"'2

then (b- I • a) '" PI • (iJ- 1 • a)-I = 1'2, with b- I • a E 1', contrary to the choice of P 2 • If nce(,HHary, n'p('at thiH argument onee mor(', Since (G, .) is a finilc grollp, all the eonjllgates of (I',.) are found af\'er a finite number of HI('pH, Hay n sl('ps. Tlw ('onjllgate subgroups of (1', .) Ilrc these: (1', .) itself, the pk, eonjllgal('s of (/'), .) by elements of P, \'he 11 k• eonjuglltes of (1'2,.) by eleml'Tlts of P, ('Ie'. 'I'll(' t.otal numlwr of distinet (!onjugntcs is therefore

I I 1/" I 1/" I , .. I pk.

=

I ·1 mp == I (mod p),

k.

>

0,

At this st.age, an ohvious qucstion arises: are then! allY Sylow p-subgroups whieh nrc not. ('onjllgate suh!l;roups of (1', .)? The unswer will be sccn to be no. Let us llSSlIllW slwh It subgroup (R, .) does exist, anel arrive at a contradid.ion. As h('fOl"(, , we eount the eonjugates of (R, .). This is done by first lilldillJ,!; tI)(' l·olljllJ.!:at('~ of (H,.) indlJ('l'c\ hy ('11'1lH'nt.s of P; t.here nre (I(J»/o(N

n 1')-=

pi,

of them, whel'(> j. > 0 (if .il = 0, then R = 1', n eontradiction). Note that (R, .) i" its('lf jlll'huh-d in this eOllnt, sinee R = e • R • e-t, eEl', Following Ihc proc'('dllrl' ahov(', II\(' I.olnl III1111bN of (~onjugat(' subgroups of (R,.) is found to be j; > O.

2-9

SYLOW THEOREMS

137

Thc ilJlport.:Illt. point 1l('I'(' is t.hat no l('rll1 of this SIIIII /'('(ilw(,1-l to 1; if some j. = 0, t.Il(~n (ll, *) wOllld I)(~ a (~()njllgat.(! of (I', .), whidl is impoH!>iblc, But we havl' Jll'('viollsly dl'tl'rJIlilH'd Ihat. I hI' nllllll)('1' of ('I)njllgah's of any Sylow P-HIlItJ,!;I'OIiP i" (,OIlJ,!;I'tIl·/tt. to I IIlo
Corollary. AllY two Hylow p-Huhgroups of (0, .) (!()ITI'sJ!ollding to till! prime p arc conjugate, hence isomorphic,

SlllIIl'

Corollary. The number of distinct Sylow p-subgroups is [G:N(P)I, where (P, "') is any particular flylow p-subgroup of (G, "'), This leads directly to the second major result of the section,

Theorem 2-68. (Sylow), Let (G, "') be a finite group of order pkq, where p is a prime not dividing q, Then (G, .) has a Sylow p-subgroup of order pk, Proof, Suppose (P, "') is nny Sylow p-suhgroup of (G, tit), Our (!ontention is that 0(1') = 11k, For thiK, it is enough to Khow [G: PI has 110 1)-power' divisor. Sinee (I', .) iH It p-subgroup Qf (G, tit), 0(1') = pi, with j ~ k; henee,

and, if one knows that [G: P] is divisible by no pow('r of p, then j = k, The strate~ we employ is to prove thnt both flu-tOrs 011 the right of the equation IG:P) = [G:N(P»)[N(l'): 1') ~ [(]:N(l'»)()(N(P)/P)

nre prime to p, Now, [a: N(P)] is equal to the number of eonjugntc suhgroups of (1', tit) IUIII, hy our last. r('su It. , this valuc must. be (·(lIIgrUl'nt. to 1 modulo p; thereforI', HI(' possibility 1'1 [a:N(p)) eannotoeeur, To :see that o(N(P)/1') ha:s no p-power divisor, let us assume the coset a • P E N(P)/ P is of order 1', Then, as before, a P • I' = (a'" P)P = e tit P, whieh 1l1l'IlnS a P E 1'. As (I', tit) is a p-group, the element a P has order a pow('r of 1', HO t1mt a itself has [I-power o rill' r, Because a E N (I'), the Inst lemma illlpliml a E J'; hp/wl', a'" J' -= e • 1', or o(a. 1') = 1. From thiR eontradidion and Cmll'hy'H Theorem, we illfl'r t.hat p t o(N(I')/I'), Since p is a prime and divides neither of the factors of [G: 1'], it cannot divide [G: PJ either, thereby eompleting the argument,

,

Example 2-64. There call be 110 simple group (a, *) with o(G) = 42, Sillee 42 = 2, 3, 7, the preceding theorem implies the group ('olltains a :subgroup (II, .) of order 7, By Throrem 2-()7, the lIumber of (!onjugate subgroups of

2-9

mWlJl' TJlEOHY

(//, *) ii'! of til(' fOl'lll J I 7k, k E Z, anti tlivid('"42; k = 0 i~ the only pO!l8ibilit.y.

11 .. ,11''' (1/,

«(1,

*)

*)

('allllOl

il'l (·.Fli(· lind 'L

1)('

1l01'1I11t1

Hllill(I'Oll\l (i .... ,

HIM-I~onjlll(lLt(l),

"0 t.JIILt.

~illlpl('.

From Thl'orem 2-(i)';, WI' can obtain a partial converse to Lagrange's Theorem: If (0, *) is a fillite group whose order is a produet of powers of distinct primes,

Ul('n (G, *) ha~ subgroup~ of ordl'rs p~i (i = 1,2, ... ,n). As a matter of fact, it enn 1)(' ~howlI t.hat, for I'l\('h prime Pi, (G, *) eontains subgroups of orders {Ii, {I;, /1;1, ... ,{I~" Thlls, for im,t.ane(·, ILIlY p;rollJl of ord..r ]2 = 22. :J will hav(' Huhgl'llups of ol'deri'! 2, 2~ alld :1; nothing, how(wer, can be concluded ill regard to IL subgroup of order n. Example 2-65. Our work up t.o now hus given us sufficient tools to show that any group (G, *) of order 1.') is cycliC'. Since 15 = 3· 5, we first note that (0, *)

has at least, one Sylow :l-suhgroup and at least one Sylow 5-subgroup. The number of Sylow 3-suhgrollps must, by Theorem 2-67, divide 15 and yet be == 1 (mod 3). As the only such integer is 1, there is a unique Sylow 3-subgroup, call it (SYa, *); in other words, (SY3, *) is cyclic (being of prime order) and normal in (G, *). By the same reasoning, there is precisely one Sylow 5-subgroup, label it (SYIi, *), whieh is also cyC'lic and normal. We propose m show that (G, *) ~ (Sy:! X BY5, .). Detouring slightly, let us record several preliminary observations: Because (SY3 n SYs, *) is n xuhgroup of both (SY3, *) and (SYI;, *), it follows that o(Sy:! n By ... ) divid('s hoth :land !i. The only possibility is for o(SYa n SYa) = 1, whelll'e BYa n SY5 = {e}. In particular, this menns each element of (SYa, *) rommutes wit.h eReh clement of (SYs, *) [problem 10, Seetion 2-.')]. Next, we considl'r ('osds a * SY5, where the clement a E SY3' These coscts arc evidently nil distilll't,; for, if a * RY5 = 1) * RY5 (a, b E SY3), then

so t.hat a ~ h. However, [G: BY51 = 3 and SY3 contains just 3 elements, whieh implil's the (·oliet·s of RY5 in G must be of the form a * Sy/S, a E SY3. Since each element of G lies in one and only one such eoset, we infer at once that. cadI clement of G is uniquelY expressible as a product a * b, with a E SY3, bE SYs. The results of thes(~ last two pamgmphs permit us to UHe Problem 19, Section 2-7, amI therl'by ('0111'1,,<1(' (G, *) ~ (HY3 X SYs, .); but, on closer scrutiny, we not(~ that

It .,ydi(~

group of or
nl'('
is Prohlem 22, Section 2-7).

2-9

RYLOW THEOREMR

139

'fhii'! filliKhl'i'! our Kt.lldy of KrullP f.1l1'ory. WI' h/\VI' lint nt,\,l'lllpl.l'd to rnlll/:" live!" till! whol(~ UU'o!"Y 1101' tu cXlUuilW ill uepth allY plu·ticular Wlpet:t of it.

Instead, we have hwrely seratched the surface, introducing the reader to a few of the high points. Needless to say, current research in this branch of mathemnties is both vigorous and extensive. A variety of dassical problems still remain unsettled, while in some directions the research has only recently begun.

PROBLEMS

In the following set of problems, p alwaYR denotes a prime number.

t. Let (/I, *) and (K, *) be two subgroups of the group (0, *). Show that (/I, *) is normal in (K, *) if and only if 1/ k K k Na(1l). 2. For a finite group (G, *), prove that a) if (G, *) has exactly two conjugacy classes, then o(G) = 2; b) if there exists an element a E G with exactly two conjugates, then (G, *) contains a nontrivial normal subgroup. 3. Prove that a subgroup (II, *) of (G, *) is a normal subgroup if and only if the set /I is the union of conjugacy classes of (G, *). 4. Suppose (II, *) is a proper'subgroup of the finite group (G, *). Verify that G contains an element which belongs to no conjugate subgroup of (H, *). 5. Let (G, *) be any group, a a fixed element in G. Define the function fby f{x * C(a») = x * a * X-I, X E G. Prove that f is a one-to-one mapping from the left cosets of C(a) onto the set of distinct conjugates of a, thereby deducing Theorem 2-63. 6. a) Describe the class equation of the symmetric group (Sa, .). b) Show that, in a p-group, the number of self-conjugate clement'! (that is, elements whnse eonjugaey daSSel! are singleton>!) mUtlt be a multiple of p. 7. For a finite p-grollJl (G, *), prove the following; a) Any homomorphic image of (G, *) is again a p-group. b) (G, *) has a normal subgroup of order p. [/lint: cent G contains an element of order p.) c) (G, *) is a solvable group. [Hint: Induct on o(G); show that (cent G, *) and (G/cent G, ®) are hoth !iolvable.) 8. Prove that if o(G) = p", then the group (0, *) has at least one normal subgroup of order pk for aU 0 :::; k :::; n. [llint: Proeeed hy induction on n and utilize part (b) of the last problem.] 9. Let (11, *) be a normal subgroup of a finite p-grollp (G, *) and suppose that 0(1/) = p. Establish the indus ion 11 ~ cent G. 10. Prove t.hat if 0(0) = p2, then (G,.) is a commutative group. (llint: Recall Problem 21, He(~tion 2-5.)

140

(mOlll' TIU;ORY

11. 1.et (G, *) Ill' a

2-9

/(1'11111' such that 0(0) - lIfO. Verify the following assertions: a) o(cent G) ,.. p .. -I. b) cent G contains at least p elements. c) Every subgroup of order p .. - l is normal in (G, *). 12:-netepnine all Sylow p-subgroups of the group (Z24, +24). 13. Let (P, *) be a Sylow p-subgroup of the finite group (G, *). Prove that if (II, *) is any subgroup of (G,*) with P<;;N(P)<;;ll, then N(l/) = H; in particular, (leduce that N(N(P» = N(I'). 14. !'mve t.hat if (II, *) is a. f}-subgroup of (0, *) whi(,h is contained in exactly one Sylow THllIhgrllup (/" *) of (0, *), then N(l/) <;; N(P). 15. Huppose (II, *) is a Il"rmal p-!Suhgroup IIf the finite group (G, *). Show that (II, *) is a subgroup of each Sylow p-subgroup (for the same prime p) of (G, *). 16. Prove that a) any group of order 2p ha.'1 a normalllubgroup of order p, b) there arc no lIimple groups of orders 20, 30, or 56. 17. Let (G, *) be a group of order pq, where p < q are both primes. a) Show that (G, *) ha.'1 precisely one normal subgroup of order q. b) If q ¢ I (mod p), prove that (G, *) is a cyclic grollI', 18. Assume (G, *) is a f!;rollp of ordflr p"q, with p ancl q relatively prime. Prove that a Huhgroup (II, *) of (G, *) is a Sylow p-sub/(l'oup if and only if 0(11) = p",

CIIAPTEH

a

RING THEORY

3-1 DEFINITION AND ELEMENTARY PROPERTIES OF RINGS

In thi!! dlllpler, we shall illve!!tigate algebraie sysh~lII!! having two suitably re!Stricted binary operation!!. Obvious examples are the familiar number systems of elementary mathemati('s (the integerM, the rational numbers, etc.) and the algebra of sets. Using these systems as models, we shall presently define an algebraic structure known as a ring. Inasmuch as a ring is basically a combination of a commutative group and a semigroup, our previous experience with groups will prove to be of ('onsiderable help. As the reader will see, many of the important notions in group theory have natural extensions to sYlltems with two operations. When there are two binary operations present, it is real!Onable to expect them to be related in some way. The usual requirement is that one of the operations be distributive over the other. Definition 3-1. Let (8, "', .) be a mathematieal system with binary operations ... and.. The operntioll • ill !!aid to be distributive over the operation ... if a. (/) ... c) = (a. b) ... (a. c) (left distributive law)

and (b ... c) • a = (b. a) ... (c. a)

(right. distributive law)

for every triple of elpments a, b, c E 8. If the operation. happens to be commutative, then whenever. is left distributive over the operation "', it is all!O right distributive (and conversely), since (/) ... c) • a = a. (IJ ... c)

=

(a. b) ... (a. c)

=

(b. a) ... (c. a).

Cert.ainly, ordinary Illultiplication is distributive over ordinary addition, and we usc this fact in the next example. Example 3-1. Consider two biliary operat.ions ... and defined on the set Z of integer!! by a'" b = a + 2b, a b = :!ab. A !Simple calculation shows the opemtion to be left distributive over "', for 0

0

0

a • (b'" c)

=

a

0

(b

+ 2c) = 141

2a(b

+ 2c) =

2ab

+ 4ac,

142

3-1

lUNG TIIJWlty

while (a b) • (a c) 0

0

=

(2ab) • (2ac)

=

2ab

=

2ab

+ 2(2ac)

+ 4ac.

Because is a commutative operation, we (!onclude that it is also right distributive over •. 0

Definition 3-2. A rina iH a mathematiell.l system (ll,., 0) eonsisting of a 1J0nempty !let R aud two hinary operntions • and. ddined on R such that 1) (H,.) is a commutat ive group, 2) (R, 0) is a scmigroup, :J) t.he !I(,JIIigrollp olll'ratioll • iR distrihutive over the group olwrll.tion •.

It is convenient and customary to usc + for the group operation and· for the semigroup operation, rather t.han the symbols. and o. This convention is long-standing and is parti('ularly helpful in emphasizing the analogy between result.'i obtained for rings and those of the familiar numher systems. The above definition then takes t.he following foml.

+, .)

Definition 3-3. A ring is an ordered triple (R, consisting of a nonempty set R and two binary operations + and· on R such that 1) (R, +) is a commutative group, 2) (R,') is a semigroup, 3) the two olwrations arc related by the distributive laws

+ (a· c), (b + c) . a = (b· a) + (c· a)

a· (b

+ c) =

(a· b)

for all a, b, c E R. The reader should clt'arly understand that + and . represent abstract unspecified operations 1\111\ not ordinary addition and multiplication. For convenience, however, we shaII refer to the operation + all addition and the operation as multiplication. In the light of this terminology, it is natural then to speak of the commut!Ltive group (R, +) as being the additive group of the ring and (R, .) th(' multiplicative semiyroup of the ring. The unique identity elem('nt for addition is calIed the zero element of the ring !Lnd is denoted by the usual symbol for zero, O. The unique additive inverse of an clement a E H !lhalI he written as -a. In order to minimiz{' the use of parentheses in expressions involving both operations, W{' make the Rtipulation that multiplication is to be performed h('fore addit.ion. Thlls th(' ('xpression a· b + c IItands for (a· b) + c and not fOI'

a . (b

I c).

With this notation in mind, we {'nn now amplify our eurrcnt definition of a ring. A ring (R, +, .) ('onsists of a nOllempty set R and two operations, called addit.ion llnd Illult ipli('at iOIl and denoted hy + and " respectively, satisfying

a-I

UEHNITION AND ELEMENTAUY PItOPEltTn.;S OF lUNGS

the requirements: 1) R iH closed under audition,

+ b = b + a, + II) + C = a + (b + c),

2) a 3) (a

4) t.here existl:! an element 0 in R sueh that a + 0 = a for every a E R, Ii) for
+

+

+ c) . a =

b· a

+ t: • a,

wlJl'rl' it, is ululerHtood t.hat a, I}, t: represent arbitrary elements of R. By pl:willl!; fur·ther rl'sl.ri(:tionH Oil the opemLion of Illultiplieation, !!everal special types of rings arc obtained. Detlnition 3-4. 1) A commutative ring is a ring in which (R,·) is a commutative scmigroup; that is, the operation of multiplication is commutative. 2) A rina with identity is It ring in whieh (R,·) ill a semigroup with identity; that is, there exists an identity element for the operation of multiplication, customarily denoted by the symboll.

In a ring (R, +,.) with identity, we sayan element a E R is invertible if it possesses an inverse relative to multiplication. Multiplicative inverses are unique, when they exist, and shall be denoted as before by a-I. The symbol R* wiII be used subsequently to represent the set of all inyertible clements of the ring. Theorem 3-1. Let (R, +,.) be a ring with identity. Then the Jlllir (R*,·) forms a group, known aI:! the group of invertible elements.

Proof. The set R* is nonempty, for at thc very least the multiplicative identity 1 E R*; moreovcr, 1 serves as the identity clement for the system (R*, .). If a, bE R*, the equation!! (a· b) . (b- I . a-I) = a· 1· a-I = 1,

(b- I

.

a-I) . (a· II) = 1>-1·1· b = 1

show the product a· b is also a member of R*. Whenever a is invertible, then we have a· a-I = a-I. a = 1, indicating that a-I is in R*. The allllociative law for multiplication is obviously inherited since R* S;;; R, so (R*, .) is a group. Before proceeding to the proofs of the basie results of ring theory, we shall consider several exampleH. Example 3-2. Each of the following familiar sYHtems, where ordinary addition and multiplication, is Il. commutative ring:

(R',

+, .),

(Q,

+, .),

(Z,

+, .),

(Ze,

+ and·

+, .).

inuieate

3-1

144

The first three of t.he!!C rings have an identity element, the integer 1, for multiplication. Exampl. 3-3. Anotlll'r ('xlunple of a rillK is provided hy the sct R = {a

+ bval a, b E Z},

and the operations of onlinary addition and multiplication. The set R is obviously closed under these operations, for

+ bva) + (c + elva) = (a -I- bv'3) . (c + dva) =

(a

+ c) + (b + d)v3 E R, (ac + 3bd) + (ad + bc)va E (a

R,

whenever a, b, c, dE Z. We omit the details of showing that (R, +,.) commutative ring with identity element 1 = 1 + ova E R.

IS

a

Example 3-4. If P(X) is the power Het of HOnle nonempty !!Ct X, then both the triples (P(X), u, n) and (P(X), n, u) fail to be rings, Kince neither (P(X), U) nor (/,(X), n) forms a group. However, the reader may recall that

in Example 2-20 we showed the pair (P(X),~) to be a commutative group; here, the symbol ~ indi(~at.es the symmetri(~ diff('rence operation A

~

B

=

(A - B) U (B -

A).

Since (P(X), n) is dearly a nommutative semigroup, in order to establish that the triple (p(X) , ~, n) constitutes a ring it is only necessary to verify the left distributivity of n over~. For this, we require the set identity (Problem 6, Section 1-1) A

n (B

- C)

=

(A

n B)

-

(A

n C).

The argument now proeeedK as follows: for all subsets A, B, 0 A

n (B

~

C)

=

A

n [(B

- C)

u (0

~

X,

- B»)

IA n (B - C») u [A n (0 - B)] = ((A n B) - (A nO») u [(A n C) = (A n B) ~ (A nO). =

-

(A

n B»)

Example 3-5. Let R denote the tlCt of all functions I: R' -+ R'. The sum I g and product I . (f of two functions I, g E R are defined as usual, by the

+

equations (f

+ g)(a) = I(a) + !/(a),

(f. g)(a)

= I(a) . g(a),

a E

R'.

In other words, we specify the functional value of these combinations at each point in tlwir domain. That (R, +,.) is a (~ommutat.ive ring with identity

J)";~'INITION

AND ELJo:MENTAUY PROPEIlTlJo:S OF RINGS

145

follows from the fact that the real numbers with ordinary addition and multipliention comprisc 8u(!h a system. In particular, the multiplicative identity d('lIwnt ill the NIIIHtUllt fUllctioll whoHe value at each real number is 1. It iH int('rpsting to nof.(! that the triple (R, /-,0), where 0 indicates the operation of fUllctional eompositioll, fails to be a ring. The left distributive law

f

0

(g

+ /t) =

(f. g)

+ (f oIt)

does not hold in this case. Example 3-6. For a more interesting example, let (G, *) be an arbitrary commut.ative group and hom G be the set of all homomorphisms from (G, *) into itself. From Theorem 2-36, it is already known that (hom G, .) is a semigroup with identity. We propose to introduce a notion of addition in hom G such that the triple (hom G, +, .) forms a ring with identity. To achieve this, simply (lC'fillc the sum f + (J of two fUlletions f, (J E hom G hy the rule

(f

+ g)(a) = f(a) * g(a),

aE G.

With the possible exception of the closure condition, it is fairly evident that the pnir (hom G, +) is a commutative group; the trivial homomorphism acts as the zero element, while the negative of each function I E hom G is obtained hy taking (-- I)(a) = f(a)-l fOr nil a E G. To estnhlish (llmlllm, (!hooBrlnrbitmry elt!lIlcnts a, b E (J and fUllctiollti I, y ill hom G. Theil (f

+ g)(a * b) = I(a * b) * g(a * b) = I(a) * feb) * g(a) * g(b) = (j(a) * Il(a») * (f(b) * {f(b») = (f + g)(a) * (f + y)(b) ,

so that the sum J + g is also t\ homomorphitllll of (G, *) into itself. In J'(·gard to UI(l left distributive law, we lllwe, fol' /Lny a in 0, [f. (g

+ h)](a) = f«g + h)(a») = J(g(a) * /t(a») = J(g(a») * f(h(a») = (f. g)(a) * (f. h)(a) =

(f. g

+ I· h)(a).

Therefore, 1 «(J - f- It) = I· (J - f- I . It. The right distributive law follows in a similar manner, showing (R, +, .) indeed to be a ring. Necdless to say, eommutativity of the underlying group (G, *) is essential to our discussion; ill its absence, we could not even prove addition to be a. binary operation in hom G. 0

146

3-1

RING THEORY

Example 3-7. In Section 2-3, we considered the group (Z.. , +..) of integers modulo n. This group was obtained upon defining in the set Z.. the notion of addition of congruence classes:

fa]

+.. fb] =

[a

+ b].

A binary operation '" of multipliHation of e1a.~SC!! may f!qually well be introduced in Z .. by speeifying that for each pilir of clements [a], [b] E Z .. , [a]· .. fb]

=

fa· b].

This lat,t(1r definition pre!!l1ntR R. problem Rimilar to that of addition in that we InUloJt Hhow Umt !Jill rmmll.ing produd leI' 1/) iH illlil!)llm
We first observe that a' E [a'] a'

==

=

=

[a· b].

[a] and b' E [b']

a (mod n),

=

[b] imply that

b' == b (mod n).

From Theorem 2-10 we are then able to conclude that

a' . b' == a· b (mod n), and consequently a' . b' E [a· b]. This means, in view of Theorem 2-12, that [a' . b'] = fa· b], as desired. For each positive integer n, the system (Z.. , +.. , '.. ) is a commutative ring with identity, known as the ring of integers modulo n. The verification that the ring axioms 11.1"(1 satiRfi(1c1 is very straightforward, depending only on the definitions of the operations +.. and '" and the fact that (Z, +, .) itself is a ring. For instance, to show the left distributivity of '" over +'" we choose [a]; [b], [e] E Z" and obtain [a]

'n

([b]

+.. [e]) =

+ e] = [a· (b + e)] = [a· b + a· e] [a]'n [b

+.. [a· e]

=

[a· b]

=

[a]'n [b]

+.. [a] '" fe].

Clearly [1] is t,he multiplicative identity clement. We shall leave the confirmation of the remaining ring axioms and of commutativity to the reader. As mentioned earlier, it is convenient to remove the brackets in the designation of the congruence classes of Z", and this shall he our practice subsequently.

3-1

147

DEFINITION AND ELEMENTARY PROPERTIES OF RINGS

Given a ring CR. +.. ), all the results of the previous chapter on groups apply to the system (R, +). For instance, we know that the zero element a.nd additive inverses are unique, that the cancellation law holds for addition, and so on. In the theorems to follow, we shall establish ;;Dme of the fundamental properties which depelld 011 hot.h ring operations. As usual, we may aSl"lUme nothing about t.he Hpm:ific IIUI.III"" of t.he KYHt."1lI (R, I,·) «·x«:ept. t.hut. it. Hat.iHfies the pOld,ulates presented in Defillition :J-a. Theorem 3-2. In any ring (R,

+, .), if a E R, then a· 0 = o· a = O.

Proof. First note that an application of the left distributive law (and the fact that zero is t.he idl'llt,ity element for addition) yil'lds It . (I

+ It • () -

a· (0

-t- 0) =

By the eancellation law for addition, a· 0 show that 0 . a = O.

II . () ::: I l ' ()

=

O.

-+ n.

In similar fashion, one can

The reader may have speculated as to whet.her, in a ring with multiplicative identit.y, the identity !\Ild zero elements of the ring are Elver equal. It follows at ollce from Theorem 3-2 that this situation can only occur in the one-clement ring ( {O}, the so-called zero ring; the proof is given below:

+, .),

+, .).

Theorem 3-3. Let CR, be a ring with identity such that R Then the elements 0 and 1 are distinct.

Proof. Since R ¢ {O}, there exists some nonzero element a E R. 1 = 0, it would follow that

a

=

a· 1

=

a· 0

=

¢

{O}.

Now, if

0,

whieh is nn obvious contradiction. Unless stated to the eontrary, we shall tacitly assume that any ring with idenUty conta7:ns more than one dement,. this will I'xdll
= R' X R' of ordered pairs of real number:;. We define addition and multiplication in R by the formulas

Example 3-8(a). Consider the set R

(a, b)

+ (e, d)

=

(a

+ e, b + d),

(a, b) . (e, d) = (ae, bd).

Theil stmightfOl'ward ealc'lIlat.iolls will Hhow that (R, +,.) is a commutative ring with identity CICllltJllt (1, 1). Here the zero element is the pair (0,0).

148

3-1

RING THEORY

Observe that while (1.,0) • (0, 1)

==

(0,0),

neither (1,0) nor (0, 1) is the zero of the ring. Exampl. 3-1(1)>). Another example in which this situation occurs is the ring (Z4, +4, '4) of integers modulo 4. The addition and multiplication tables are shown below: +4

o 1 2 3

0 1 2 3

'4

0 1 2 3

0 1 2 3

o

0 0 0 0

1 2 3 230 3 0 1 0 1 2

1 2 3

0 1 2 3

0 2 0 2

0 3 2 1

Here, we have 2'42 == 0, the product of nonzero elements being zero. Note also that 2 . 4 1 = 2· 4 3, yet it is clearly not true that 1 == 3. The multiplicative semigroup (Z4, '4) does not satisfy the cancellation law.

+, .)

Deftnhlon ~5. A ring (R, is said to have diviBor, oj Z6TO if there exist nonzero elements a, b e R such that the product 0 • b == O. We exhibited two rings in the above example which possess dWisors of zero. The second ring, in particular, suggests a relationship between the existence of divisors of zero and the failure of the cancellation law for multiplication. We shall see shortly that this is indeed the case. First, several preliminary results concerning additive inverses are required. We shall adopt the usual convention of writing a (-b) as a - b and refer to this expression as the difference between a and b.

+

Th.....m 3-4. Let (R,

+, .) be a ring a.nd a, b e R. -(a· b)

==

a· (-b)

==

Then

(-a)· b.

Proof. From the definition of additive inverse, we know that b + (-b) == Using the left distributive law,

a· b + a· (-b) == o(b + (-b»

o.

== a· 0 == o.

Inasmuch sa the inverse of an element under addition is unique, this last equation implies that

== a· (-b). (a • b) == (-a) . b.

-(a· b) A similar argument shows that -

Corollary. For any elements a, b e R, (-a)· (-b)

== a· b.

3-1

DEFINITION AND ELEMENTARY PROPERTIES OF RINGS

149

Proof. It follows from the above theorem that

(-a)· (-b)

=

-«-a)· b)

=

-(-(a· b»)

The last equality stems from the fact that (X- 1)-1

=

a· b.

= x in any group.

Corollary. If a, b, c E R, then

a· (b - c)

=

a· b - a· c,

(b - c) . a

= b. a -

c • a.

''''hat is, multiplication is distributive over differences. Proof. Since multiplication is left distributive over addition,

a· (b - c) = a· (b

+ (-c»)

= a· b

+ a· (-c)

= a· b+ (-(a· c») =a·b-a·c. In a like manner, a ring distributive law for differences is obtained. Having thus laid the groundwork, we are now able to establish the following result. 3-5. A ring (R, +, .) is without divisors of zero if and only if the cancellation law holds for multiplication.

Theorem

Proof. First, we assume the ring (R, +, .) contains no divisors of zero. Let a, b, c be elements of R s~ch that a ¢ 0 and a· b = a· c. Then,

a· (b - c) Since a

¢

0 and (R,

b - c = 0 or b ~ith

=

a· b - a· c =

o.

+, .)

= c. In 0, yields b = c.

has no zero divisors, this last equation implies a similar manner, we can prove that b· (I = c· (I,

a¢ Conversely, suppose that the cancellation law holds in the semigroup (R,·) and the product (I. b = O. If the element (I is nonzero, then it may be can\eeled from the equation (I. b - a· 0 to conclude b = O. On the other hand, b ¢ 0 implies (I = 0 by the same argument. This shows (R, +, .) is free of divisors of zero.

+, .) be a ring with identity which has no zero divisors. Then the only solutions of the equation a2 = a arc a = 0 and a = 1.

Corollary. Let (R,

Proof. The proof is easy. If the cancellation law a = 1.

(12

= a = a· 1, with

(I

¢

0, then by virtue of

Definition 3-6. An integral domain is a commutative ring with identity which does not have divisors of zero.

We have just shown that the cancellation law for multiplication is satisfied in any integral domain. .

150

3-1

RING THEORY

A word of (,llutiOIl: Some authors omit the requirement of an identity and usc the term integral domain to indieate any commutative ring without zero divisors. For our JlUrpOKCB, the preceding definition if! more appropriate. In Section 2-4, we discussed the concept of a subgroup of a group. It is natural that there should be a corresponding notion of subsystems for rings. Deftnitlon 3-7. Let (Il, +,.) be a ring and S ~ R be II. nonempty subset of R. If th(> triple (8, +, .) is itHClf a ring, then (8, +, .) is said to be a 8ubriny of (R,l ,.).

An examination of the definition of ring as given in Definition 3-2 shows that

+, .)

+, .)

+),

(8, is a subring of (R, provid(>d (8, +) is II. subgroup of (R, (8, .) is a suhsemigroup of (R, .), and til(' two dist.ributive laws hold for ele-

mellts of S. But hoth t,he distributive alld assoeiativ(> laws hold automatically ill S as a ('Ollsequl'll('(> of their validity in R. Sinee these laws arc inherited from H, there is 110 partirular n('('(>ssity of requiring them ill the definition of II. subring. In view of this observation, II. subring eould alternatively be defined as follows: the t.ripl!' (8, -1-, .) is It suhring of the ring (R, +, .) whenever 1) S iF; II nOIl(,llIpty HuhHet of Il, 2) (8, +) is a subgroup of (R, +), 3) S is closed undcr multiplication. Even this definition may be improved upon, for the reader may recall that if 0 ~ II ~ G, then the pair (II, *) is a subgroup of the group (G, *) provided that a, bEll implies a * b- 1 E H. Adjusting the notation to our present situation, we obtain a minimal set of conditions for detennining subrings.

+, .) be a ring and 0 ~ S ~ R. +, .) if and only if

Definition 3-8. Let (R, (S, is a 8ubring of (R,

+, .)

Then the triple

1) a - bE S whl'll('ver a, b E S (closed under differences), 2) a· b E .-; wiwlwVI'r a, 1) E'-; (closed under multiplication).

(R, +, .) has two trivial subrings; for, if 0 denotes the zero element of the ring (R, -1-, .), then both ({OJ, +, .) and (R, +, .) are sub rings of (R, +, .).

Example 3-9. Evpry ring

+, .), the triple (Z" 1-,') is U Hubrin!!:. while (Zo, +, .) i~ not. In particular, we infer that in It ring with identity, a suuring does not n('('11 to contain the identit,y element. 0 2 4 '0 +6 0 2 4 Example 3-11. COllsi(\l'r (Zo, I 0, '6), the - - - - - 0 0 0 0 0 2 4 0 ring of int('gl'rs modulo (i. IfS= ~O, 2, 4) , 2 0 4 2 2 2 4 0 then (8, +6, '6), whost, op<'mt.ioll tables arc 4 0 2 4 4 0 2 4 given at the right, is a subring of (Zo, -h, '6)' Example 3-10. In thl' ring of integers (Z,

3-1

DEFINITION AND ELEMF.NTAUY PltOl'lmTIES OF RINGS

Example 3-12. Let S= {a+bV3la, bEZ}. of (R, +, .), since for a, b, c, d E Z, (a

+ bV3) - (e + dV3) = (a + bV3) . (e + dV3) =

151

Tlwn (S,+,') is a subring

+ (b - d)V3 E S, (ac + 3bd) + (be + ad)V3 E S.

(a - c)

This shows that S is closed under both differences and products. It has already been pointed out that, when a ring has an iuentity, this need not be true of its subrings. Other interesting situations may arise: 1) Some subring has multiplicative identity, but the entire ring does not. 2) Both the ring and one of its sub rings possess identit.y elements which are distinct. In each of these cases, the identity for the subring must be a. divisor of zero in the parent ring. To justify this last assertion, let l' denote the identity clement of the subring (S, +, .); we assume l' is not an identit.y for the entire ring (R, +, .). Accordingly, there CXiHLs an clement a E II for whil·h a· I' ;a! a. It is clear that (a· 1') . .I' = a· (I' . I') = a· l' or (a· l' - a) . l' = O.

Since neither a· I' - a nor I' is zero, the ring (R, +,.) has zero divisors; in particular, I' is a zero divisor of (R, +, .). To give a simple illustration of a ring in which possibility (2) occurs, we may refer to Example 3-8. There, the system (R ' X R', +, .) was shown to be a ring provided addition and multiplication are defined by (a, b)

+ (e, d) =

(a

+ c, b + d),

(a, b) • (c, d)

=

(ac, bd).

It is a routine matter to verify that the triple (R' X 0, +, .) forms a subring ·with identity element (1,0); in this case, (1,0) differs from the identity for the parent ring which is the ordered pair (I, 1).

Regurding ('neh elemcnt of the ring (ll, +, .) a.~ Rimply n. member of the additive group (R, +), the familiar laws pertaining to integral exponents (Theorem 2-!) mf.y be translated direetly into properties of integral multiples. Thus, for a, bE R :,nu arbitrary integers n anu m, the following hold: (n

+ m)a =

na

+ ma,

(nm)a = n(ma), n(a

+ b) = na + nb.

152

~1

RING THEORY

In addition to these rules, there are two further properties resulting (rom the distributive law, namely n(a· b) = (na)· b = a· (nb), (na) . (mb)

=

(nm)(a· b).

We should emphasize that the symbol na does not neeelJlJ8J'ily represent a ring product; indeed, the integer n may not even be a member of R. For n > 0, the notation na is merely an abbreviation for the finite sum a + a + ... + a, n summands. However, when there is an identity element present, it is possible to express na &8 the product of two ring elements: na = (nI)· a. DefInition 3-9. Let (R, +, .) be an arbitrary ring. If there exists a positive integer n such that na = 0 for all a E R, then the least positive integer with this property is called the characteristic of the ring. If no such positive integer exists (that is, na = 0 for all a E R implies n = 0), then we say (R, +,.) has characteri8tic zero.

The rings of integers, rational numbers and real numbers are standard examples of systems having characteristic zero. On the other hand, the ring (P(X), !:l, n) is of characteristic two, since 2A

=

A !:l A

=

(A - A) U (A - A)

=0

from every subset A of X. Let (R, +,.) be a ring with identity. Then (R, +,.) has characteristic n > 0 if and only if n is the least positive integer for which ni == O.

Theorem 3-6.

Proof. If the ring (R, +, .) is of characteristic n > 0, it follows trivially that = O. Were mi = 0, where 0 < m < n, then

nl

ma

=

m(I . a)

=

(mI) . a

= o· a =

0

+, .)

(or every element a E R. This would mean the characteristic of (R, is Jeaa than n, an obvious contradiction. The converse is established in much the same way. Corollary. In an integral domain, all the nonzero elements have the same additive order, which is the characteristic of the domain.

Proof. To verify this assertion, suppose the integral domain (R, +, .) has positive characteristic n. According to the definition o( characteristic, any a E R (a F- 0) will then poBse88 a finite additive order m, with m S n. But the equation

o=

implies ml

=

ma = (mI)· a

0, since (R, +,.) is free of zero divisors. We may therefore con-

3-1

DEFINITION AND ELEMENTARY PROPERTIES OF RINGS

153

clude from the theorem that n ::; m. Hence m = n and every nonzero element of R has additive order n. A somewhat similar argument can be employed when (R, +, .) is of characteristic zero. The equation 100 = 0 would lead, as before, to ml = 0 and consequently m = O. In this case, each nonzero element of R must be of infinite order. The last theorem serves to bring out another point. Corollary. The characteristic of an integral domain (R, or a prime number.

+, .)

is either zero

Proof. Let (R, +,.) be of positive characteristic n and assume that n is not a prime. Then n can be written as n = nln2 with 1 < ni < n (i = 1,2). We therefore have Since by hypothesis (R, +, .) is without zero divisors, either nil = 0 or n21 = O. But this is plainly absurd, for it contradicts the choice of n as the least positive integer such that nl = O. Hence, we are led to conclude that the characteristic must be prime. Now, suppose (R, +,.) is an arbitrary ring with identity and eonsider the set ZI of integral mUltiples of ihe identity ZI

=

{nIl n EZ}.

From the relations nl -

ml

=

(n - m) 1

and

(nI) . (ml)

=

(nm)I

one can easily sec that the triple (ZI, +,.) itself forms a commutative subring with identity. The order of the additive cyclic group (ZI, +) is simply the characteristic of the original ring (R, In case (R, +, .) is an integral domain of characteristic p > 0, p a prime number, we are able to show considerably more: each nonzero element of (ZI, +, .) is invertible. Before proving this, first observe that by Theorem 2-23, the set Zl consists of p distinct elements; namely, the p sums nI, where n == 0,' 1, ... ,p - 1. Now, let nl be any nonzero element of ZI, 0 < n < p. Since p and n are relatively prime, there exist integers r, 8 for whieh rp sn = 1. Therefore,

+, .).

+

1

=

(rp

+ sn)1 =

(rl)· (pi)

+ (d) . (nl).

As pi = 0, we obtain 1 = (81) . (nl), so that 81 constitutes the multiplicative inverse of nI in (Zl, +.. ). We shall return to a further discussion of the characteristic of a ring at the appropriate place in the sequel; in particular, the value of this last result will have to await future developments.

154

3-1

lUNG TIH:
PROBLEMS

* and

1. Define two binary operations

on the set Z of integers by

0

a*b - a+ b+2, a b - ab 0

+ 2a + 2b + 2.

Show that i~ dhltributive over *. 2. Let (R, +) b(' any commutative group. Determine whl'th('r (R, +,.) forms a ring if muitipiieation is dt'finl'd by a) a' b = a. h) a' b = O. wh('re 0 is the identity element of the group (R, +). 3. Let (R, + .. ) be an arbitrary ring. In R define a new binary operation by the rule 0

0

aob = a·b+b·a

fllr all a. bE fl. Establish that (fl, +,0) ill a commutative ring. 4. Obtain the group of invertible elements (ZT2, '12) for the ring (ZI2, +12, addition, show that (Z!2, '12) is isomorphic to Klein's four-group. 5. Given that a, b, c, d are clements of a ring (R, prove that

'12).

In

+, .),

+

+

+

+

a) (a b) . (c d) = a' c b.c a' d b) -(a·b·c) = (-a)· (-b)· (-c),

+

+ b . d,

(.) if a . b = b . a = 0, then (a b)" = an (As usual, a" = a' a ... a, n times.) 6. Define two binary op('rations

* and

0

+ b" for n E Z+.

on the set Z of integers as follows:

a*b=a+b-l,

a .. b = a + b - abo Prove that th!' sYKtem (Z, *, 0) is a commutative ring with identity. 7. Let R be the s('t of all ordered pairs of nonzero real numbers. In the following Casell, dct!'rmin!' wh!'th!'r (R, is a commutat.ivc ring with identity. For those HYHWIllIi failing to he HO, indicatt~ which axiulll!! are not Hatisfil'll: a) b) c) d)

8. In a) b) c) d)

(a, (a, (a, (a,

+ (e, d) + (e, d) b) + (e, d)

+, .) (ae, be + d), (a + e, b + d), (ad + bd, bd),

b)

(a

IJ)

b)

+ (e, d)

+ c, b + d),

(a, (a, (a, (a,

b) . (c, b) . (e, b) . (e, b) . (e,

d)

d) d) d)

(ae, bd), (ae bd, ad (ae, bd), (ae, ad be).

+

+ bd),

+

a ring (R, +,.) with identity, prove that the Illultiplicative ili!'ntity element ill unique, if a E fl has a multiplicative inverse, then a-I is unique, if the !'l!'mcnt a is invertible, so also is -a and (_a)-1 = -a-I, no divisor of z!'ro can llOSS!'SS a multiplicative inverse.

9. Let (fl, +,.) be a ring which has the property that a 2 = a for every a E R. Prove that (R, +,.) is a commutative ring. [Hint: First show a + a == 0 for any aER.)

3-]

155

IlJ<:FINITION ANI> ELEMENTARY PUOPEUTIES OF RINGS

10. Prove that a ring (R,+,') is commutative if and only if (a + b)2 = a 2 + 2(a . b) + b2

for every pair of clements a, bE R. 11. l>iHllover divil!orll of zero to show that (Zo, +0, '0) iH nut an intogral domain. Mnrel Kenl'raliy, Hhow that (Z., contains divis(ml of zero if n is not prime. 12. a) In an integral domain, show that the only HolutionK of the equation a 2 - 1 are either a = 1 or a = -1. b) If the set X containH more than one element, prove that every nonempty proper subset of X is a divisor of zero in the ring (P(X), A, n). 13. An element a of a ring (R, is said to be nilpotent if a~ = 0 for Boma n E Z+. Prove that in an integral domain the zero element is the only nilpotent element.

+., '.)

+, .)

14. Prove that the system ({O, 3, 6, 9},+12, '12) is a subring of (Z12, +12, '12), the ring of intRlI;!'rH modulo 12.

-1-, .) are buth Huhrings of tlw ring

15. D('rive the r('Hult: if (S, h,) and (7', then so also is the triple (8 n T,

+, .).

(R,

+, .),

16. Prove that in an integral domain any suhring which containl:! the identity element is again an integral domain. 17. The center of a ring (R,+, .), denoted by cent R, is the set cent R

= {c E Ric· x = x' c for all x E R}.

+, .)

+, .).

Prove tJlat (cant R, is a /lubring of (R, 18. For every n > ], show that thera exists at leBllt one ring of characteristic n. [/lint: Consider the ring (Z", 19. a) If a and b are elements of a commutative ring with identity and n E Zh obtain fihe analog of the familiar binomial expansion for (a b)":

+..,.,,).]

+

t

(a+ b)" =

G)an-V.

k-O

As usual, the hinomial co!'/fieicnt.

(~)

= (n

~~) !k-! .

b) From this, delduee that in an integral domain of eharuetl'ristic p

+

>

0,

+

(a b)1' = a" b" for all a and b. 20. Suppose (R, ., 0) and (ll', .', .') urn t.wo rinll;H. J)('lim' hi nary operations lin the Cart.c>sian procluet Il X Il' as follow!!: (a, b)

+ (e, d)

=

(a. c, b.' d),

+, .)

(a, b) . (e, d) = (a

0

+ and·

c, b.' d).

a) Prove t.hat th!' system (R X R', forms a rin!!;, "ailed t.he direct product of thc rinll;s (ll, *,0) and (R', .', .'). b) If the original rings are commutative with idl'ntity, show that the same must bc truc of (R X R',+, .).

156

3-2

RING THEORY

~2 IDEALS AND QUOnENT RINGS

In this section we introduce an important class of subrings, known as ideals, whose role in ring theory is similar to that of the normal subgroups in the study of groups. As shall be seen, ideals lead to the construction of quotient rings which are the appropriate analogs of quotient groups. Deftnltion 3-10. A subring (I,

+,.)

of the ring (R,

(R, +, .) if and only if r e R and a e I imply both r·

+,.) eI

0

is an ideal of and o· reI.

Thus we require that whenever one of the factors in a product belongs to I, the product itself must be a member of I. In a sense, the set I "captures" products. If (1, is a subring of (R, I is already closed under mUltiplication. For (1, +, .) to be an ideal, a stronger closure condition is imposed: 1 is closed under multiplication by an arbitrary element of R. In view of Definition 3-8, which gives a minimum set of conditiona on I for (I, +,.) to be a subring, our present definition of ideal may be rephrased as follows:

+, .)

+, .),

Definition 3-11. Let (R, +,.) be a ring and I a nonempty subset of R. Then (1, +,.) is an ideal of (R, +,.) if and only if 1) 0, bel imply 0 - bel, 2) r e Rand 0 e I imply both r· 0 e I and o· reI.

In the case of a commutative ring, of course, we need only require r·

0

e

I.

Before proceeding further, we shall examine this concept by means of several specific examples. Example 3-13. In any ring (R, +, .), the trivial subrings (R, +,.) and ({O}, are both ideals. A ring which contains no ideals except these two is said to be simple. Any ideal different from (R, is termed proper.

+, .)

+, .)

Example 3-14. The subring ({O, 3. 6, 9}, +12) is an ideal of (Z12. +12, '12), the ring of integers modulo 12. Example 3-15. For a fixed integer 0 e Z, let (0) denote the set of all integral multiples of o. that is. (0) = {no I n e Z}.

+.. ) to be an ideal of the ring of

The following relations show the triple «0). integers (Z. +, .): no - rna = (n - m)o. m(no)

=

(mn) 0,

n, m eZ.

In particular, since (2) = Z•• the ring of even integers (Z•• (Z, + .. ).

+..) is an ideal of

3-2

IDEALS AND QUOTIENT RINGS

157

Example 3-16. Suppose (R, +,.) is the commutative ring of functions of Example 3-5. Define 1= {fE R If(I) = O}.

For functions f, gEl and hER, we have (J - g)(l)

= J(I)

- gel)

=

0 - 0

=

0

and also (h· /)(1)

==

h(I) . f(I)

=

h(I) • 0

= o.

Since hothf - g and h· g belong to I, (1, +,.) is an ideal of (R,

+, .).

If condition (2) of the definition of ideal is weakened so as to only require that the product r· a belongs to I for every a E I and r E R, then we arrive at the notion of a left ideal (right ideals are defined in a similar way). For commutative rings, it is plain that every left (right) ideal is an ideal or; as it is sometimes called, a two-Bided ideal. We next derive several interesting and useful results concerning ideals of arbitrary rings.

3-7. If (I, +,.) is a proper ideal of a ring (R, +, .) with identity, then no element of I has a multiplicative inverse; that is, I n R* = 0.

_ Thearem

Proof. Suppose to the contrary that there is some member a ~ 0 of I such that a-I exists. Since I is closed under mUltiplication by arbitrary elements of R, a-I. a = 1 E I. It then follows by the same reasoning that I contains

r

r· 1 = r for every E R: "That is, R !; I. Inasmuch as the opposite inclusion always holds, I = R, contradicting the hypothesis that I is a proper subset of R. 3-8. If (It, +, .) is an arbitrary indexed collection of ideals of the ring (R, +, .), then IJ!> also is (nit, +, .).

Theorem

Proof. First, observe that the intersection nli is nonempty, for each of the sets Ii must contain the zero element of the ring. Suppose the elements a, bE nIi and r E R. Then a and b are members of It, where i ranges over the index set. As the triple (It, +,.) is an ideal of (R, +, .), it follows from Definition 3-11 that a - b, r· a and a· r all lie in the set h But this is true for every value of i, so the elements a - b, r· a and a· r belong to nI., which implies that (nI., +,.) is an ideal of (R, +, .).

Consider, for the moment, an arbitrary ring (R, 8 of R. By the symbol (8) we shall mean the set (8)

=

n{I I 8!; I; (1,

+, .) and a nonempty subset

+,.) is an ideal of (R, +, .)}.

The collection of all ideals which contain S is not empty, since the improper

158

3-2

RING THEORY

+, .)

id<>al (R, clearly belongs to it; thus, the set (8) exists and is such that 8 ~ (8). Theorem :l-S leads directly to the following result. Theorem 3-9. The triple «8), the ideal ymeraled by Ihe set S.

+,.) if! an ideal of the ring

(R,

+, .), known as

It is not.eworthy that whenever (I, +, .) is any ideal of (R, +, .) for which S ~ I, t.hell (8) ~ I. III view of this, olle frequently speaks of «S), +,.) ILS being the smallesl ideal to contain the set S. An ideal generated by a single ring clement, say a, is called a principal ideal and is designated by «a), +,.). A natural undt'rtaking is to determine the precise form of the members of (S). If we impot«) the requirement that (R, +, .) he a commutativo ring, it is a fnirly Hilllpl!' matter to e/wek Umt (8) is givell by (8) =

{1:ri"~i+ 1:njlljlriER;si,l!jE8;njEZ},

where the symbol 1: illr as all exprcil:!C. In the eal:!C of the prineipal ideal «a), this description of (8) reduces to

+, .),

(a)

=

{r· a

+ na IrE R, n E Z}. +

Obl:!Crve, incidentally, that the clement a is contained in (a), since a = O· a la. When there ia an identity clement present, the term na becom~ superfluous. For, in this situation, we mfty write the expression r· a + na as l' .

a

+ na =

+ n(t . a)

r' a

=

r· a

+ (nl) . a = (r + nl) . a,

with r·1 III a rilll( demellt. ThuH, the Het (a) merely consists of all ring multiples of a: {r· a IrE R}.

(a) =

III actual fnet, the elements 1" a (r E R) compril:!C the set of elements of an ideal of (R, +,.) evell whell the ring doesllot possess an identity; the difficulty, however, iH that thil:! i
then thc prineipal ideal «a),

1\

('ommutative ring with identity and a E Il,

+, .) gencrated by a is such that

(a) =

{r· a 11' E R}.

The prirll'ipnl ilil'all:! are the only il\l'als of the ring of integers, as the following theorem will :;how. Theorem 3-11. If (/, t-,.) is an idl'al of the ring (Z,

for some Ilollll('gative integer n.

+, .),

then 1= (n)

3-2

159

IDEALS ANO QUOTIENT RINGS

+, .)

Proof. If I = {O}, Ill(' th{'orpm is trivially tmc, for till' zpro ideal ({O}, il:! the print"ipal idml g('ncrateu by O. Suppose then t hal I uoel:! not (~()nsist of the zero clement alone. :\'ow, if mEl, -m also belong;; to I, so that I contains pOliit.iv(' illtq(crli. Let It deHignlLte the lellst pmdtiv(' intl·ger in I. As (I, +,.) il:! lUi ideal, eaeh intel!;rul multiple of n must. be in I, UlIlt is, (n) ~ I. On the otIl('r hand, any integer k E I may be exprel-lscd as k = qn + r, whpre q, 1" E Z and () ~ r < n. Hinee k and qn are nwmbers of I, it follows that k - qn = TEl. Our definition of the integer n implies T = 0, and consequently k = qn. Thus every member of I is a multiple of n implying that I ~ (n). The two inclu!!ions !!how I = (n), eompleting the argument. Actually, It !lllll'h shortl'r proof of thl' fOl·l'l(oilll( '·I·HUIt. ('lIuld hI' lIiltnilll'(\ using Tlworclll :!-:!4; till' proof, as given, hILS till' advantage of hl'ing HI·lf ('ontainI'd. By It principal ideal Tina is meant a commutative ring with identity in which every ideal is principal. It is apparent from Theorem ;~-11 that the ring of integers constitutes IL principal ideal ring. Now, I:!Uppose that ai, a2, ... , an arc nonzero c1ement.s of (R, a commutative ring with identity. An clement a E R il:! said to be a common multiple of aI, a2, ... , an provided a is a ring multiple of Clwh of t.hese. For instance, the products al . a2 ... an and - (al . a2 ... an) are both common multiples of all a2, •. . , an. We shalL cull the element a It least common multiple of all a2, .•. ,an if (1) a is It common multiple of these clements and (2) any other common multiple of a" a2, ... , an is a multjp!l~ of a liS well. We make immediate use of this terminology to prove the next theorem.

+, .),

Theorem 3-12. Let aI, a2, ... , an be nonzero elements of ring (R, I,·). Theil

It

principal ideal

(n(ai),+,·) = «a),+,·), where a is a lea!!t eommon multiple of ai, a2, ... , an.

+, .).

PToof. According to Theorem 3-9, the triple (n(ai), +,.) is un ideal of (R, But every ideal of (R, is a principal ideal; hence, there exists an element a E R for which (a) = n(a;). Since (a) ~ (aj) [i = 1,2, ... , nl, a = T;· a; for some T; E R. We t.hus I"ondude that a is a common multiple of all a2, ... , an· Next, assume b il:! any common multiple of aI, a2, . .. , an, say b = s;· ai, where S; E R [i = 1,2, ... , n]. If r E R, then

+, .)

r· b = r· (s;· ai) = (r· s;)· a; E (a;), which shows (b) ~ (aj) for eaeh value of i. TheJ"{·fo!"1' (b) ~ n(aj) = (a) aAd accordingly b must be a multiple of a. Our argllntellt. establi!;hes that a is a least common mult.iple of ai, a2, ... , a".

+, .)

To illustrate thi!; theorem, eonsider the prinl"ipal ideals «4), and «6), +,.) generated by the integer!; 4 and 6 in the ring (Z, +, .). The reader

160

3-2

RING THEORY

can easily verify that (4)

n (6), +, .) = (12), +, .),

where 12 is the least common mUltiple of 4 and 6. We now turn our attention to the matter of cosets in a ring. If (1, +, .) is an ideal of the ring (R, +, .), then, since addition is commutative, the system (I, +) is a normal subgroup of (R, +). Thus by the results of Section 2-5, we may construct the quotient group of R by 1. In our present notation, the cosets of 1 in R &88Ume the form

+ iii E I}, By Thcorem 2-27, two cosets a + I and b + I a

+1 =

{a

where a E R. are .equal if and only if a - bEl. As before, the collection of distinct cosets of I in R shall be denoted by RI I. It follows from Theorem 2-33 and Problem 14, Section 2-5, that if addition of cosets is defined by the rule (a

+ 1) + (b + I) =

(a

+ b) + I,

then (RII, +) becomes a commutative group. An operation of mUltiplication can a1ao be introduced in RII in a natural way with the result that a ring is obtained i all we need to do is to specify (a

+

I) • (b

+ I) = (a· b) + I.

Because (I, +,.) is an ideal, this definition of coset multiplication is well defined and does not depend on the particular representatives of the cosets used. Indeed, suppose that

a+I=a'+1 and

b+ 1= b'+I. Then, as observed above, a - a' this, we conclude that

a· b - a' . b'

= a· (b -

= it and b -

b' = i 2, where i" i2 E I. From

b/) + (a - a') . b'

= a· i2 + it . b' E

I,

since both the products a· it and i 2 • b' are in I. Consequently,

a•b

+1 =

a' • b'

+ I.

The closure of I under multiplication by arbitrary elements of R thus leads to a meaningful definition of coset multiplication j indeed, this is the principal reason for defining an ideal as we did.

+, .) is an ideal of the ring (R, +, .), +, .) is a ring, known as the quotient ring of R by I.

Theorem 3-13. If (l,

(R/ I,

then the system

3-2

IDEALS AND QUOTIENT RINGS

161

We omit the details of the proof and merely point out that the zero element of (R/I, +,.) is the coset 0 + I = I, while -(a + I) = (-a) + I. Example

«n), +,

3-17. In the ring (Z,

+,.) of integers, consider the principal ideal

.), where n is a nonnegative integer.

The cosets of (n) in Z take the

form a+ (n)

=

{a+ 1m I k eZ}

=

[a].

That is, the cosets are precisely the congruence classes modulo n. It follows from the definition of coset addition and multiplication that the quotient ring of Z by (n) is merely the ring of integers modulo n: (Z,.,

+,., .,.) =

(Z/(n), +, .).

A homomorphism between two rings (R, +,.) and (R', +', .'), as one might expect, is a function f: R -+ R' which preserves both ring operations. This amounts to applying the familiar homomorphism concept to the additive groups (R, +) and (R', +'), and to the multiplicative semigroups (R,·) and (R', .'). The precise definition follows. Definition 3-12. Let (R, +,.) and (R', +',.') be two rings and f a function from R intO R'; in symbols, f: R -+ R'. Thenf is said to be a (ring) h0momorphism from (R, +,.) ~to (R', +',.') if and only if f(a

+ b) =

f(a)

+' feb),

f(a· b) = f(a) .' feb)

for every pair of elements a, b e R. Before proving any theorems concerning homomorphisms between rings, we pause to examine a few examples. Example 3-18. Let (R, +,.) and (R', +', .') be arbitrary rings and f: R -+ R' be the function that maps each element of R onto the zero element 0' of (R', +', .'). A simple calculation shows thatf is operation-preserving:

f(a

+ b) =

f(a· b)

0' = 0' +' 0' = f(a)

+' feb),

= 0' = 0' .' 0' = f(a) .' feb),

a. b e R.

As with the case of groups, this mapping is called the trivial homomorphism. 3-19. The mapping f: Z -+ Z. defined by f(a) = 2a is not a homomorphism from (Z, +..) into (Z., +, .), for while addition is preserved. multiplication is not:

Example

f(a

+ b) =

2(a + b)

=

2a + 2b

= f(a) + feb),

but f(a· b)

=

2(a· b) ~ (2a). (2b) = f(a) • feb).

162

3-2

lUNG THEORY

+, .),

+..,

Example 3-20. Consider (Z, the ring of integers, and (Z", 'n), the ring of integers modulo n. Define I: Z - Zn by taking I(a) = [a]; that is, map each integer into the congruence class containing it. Then I(a

+ b) = [a + b] =

I(a· b)

+n [b) = f{a) +.. feb), = [a· b) = [a]'n [b] = I{a) '"/(b), [a]

so that I is a homomorphic mapping. Example 3-21. Let (R, +,.) be any ring with identity. For each invertible ('Iement a E R·, the function la: R - R given by

la(x) = a· X· a-I

is a homomorphism from (R, +,.) into itself. Indeed, if x, y E R, we see that la(x

+ y) =

la{x, y)

=

a· (x

+ y) . a-I =

a· (x· y) . a-I

=

a· X· a-I

+ a· y. a-I = la(x) + la(Y) ,

(a· x . a-I) . (a· y. a-I)

=

la{x) . la{Y) ,

showing that la has the asserted property. The next theorem gives the ring-theoretic ana.logs of Theorems 2-38 and 2-39. We shall give no details, since the proof follows the lines of the corresponding rcRultll obtained for groups. The parts of the theorem concerning addition (~arry over with just a dlUngc in notntion. Theorem 3-14. Let I be a homomorphism from the ring (R, ring (R', -1-', .'). Then the following hold: . 1) 1(0) = 0', where 0' iR the zero element of (R', 2) I( -a) = -f(a) for all a E R. 3) The triple (f(R), +', .,) is a subring of (R', +', ").

+,.)

into the

+', .').

If, in addit.ion, (R, +,.) !\nd (R', -+', .') a.rc rings with identity elements 1 and 1', respectively, and I( R) = R', then 4) 1(1) = 1', Ii) J(a- I ) ~ J(U)-I fll/' I'llI'h invt'rtihll,I'II'llI!'n\. 11 Ell.

Two COlllllwntll regarding pnrt (4) of the theorem are in order. evident that J(a) .' l' = J(a) = J{a· 1)

First, it is

= J(a) .' J(I)

for all a in R. From this, one might be tempted to (incorrectly) invoke the cancellation law to mme\llIle that I( 1) = 1'. What iR uctunlly required is the fact thut Il\ul\.iplicntive idl'ntit.ies lUI' uni1lue. Secondly, if thl' hypo\.h!'sis tlmt. I mups ont.o t.he 1I('t. Il' ill omitted, then it can only be inferred t.lmt J(1) is t.he idl'ntity of the subring (I(R), f-', "). Tltl'

:J-2

IDEALS AND QUOTIENT RINGS

clement fell need not serve as an identity for the entire ring (R', faet, it may very well happen thatf(I) ~ 1'. We also observe, in passing, that by statement (2), I(a -

b)

= I(a)

163

+', .');

in

a, b E R.

- feb),

That is to say, a homomorphism preserves differences as well as sums and products. We shall need this fact presently. If f is a homomorphism from the ring (R, +,.) into the ring (R', +', .'), then the kernel of f is the set ker (f)

=

{a E R I f(a)

=

O'},

+', ").

where, as usual, 0' designates the zero clement of (R', Ignoring the multiplication operations in the rings, this is just the usual definition of the kernel of a homomorphism between the additive groups (R, +) and (R', +'). As before, f is a one-to-one mapping if and only if ker (f) = {O}. For our analogy between ideals and normal subgroups to be meaningful, one would anticipate that the kernel of a homomorphism is an ideal. This is indeed the content of the following theorem. Theorem 3-15. If , is a homomorphism from the ring (R, +,.) into the ring (R', +', .'), then the "triple (ker (f), is an ideal of (R, +, .).

+, .)

:~-14, 0 E ker (f), liO that the kernel is Ilonempty. any two elements a and b in ker (f); by definition, f(a) = 0' = feb). Since any homomorphic mapping between rings preserves differences, it follows that f(a - b) = f(a) - feb) = 0' - 0' = 0',

Proof. By part (I) of Thcor(!m

Now,

COli Hider

and consequently a - b E ker (f). If r is an arbitrary member of R, then f(r· a)

=

f(r) .' f(a)

=

fer) .' 0'

= 0'.

Accordingly, the product r· a E ker (f). In a like manner, we also conclude that. r· a IiI'S ill J.;('r (f). This is Imfficdellt for (k('r (f), -f-,.) to be an ideal of (Ii, I,,),

Example 3-22. COIIHiJer :til lu'bitmry rillg (Ii, -f-,') with identity clement 1 and the mapping f: Z -+ R given hy fen) = nI. A simple eomput.ation shows that f, so ddiIlC>d, is a homomorphism from the rillg of int.egers (Z, +, .) into the ring (R, +, .): fen I 11/) -

f(nm)

=

(n I 111)1 (nm)l

=

= nl

+ ml

= fen) I f(m) ,

(nm)12 = (nl)· (mI)

Since (ker (f),J , ,) is then an ideal of (Z,

+, ,),

=

fen) ·f(m).

it follows at once from

164

3-2

RING THEORY

Theorem 3-11 that

ker (J) == {n E Z I nl

=

O}

-=

(tn)

for some nonnegative integer m. A moment's reflection will convince the reader that the integer m must be the characteristic of the ring (R, +, .). In other words, the ideal (ker (J), +, .) is nothing more than the principal ideal generated by the characteristic of (R,

+, .).

In agreement with our previous use of the term, two rings (R,

+, .)

and

(R', +', .') are said to be iBOmOrpilic if there exists a one-to-one homomorphism

from the ring (R, +,.) onto the ring (R', +', .'). We indicate this by writing (R, +,.) ~ (R', +', .'). The last example, for instance, implies that any ring (R', +',.') with identity which is of characteristic zero contains a subring isomorphic to the integers; more specifically, (Z, +,.) ~ (Zl, +', .'), where 1 is the identity element of (R', As we have seen, many aspects of ring theory are considerably simplified when a multiplicative identity exists. The next theorem shows that there is no real loss in generality in assuming the presence of such an element, for every ring is isomorphic to a subring of a ring with identity. Because this result is often phrased differently, we require another definition.

+', .').

DeflnlHon 3-13. A ring (R, +,.) is imbtdded in a ring (R', +', .') if there exists some subring (S, +', .') of (R', +', .') such that (R, +,.) ~(S, +', .').

Theorem 3-16. Any ring can be imbedded in a ring with identity.

Proof. Let (R,

+, .) be an arbitrary ring and consider the Cartesian product RXZ

=

{(r, n) IrE R, n E Z},

where, as usual, Z designates the integers. If addition and multiplication are defined in R X Z by means of the equations

+ (b, m) = (a + b, n + m), (a, n) . (b, m) = (a· b + 1M + nb, nm),

(a, n)

then it is a simple matter to verify that the enlarged system (R X Z, +, .) forms a ring. This ring has a multiplicative identity, namely the pair (0, 1); for (a, n) . (0, 1)

=

(a· 0

+ la + nO, n 1) =

(a, n),

and, similarly, (0, 1) . (a, n)

=

(a, n).

Next, consider the subset R X 0 of R X Z consisting of all pairs of the form (a, 0). Since (a,O) -

(b,O)

=

(a - b,O),

(a,O) . (b,O)

=

(a· b, 0),

it foUowathat the triple (R X 0, +, .) constitutes a subring of (R X Z, +, .).

3-2

165

IDEALS AND QUOTJlIINT RINGS

The proof is completed by showing (R X 0, +,.) is isomorphic to the gi~en ring (R, To this end, define the functionJ: R - R X 0 by taking

+, .).

=

J(a)

(a, 0).

Evidently, J is a one-to-one mapping of R onto the set R X O. Furthennore, this function has the property of preserving algebraic structure: J(a

+ b) = (a + b, 0) =

J(a· b)

=

(a· b, 0)

=

(a,O)

+ (b,O) = J(a) + J(b) ,

(a,O) • (b,O)

= f(a) . J(b).

Whence, (R, +,.) ~ (R X 0, +,.) and we may regard the ring (R, imbedded in (R X Z, a ring with identity.

+, .),

+,.)

as

A point to be emphasizl..'
= a+1

for all a E R. Paralleling our work in group theory, it is possible to prove: Theorem 3-17. The natural

mapping nati is a homomorphism from

(R, +,.) onto (RII, +,.) with kernel I. Proof. By the corresponding theorem for groups, Theorem 2-42, natl is known to be a group homomorphisni of (R, +) onto (RI I, +) such that ker (natI) = I. Thus, we need simply show that products are preserved by nat/. This follows immediately from the fact that, for any a, b E R, nat/(a· b)

=

a· b

+1=

(a

+ I) • (b + I) = natl (a) . natl (b).

Hence, natl is a ring homomorphism. Without further delay, let us now derive the ring-theoretic version of the Fundamental Theorem for Groups.

166

3-2

RING THI';ORY

f is a homomorphism from the ring (R, +,.) onto the ring (R', +', .'), then (Tl/ker (f), +,.) ~ (R', +', .').

Theorem 3-18. If

Proof. Just as in the group case, we define a function 1: R/ker (f) --+ R', the induced mapping, by taking 7(a + ker (f») = f(a); it is this function which establishes the desired isomorphism. From the proof of Theorem 2-47, we already know that (R/ker (1), +) ~ (R', +') by]. To complete the argument, it remains only to settle the question of whether 1 preserves the multiplication operation in (R/ker (f), +, .); but this is straightforward:

](a+ker(f»' (b+ker(f)) =](a·b+ker(f)

= f(a· b) = f(a) .' f(b) = ](a + ker (f) .'](b +

ker

(f».

Incidentally, note that by virtue of t.he definition of the indu('oo mapping 1, any ring homomorphism f admits the factorization

Example 3-23. For a simple, but useful, example illustrating some of these

ideas, eonRider the rin~R (Z., +., .•) and (Z2, -h, '2)' We define the function Z. --+ Z2 as follows:

I:

f(O) = 1(2)

=

0,

1(1)

=

f(3)

=

1.

It is a lengthy procedure to verify that f is a homomorphism and we pass over the detni\s. ' The reader should cheek several possibilities to ·satisfy himself that this property actually holds. In the present case, the kernel of 1 is the two-element set to, 2}. Moreover, we see that Z./ker (f) = {{O, 2}, {I, 3}}.

The operation tables for the quotient ring (Z./ker (f), +, .) are as shown:

-+-

{0,2}

{I,3}

{0,2} {1,3}

{O,2} {I, :J}

{1,3} {0,2}

{0,2} {I,3}

{O,2}

{1,3}

{0,2} {0,2}

{0,2} {1,3}

Theorem 3-18 asserts that the Ryst('m (Z./ker (f), +, .) must be isomorphic to the ring of integers modulo 2; indeed, this is reasonably evident from the nature of the foregoing tables. As an applicntion of Theorem 3-JR, we propose to show that each homomorphism onto the ring of integers (Z, +, .) is entirely determined by the set

a-2

IDEALS AND QUOTIENT RINGS

167

where it 1l1lSIIInes t.he value O. Contrll.!lt this with the case of groups: both iz and -iz nrc homomorphisms from the additive group of integers (Z, +) onto itself. The kernel of each of these mappings is {O}, but clearly iz "'" -iz. We begin hy establishing a lemma which is of independent interest. Lemma. The only nontrivial homomorphism from the ring of integers (Z, +,.) into itself is the identity map iz.

Proof. Consider any homomorphism f having the asserted properties. Because I 1 (n summands), the each positive int.eger n may be written II.!! I operation-preserving nature of f implies fen) = nf(l) for each n E Z+. On the ot.her hand, if n ill lUI arhitmry negative integer, -n E Z+, and accordingly,

+ + ... +

fen)

= f(-(-n») =

-f(-n)

=

-(-n)f(1)

=

nf(l).

Finally, f(O) = 0 = Of(1). The net result of all this is that fen)

=

nf(l)

for every n in Z. As the function f is, by hypothesis, not identically zero, we must have f(l) = 1. Therefore, fen) = n = iz(n) for a.ll n, so that f is just the identity function on Z. Corollary. There is at most one homomorphism under which an arbitrary ring (R, +,.) is isomorphic to (Z, +, .).

Proof. Suppose the rings (R, +, .) and (Z, +, .) are isomorphic under two functions f and 0, where f. g: R --+ Z. Then the composition f. g-1 is a homomorphic mapping from the ring of integers (Z, +, .) onto itself (we leave the verification of this fact as an exercise). It follows at once from the lemma just proved that f. g-1 = iz or f = g.

We now have all the necessary information to prove the following result. Theorem 3-19. Any homomorphism from an arbitrary ring (R, +,.) onto the ring of integers (Z, +,.) is uniquely determined by its kernel.

Proof. Letf and g be two homomorphisms from the ring (R, +,.) onto (Z, +,.) with the property that ker (I) = ker (g). Our aim is to show that f and g must, in actual fact, be the same function. Now, by Theorem 3-18, both the quotipnt. ringll (Riker (I), +, .) and (Riker (g), +, .) are isomorphic to the' ring of int('gers under the illlltH'ed mappings 7and g, respectively. The assumption thatf lLnd g have a common kernel, taken in
=

g • natker(Q)

that the funetiolls f and g arc themselves identical.

168

3-2

RING THEORY

As a final topic in this section, we look briefly at the problem of extending a function from a subsystem to..the entire system. In practice, one is usually eoncerned with extensions which retain the characteristic features of the given function. The ncxt theorem, for instance, presents a situation where it is possible to extend a homomorphism in such a way that its extension also has this property. But first, one more definition is needed (see Problem 17, Section 3-1). DefInition 3-14. The center of a ring (R, +, .), denoted by cent R, is the center of the multiplicative eemigroup (R, .) j that is,

cent R

=

{r E R I r· a

=

a· r for every a E R}.

We are now in a position to state and prove our theorem. Let (I, +, .) be an ideal of the ring (R, +, .) and I a homomorphism from (1, +,.) onto (R', +', .'), a ring with identity. If I s;;; cent R, then there is a unique homomorphic extension of I to all of R.

Theorem 3-20.

Prool. As a starting point, we choose u E I so that I(u) = 1', where l' is the identity element of the ring (R', +', .'). Since (1, +,.) is an ideal, the product a· " will be a member of the set I for each choice of a E R. It is therefore possible to define a new function g: R -+ R' by setting yea) = I(a· u), a E R. In particular, if the element a belongs to I, then g(a)

= I(a. u) = I(a) .' I(u) = I(a) . l' = I(a),

showing that the restriction y I I agrees with I. What we are observing, in effect, is that g extends the given function J to all of R. The next thing to establish is that both ring operations are preserved by the function g. The case of addition is fairly obvious: if a, b E R, then

g(a + b)

= I«a + b) . u)

= I(a· u + b . u) = I(a· u) +' I(b· u)

=

g(a)

+' g(b).

As a preliminary step to demonstrating that g also preserves multiplication,

note that

J«a' b) • u 2 )

= I«a· b· u) . u)

== I«a' b) • u) .' I(u) == I«a· b) . u).

In licht of this, we are able to conclude

g(a· b)

= I«a' b)· u) = I«a· b)· u2 ) = I«a· u)· (b· u» = I(a· u) .' I(b· u)

= g(a) .' g(b).

The third equality is justified by the fact that u E cent R, hence commutes with b.

3-2

IDEALS AND QUOTIENT RINGS

Only the uniqueness of U remains unproved. Let us therefore suppose the function h is another homomorphic extension of f to the larger set H. Since f and h are required to coincide on I and, more specifically, at the element u, h(u) = f(u) = 1.

With this in mind, it follows that h(o) = h(o) .' h(u)

=

h(o· u) = f(o . u)

= g(o)

for all 0 in H, 80 hand 9 must be the same function. Hence there is one and only one way of extending f from the ideal (I, +, .) to the entire ring (H,

+, .).

PROBLEMS

1. Determine all ideals of (Z12. +12. '12). the ring of integers modulo 12.

2. Show by example that if (11,+.') and (12,+,·) are both ideals of the ring (R.+ • .). then the triple (11 U 12,+.') is not necessarily an ideal.

3. For any ideal (1.+,,) of the ring (R,+, .). define C(l) to be the set

C(I) - {r e R I r . a' - a . reI for all a e R}. Determine whether (C(l),+.·) forms a subring of (R,+ • .).

4. If (11,+,') and (12,+,') are ideala of the ring (R,+,') such that 11 n 12'" {O}, prove a· b = 0 for every a E 11, bE 12.

+, .). is a simple ring. ,+..,'..) of intepre modulo" It a

5. a) Verify thanlienng--orreal numbers (R', b) Prove that for each n e Z+, tho rin, (Z .. principal ideal ring.

6. Let (1,+,,) be an ideal of the ring (R,+.·) and define ann 1 :- {r E R ! r • a - 0 for all a E l}. Prove that the triple (ann 1,+.,) constitutes an ideal of (R,+ • •). called the

annihilator ideal of 1. 7. Let (1,+.') be an ideal of (R.+, .), a commutative ring with identity. For an arbitrary element a in R, the ideal generated by 1 U {a} is denoted by «(I, a), Assuming a Ii! I. show that

+, .).

(I,a) - {i+r·a!iEI.rER}.

8. Suppose (II, +,.) and (12,+,') are ideals of the ring (R.+, .). Define

/11+12 - {a+blaElt,bE I 2}.

+

Show that (11 12,+.·) is an ideal of the ring (R.+ • .); in Tact, (11 is the ideal generated by 11 U 12.

+ 12.+,·)

170

3-2

RING THEORY

9. In the ring of integers, consider the principal ideals «n),+,·) and «m),+,·) generated by nonnegative integers nand m. Using the notation of the previous two problems, verify that «n), rn) = «rn), n) = (n)

+ (m)

= ({m, n}) -

(d),

where d iR !.he grl'at.l'lit common rlivillor of nand m. 10. ConRider th,' ring (I'(X), ~,n) of ExamJllll3-4. For a fixed BuhllCt.~!;; X, definll the function I: I'(X) -+ P(X) by I(A) = Ii noS.

Show that I ill a homomorphism and determine its kernel. 11. THilizI! Prohl/'1Il 19, Hl'dinll 3-1, to Rhnw that in an integral domain (R, characteristic p > 0, the mapping

I(a) = aP ,

+, .) of

aE R,

+, .)

is a homomorphism of (R, into itself. 12. Given thatf is a homomorphism from the ring (R, onto the ring (R', prove that a) if (1, is an ideal of (R, then the triple (f(l), is an ideal of

+, .)

+', .'),

+, .) +, .), +', -') b) if (1', +', .') is an ideal of (R', +', -'), then the triple (f-l(1'), +,.) is an ideal of (R, +, .) with ker (f) !;;f- 1 (1'), c) if (R, +, .) is a principal ideal ring, then the same is true of (R', +', -'). [Hint: ForaER,f«a» = (f(a».] 13. Let I be a homomorphism from the ring (R, +, .) into itself and oS be the set of (R',+',·'),

elements that are left fixed by /:

oS = {a E R I f(a) = a}. Establish that (oS,+,·) is a subring of the ring (R,+, .). 14. For a fixed element a of (R, +, .), a ring with identity, define the le/t-muUiplicatio7J /unction/.: R -+ R by taking fG(x) = a· x,

xE R.

If F H denotes the set of all sllch functions, prove the ring analog of Cayley's theorem: a) The triple WH, forms a ring. where denotes the usual pointwise addition of funct.ions and. denotes functional composition. b) (R,+,·) ~ (FH,+, .). [Hint: Consider the mappingf(a) = fG.] 15. Let CR, be an arbitrary ring and (R X Z, be the ring constructed in Theorem 3-16. Establish that a) (R X 0, +',.') is an ideal of (R X Z, +', .'), b) (Z, ~ (0 X Z, e) if a is an idempotent element of R, thl'n the pair (-a, 1) is idempotent in the and (a, 0) .' (-a, I) "" (0,0). ring (R X Z,

+..)

+, .)

+

+', .')

+, .)

+', .'),

+', .')

3-2

IDEALS AND QUOTIENT RINGS

171

16. Suppose (/,,+,.) and (/2,+,') are both ideals of the ring (R,+, .). Define the set 1,·12 by

+, .)

where:E denote!! a finitl~ Rum with olle IIr IlIOfe term",. Prove that (h, 12, is an ideal of (R, -+, '). 17. GiVl'II that. (/. -I, .) iH all idllal of the ring (R, -/-, .), Khow that a) whenevllr (ll, -I,,) ill Cllllllllutative with idellt.ity, then 1111 ill the quotiont ring

(Rl1, +, .),

h) the ring (Rl1, +,.) lIIay have ,livitlorll of Z(,fO, I,vell though (R, -t,.) does not hav" a.ny, c) if (R, +,.) is a principal ineal rinJ!;, then so is the quotient ring (RIT,+, .). IH. IAlt (ll, I , .) till " (!IIlIIlIIutu.tive rillK wit.h hlt'lltity u.nl! I"', N ,It'notl' the 81~t of nilpotent clements of R. Verify that a) the triple (N, is an ideal of (R, [lJint: If an = b'" = 0, consider

+,.)

+, .).

(a - b)n+"'.)

b) the quotient ring (RIN, +,.) has no nonzero nilpotent elements. 19. Assume (R, is a ring with the property that a 2 a E cent R for every element a in R. Show that (R, is a commutative ring. [Hint: Make use (a b) to prove, first, that a' b b . a lies in the of the expression (a b)2 center.) 20. Illustrate Theorem 3-18 by considering the rings (Zo, +6, '0), (Za, +a, 'a), and the homomorphismf: Zo -> Za defined by

+, .)

f
+

+, .) + + +

= f(3) = 0,

1(1)

= f(4)

+

= 1,

f(2) = f(5)

=

2.

+, .) be a subring and (1, +, .) an ideal of the ring (R, +, .). Assuming +, .) is isomorphic to a subring of the quotient ring (Rl1, +, .). [llint: Use the mapping f(a) = a + I, where a E S, in conjunction

21. Let (S,

S n 1 = {O}, prove that (S,

with Theorem 3-18.) 22. a) Let f be a homomorphism from the ring (R, +,.) into the ring (R', +', .'). Given that a E R is nilpotent, show its image f(a) is nilpotent in R'. b) Suppose (R, is a ring which has no nonzero nilpotent elements. Deduce that all the idempotent elements of R bcJong to the center. [llint: If a2 = a, then (a· r . a - a· r)2 = (a· r . a - r' a)2 = 0 for all r E R.J 23. A ring (R, ill Raid to be the direct BUm of the two ideal!! (/ .. and (12, indicated by writing R = 11 ~ 12, if R = 11 + 12 with h n 12 = {O}. Given R = I, ~ 12, show that a) It· 12 = 12' I, = {O}, b) every element a E R can be uniquely expressed as a sum

+, .)

+, .)

+, .)

+, .),

where 24. Let (R, +,.) hI' a commut.ative ring with identity and a E R he an idempotent whil'h ill dilT('rl'nl from II IIr I. Prove that (R,I , .) is thl' direct 1111111 of the principal ideals «a),+,') and «(I - a),+,.): R = (a) ~ (I - a). [llint: Utilize the fact a' (I - a) = 0.)

170

3-2

RING THEORY

9. In the ring of integers, consider the principal ideals «n),+,') and «m),+,') generated by nonnegative integers nand m. Using the notation of the previous two problems, verify that

=

«n), 711)

«m), n)

=

(n)

+ (m)

... ({m, n}) - (d),

where d ill the grl'atl'Rt common rlivisor of nand m. 10. ConMilier t1w rinJl; (1'(:\), ~, n) of Example 3-4. For a fixed subset S ~ X, definn the function I: I'(X) -+ P(X) by I(A)

== An 8.

Show that I is a homomorphism and determine its kernel. II. Utilize Prohll'll1 10, HI'(:tion 3-1, to Hhnw that in an integral domain (H,

characteristic p

> 0, the mapping I(a) = a",

+, .) or

aEH,

+, .)

is a homomorphism of (R, into itself. 12. Given that f is a homomorphism from the ring (R, onto the ring (R', prove that a) if (1, is an ideal of (R, then the triple (f(l), ,I) is an ideal of

+, .)

+, .),

+, .)

(R', +', .'),

+', .'),

+',

b) if (I', +', .') is an ideal of (R', +', ,I), then the triple (f-I(1'), +,.) is an ideal of (H, +,.) with ker (f) ~rl(1'), c) if (H,+,') is a principal ideal ring, then the same is true of (R',+', ,f). [Hint: For a E H, f( (a» = (f(a) ).J 13. Let f be a homomorphism from the ring (H, into itself and 8 be the set of elements that are left fixed by f:

+, .)

8

=

{a E R I f(a)

=

a}.

+,.)

+, .).

Establish that (8, is a subring of the ring (R, 14. For a fixed element a of (R, +, .), a ring with identity, define the lelt-multiplication functionf,,: H -+ R by taking f,,(z)

= a· z,

zE R.

If Fit denotes the set of all such functions, prove the ring analog of Cayley's theorem: a) The triple Wit, +, 0) forlns a ring, where + denotes the usual pointwise addition of funct.ions and denotes functional composition. b) (R,+,') ~ (FR,+, 0). [Hint: Consider the mappingf(a) = f ... ) 15. Let (R, be an arbit.rary ring and (R X Z, be the ring constructed in Theorem 3-16. Estahlish that a) (H X 0, is an ideal of (R X Z, ,I), b) (Z, ~ (0 X Z, c) if a is an idempotent element of R, then the pair (-a, 1) is idempotent in the and (a, 0) .' (-a, 1) >= (0,0). ring (R X Z, 0

+, .) +', .')

+, .)

+', .')

+', .')

+', .'),

+',

3-2

IDEALS AND QUOTIENT RINGS

+, .) and (/2, +, .) are both ideals of the ring (R, +, .).

16. Suppose (/1, set h . 12 by

171

Define the

+, .)

wh('re :E llenott~1'I a finitl~ slim with 0111' IIr 1II0fll terlllH. Prove that (Ii· 12, is an ideal of (R,-t, '). 17. Give'll that. (/,1,') iH nil ideal of the ring (H,+, .), Khow that a) whcnevI~r (Ii, -I,,) ill commutative with idclltity, then 1111 ill the quotiont ring (HI I, +, .), h) the ring (RI I, +,.) may hav{, divisors of Z(,fO, nVCJl though (H, +,.) dOCB not have, II.ny, c) if (R, is a principal icieal rin!!:, then so is the quotient ring (RII, I~. (A,t (Ii, I , .) he a e'"lIIlIlutat,ive rinK with hit'ntity and II!\. N ,Ie'uotl' tho set of nilpotent clements of H. Verify that a) the triple (N,+,') is an idell.l of (H,+, .). [/lint: If an = b'" = 0, consider

+, .)

(a -

+, .).

b)n+ .... ]

+, .)

b) the quotient ring (HIN, has no nonzero nilpotent elements. 19. Assume (R, is a ring with the property that a2 a E cent R for every element a in H. Show that (H, +,.) is a commutative ring. [Hint: Make use (a b) to prove, first, that a' b b . a lies in the of the expression (a b)2 center.] 20. II1ustrate Theorem 3-18 by considering the rings (Z(l, +6, '6), (Za, +a, '3), and the homomorphism f: Z6 --+ Za defined by

+, .)

+

+ + +

f
= f(3) = 0,

+, .)

+

f(l) = f(4) = 1,

+, .)

f(2) = f(5)

=

2.

+, .).

21. Let (S, be a subring and (I, an ideal of the ring (R, Assuming S n I = {O}, prove that (S, is isomorphic to a subring of the quotient ring (RI I, +, .). [/lint: Use thc mapping f(a) = a + 1, where a E S, in conjunction with Theorem 3-18.] 22. a) Let f bc a homomorphism from the ring (R, +,.) into the ring (R', +', .'). Given that a E R is nilpotent, show its image f(a) is nilpotent in R'. b) Suppose (R, is a ring which has no nonzero nilpotent elements. Deduce that all the idempotl'nt elements of R belong to the center. [/lint: If a 2 ... 0, then (0' r . 0 - o· r)2 = (0' r . 0 - r· a)2 = 0 for 11.11 r E R.] 23. A ring (H, iH Rair\ to be the direct Bum of thc two idealll (/ .. and (/2, indicatl'd by writing H = II ~ lz. if R = I, + 12 with Ii n 12 = {O}. Given R = II ~ 12, show that a) II' lz = 12' 11 = {OJ, b) every clement a E R can be uniquely expressed as a sum

+, .)

+, .)

+, .)

+, .)

+, .),

where 24. I.Rt (H,+,') he a commutativ(' ring with identity and aE R he an idempotent whil·h i" rliff,'rl'nt fWIII () IIr I, Prove that (Ii, . f-, .) is the direct slim of t.he principal ideals «a), +, .) and (0 - a), H = (a) ~ (l - a). [llint: Utilize the fact o· (1 - a) = O.J

+, .):

172

3-3

RING THEORY

3-3 FIELDS

In the preceding two sections, a hierarchy of special rings has been obtained by imposing more and more restrictions on the mUltiplicative semigroup of a ring. One might be tempted to require that the multiplicative semigroup actually be a group. Such an assumption would be far too demanding, for this situation can only take place in the trivial ring consisting of the zero element alone. It turns out, however, that there do exist rings in which the nonzero elements form a group under mUltiplication. This leads us to the notion of a field.

+,.) is said to be a field provided the pair {O},·) forms a commutative group (the identity of this group will be

DefInition 3-15. A ring (F,

(F -

written as 1). It should. be evident that any field (F, +, .) must contain at least one nonzero element, for F - {O} is nonempty, being the set of elements of a group. Moreover, since a· 0 = 0 = O· a for every a e F, all the elements of F commute under multiplication and not merely the nonzero elements. In brief, a field is a commutative ring with identity in which each nonzero element has an inverse under multiplication. Observe also that if the zero element were allowed to possess a multiplicative inverse, then 0 = a· 0 = 1 for some element a e Fj this would imply F =' {O}, contrary to our convention that any ring with identity contains more than one member. As before, to distinguish the two inverses of a nonzero element a in F, we shall denote the mUltiplicative inverse by a-I and use the notation -a for its inverse relative to addition.

(R', +,.) and (Q, +, .), where + and· indicate ordinary addition and multiplication, are examples of fields.

Example 3-24. Both the systems

Example 3-25. Let F be the set of real numbers of the form a + by'3, with a and b rational: F = {a + by'3 I a, b e Q} . It is straightforward to check that the triple (F, +,.) is a commutative ring with identity (see Example 3-3). The additive and multiplicative identity elements in this case are

0= O+oy'3, To show that (F, +,.) is a field, we must verify that each nonzero element of F has an inverse belonging to F. Suppose then that a + by'3 e F, where a and b are not both zero. Under these circumstances, a 2 - 3b2 ,.s 0, for otherwise y'3 would be rational. This means that

a-bV'3 a+bV3a-bV3 1

::= a2 _

a

3b 2

, -b

+ a2 _

_ r.;

3b 2 v 3

e

F.

3-3

J'IBLD8

173

Since a/(a' - 3b') and -b/(a' - 3b') are both rational numbers, the resulting inverse docs have the required fonn to be a member of F. Note that if a and b were restricted simply to the set of integers, then (F, +, .) would no longer be a field, for then the element a -b a' - 3b' + a' - 3b'

va3

would not necessarily lie in F. Example 3-26. Consider the set 0 = R' X R' of ordered pairs of real numbers. To endow C with the structure of a field, we define addition and multiplication by (a, b)

+ (e, d) = (a + e, b + d),

(a, b) • (e, d)

=

(ae - bd, ad

+ be).

The reader may verify without difficulty that the triple (0, +, .) is a commutative ring with identity. Here the pair (1,0) serves as the mUltiplicative identity and (0,0) is the zero element of the ring. Now, suppose (a, b) is any nonzero member of C. Sioce (a, b) ~ (0,0), either a ~ 0 or b ~ 0, 80 that a2 + b' > OJ thus

for we plainly have a (a, b)· ( a 2 b2

+

'

a2

-b) + 00) = + = (a + + b -00 a + b2

2

2

a2

b"

l

b2

(1,0).

This shows that the nonzero elements of C have inverses under multiplication, proving the system (C, +, .) to be a field. The field (C, +, .) contains a subring which is isomorphic to the ring of real numbers. For if R' X 0 = {(a, 0) I a e R'}, it follows that (R',

+,.)

~

(R' X 0, +,.) via the mapping I defined by

I(a)

=

(a, 0),

aeR'.

(Verify this!) As the distinction between these systems is one only of notation, we customarily identify the real number a with the corresponding ordered pair (a, 0) j in this sense, (R', +, .) may be regarded as a subring of (C, +, .). The definition of the operations + and· enables us to express any element (a, b) E C a s ' (a, b)

=

(a,O)

+ (b,O) . (0, I),

where the pair (0,1) is such that (0,1)'

=

(0,1)· (0,1)

=

(-1,0).

lntto-

174

3-3

RING THEORY

ducing the symbol i as an abbreviat.ion for (0, 1), we thus have (a, b) = (a,O)

+ (b, 0) . i.

If it is agreed to replace pairs of the form (a, 0) by the first component a, this reprcsentation heeomcs (a, b) = a -1- bi,

with i 2 = - 1. In otlH'r WOrthl, the field (e, +, .) more than the familiar complex number system.

lUi

defined above is nothing

The following th('ormn shows that a field is wit.hout divisors of zero, and consequently it! a system in which the cancellation law for multiplication holds (see Theorem 3-5). Theorem 3-21. If (P,! , .) is a field ILnd a, b E F with a' b = 0, then cithcr a = 0 or b = O.

Prooj. If a = 0, the th('on'm is alreudy est.ublish(!d. So let us suppose that ;r6 0 and prove that b = O. By the definition of a field, the element a, being nonzero, must have a multiplicative inverse a-I E F. The hypothesis a· b = 0 then yields

a

o=

a-I. 0

=

a-I. (a· b) = (a-I. a) . b

=

1 . b =-b,

as desired. Since a field is a commutative ring with identity, and we have just proved that it contains no divisors of zero, we conclude that any field is an integral domain. There obviously are integral domains which are not fields; for instance, the ring of integers. However, an integral domain having a finite number of clements must necessarily be a field. Theorem 3-22. AllY finite integral domain (R,

+, .) is a field.

Pl'Ooj. Suppo!!C at, a2, ... , an are the members of the set R. For a fixed nonzero clement a E R, consider the n products a· alt a· a2, ... , a· an. These products are all distinct, for if a· ai = a· aj, then aj = aj by the cancellation law. It follows that each clement of R is of the form a· ai. In particular, there exililts some aj E R stwh t.hat a· aj = 1; since multiplication is commutative, we thus have ai = a-I. This shows that every nonzero element of R is invertible, so (R, +, .) is a field.

+.. , '..)

It was previously seen that for each positive integer n the system (Zn' is a commutative ring with identit.y. Our next result indicates for precisely what values of n this ring is a field. Theorem 3-23. The ring (Z", 1-", 'n) of integers modulo n is a ficld if and only if n is 1\ prime number.

3-3

FIELDS

175

Prooj. We first show that if n is not prime, then (Zn, +n, 'n) is not a field. Thus assume n = a· b, where 0 < a < nand 0 < b < n. It follows at once that [a)

'n

[h)

=

[a· b]

=

[n]

=

[0),

although hoth [a) .,& (0), [b) .,& [0]. This means that the system (Zn, +n, 'n) is 1I0t lUI intl'p;ral domain, and hcnce not a fidd. On Ow ol,hN hand, SIlPJlOSP that n is It prime number. To show that (Z", -I n, ',,) iH a fidd, it. sufliccs to pl'ove here that each 1l0llZt'ro e1emcnt of Z .. has a multiplicative inverse in Zn. To this end, let raj E Zn, where 0 < a < n. Aeeording to Theorem 1-1:~, Rinee a and n have no common factors, there exist intcgerH r and II lmdl tlmt

a·r+n·8=1. Thill implies tlmt. [a]· .. [rJ = [a· r] -1-.. [0] = [a· 1'] -1-.. [n· s]

= [a· r -I- n· 8] = [1], showing the congruence class r1"] to be the multiplicative inverse of [a]. Therefore (Zn, +n, 'n) is a field, aH required. There is an interesting relationship between fields and the la.ck of ideals; what we shall show is that fields have as trivia.l an ideal structure as possible. Theorem 3-24. Let (R, +,.) be a commutative ring with identity. Then (R, +,.) is a field if and only if (R, +,.) has no nontrivial ideals.

+, .) +, .)

Proof. ASHume first that (R, is a field. We wiHh to show that the trivial ideals ({O}, +, .) and (R, are its only ideals. Let us assume to the contrary that there existH some nontrivial ideal (I, +,.) of (R, +, .). By our assumption, the subRCt I is sueh that I .,& {01, and I .,& R. This means there is some nonzero clement a E I. Since (R, +, .) is a field, a has a multiplicative inverse a-I E R. By the definition of ideal, we thus obtain a-I. a = 1 E I, which in turn implies I = R, contradicting our choice of I. Conversely, suppose that the ring (R, +,.) has no nontrivial ideals. For an arbitrary nonzero clement a E R, consid!'r the principal ideal «a), +, .) generated by a: (a) = {r· a IrE R}. \

Now «a), +,.) cannot be the zero ideal, since a = a· 1 E (a), with a"& O. It follows from the hypothesis that the only other pORsibility is «a), = (R, that is, (a) = R. In particular, since 1 E (a), there exists an element ,. E R for whieh i'. a =, 1. Multiplication is eommulative, so that,. = a-I. Hence e:wh nonzero {'Iement of R hn" a lIlult.iplientive inverse in R.

+, .);

+, .)

176

3-3

RING THEORY

+, .)

In view of this last result, the ring of integers (Z, fails to be a field, since it possesses the nontriviaNdeal (Z., Theorem 3-24 is useful in revealing the nature of homomorphisms between fields.

+, .).

Theorem 3-25. Let f be a homomorphism from the field (F,

+, .)

onto the field (F', Then either f is the trivial homomorphism or else (F, and (F', +', .') are isomorphic.

+', ").

+, .)

Proof. The proof consists of noticing that since (ker (1), +, .) is an ideal of the field (F, +, .), either the set ker (J) == {OJ or else ker (f) = F. The condition ker (1) = {OJ impliesf is a one-to-one function, in which case (F, +, .) ~ (F', +', .') via f. On the other hand, if it happens that ker U) == F, then each element of the field (F, morphism.

+, .) must map onto zero i that is, f is the trivial homo-

Plainly, any ring with identity which is a subring of a field must in fact be an integral domain. We now tum our attention to the converse situation i specifically, one may ask whether each integral domain can be considered (apart from isomorphism) as a subring of some field. More formally, can a given integral domain be imbedded in a field? In the finite case, there is obviously no difficulty, since every finite integral domain already forms a field. Our concern with this problem arises from the desire to solve equations of the type a . x = b, a #- O. A major drawback to the notion of an in~l domain is that it does not always provide us with a solution. Of course, any such solution would' have to be unique for a· XI = b = a· X2 implies XI = X2 by the cancellation law. It hardly seems necessary to point out that when the integral domain happens to be a field, there is always a solution of the equation a . X = b (a #- 0), namely X = a-I. b. We begin our discussion of this question with a definition. Definition 3-16. By a aubfield of the field (F,

(F', +,.) of (F,

+,.) is meant any subrinl(.

+,.) which is itself a field.

+, .) of rational numbers is a subfield of the field Surely, the triple (F', +, .) will be a subfield of the field (F, +, .) provided For example, the ring (Q,

(R', +, .).

(1) (F',+) is a subgroup of the additive group (F,+) and (2) (F' - {O},·) is a subgroup of the multiplicative group (F - {O},·). Recalling our minimal set of conditions for determining subgroups (Theorem 2-17), we see that (F', +,.) will be a subfield of (F, +,.) if and only if the following hold: 1) F' is a nonempty subset of F with at lcast one nonzero clement. 2) a, b E F' implies a - b E F'. 3) a, b E F', where b #- 0, implies a· b- I E F'.

+, .)

It should come as no surprise that if (Fi' is an arbitrary collection of subfields of the field (F, +, -), then (n Fi, + •. ) is also a subfield.

3-3

I'IIILDS

177

The next theorem furnishes some clue to the nature of the field in which we wish to imbed a given integral domain. Theorem 3-26. Let the integral domain (R, field (F, If the set F' is defined by

, +, .).

F'

=

+, .)

{a· b- 1 I a, b E Rj b

~

be a subring of the

O},

then the triple (F', +,.) fonns a subfield of (F, +,.) such that R fact, (F', +,.) is the smallest subfield containing R.

~

F'. In

Proof. Note first that the definition of the set F' is meaningful; indeed, if a, b E H with b ~ 0, the product a· b- 1 must be in F by virtue of the fact (F, +,.) is a field. Since 1 = 1.1- 1 E F', F' ~ {O}. Now consider two arbitrary elements x, y of F'. We then have

for suitable a, b, e, d E R, where b x -

~

0, d

~

O. A simple calculation shows

y.= (a· d - b· c) . (b· d)-l E F'.

Also, if y is nonzero (that is, whenever c

~

0),

= (a· d) . (c· b)-l E F'.

X· y-l

In light of the remarks following Definition 3-16, this is sufficient to establish that the triple (F' , +, .) is .a subfield of (F, +, .). Furthennore, a =

a· 1 = a· 1-1 E F'

for each a in H, so that R S;;; F'. Any subfield of (F, +, .) which contains R necessarily includes all products a· b- 1 with a, b '" 0 in R, hence contains F'. Theorem 3-26 began with an integral domain already imbedded in a field. In the general case, it is actually necessary to construct the imbedding field. Since the expression a· b- 1 may not always exist, one must now work with ordered pairs (a, b), where b ~ O. Our thinking is that (a, b) will playa role analogous to a· b-1 • As a starting point, let (R, +, .) be an arbitrary integral domain and K the set of ordered pairs, K

=

{(a, b) I a, bE Hi b ~ O}.

A notion of equivalence may bc introduced in K as follows: (a, b)

= (e, d)

if and only if

a·d=b·e.

(We have in mind the foregoing theorem in which a· b- 1 if a· d = b· c.)

= c· d- 1

jf and only

178

3-3

RING THEOHY

It is not difficult to verify that the relation rl'lation in K; that is to sny,

=,

thus defined, is an equivalence

1) (a, h) = (a, h), 2) if (a, h) == (e, d), then (e,l1) == (a, h), 3) if (a, h) == (e, Ii) and (e, II) == (e, f), then (a, h)

==

(e, f).

The leaRt obvious statenlC'nt is (3). In this case, the hypothesis (a, h) and (e, rJ) = (e, f) implies that

e· f

a· d = h· e,

= d·

==

(e, d)

e.

l\Iultiplying the first of these equations by f and the second by h, we obtain

a . d . f = h . e . f = b· d· e, and, from the eomlllutativity of multiplication, a· f· d = b· e· d. Since d ~ 0, this flletor may be clUwelled to yield a· f = b· e. But then (a, h) == (e, f), as required. Next, we lahel those elements which are equivalent to the pair (a, b) by the symbol la, bl; in other words, la,

hI =

{(e, d) E K

=

{(e, d) E K

I (a, b) == (e, d)} I a . d = b· c}.

To empha.'1ize the similarity between what follows Ilnd the familiar construction of thc rational lIumi)('rs, many nuthors prefer to write alb in place of la, bl; the rcader will realize the difference is merely a matter of notation. The collection of all equivalence classes la, b] relative to == will be designated by F. From Theorem I-!i, we know that the elements of F constitute a partition of the set K. That is, the ordered pnirs of K fall into disjoint classes, with eneh dnss ('onsist illl( of (·quivnll'nt pairs, Ilnd noncquivalent pairs belong to different classes. Further, two Hueh classes la, hI and Ie, rll are identical if and only if a . d = h· e. },('t us proceed to introduee suitable operations of addition and multiplication in F. We do these by nlC'ans of the equations

la, hI

+' Ie, d]

=

la·

la,

bl·' Ir, til

=

[a· e, b ·11).

d

+ b· e, b . ri],

Note, incidentally, that sin('e b ~ 0 and tl ~ 0 imply b· d ~ 0, the right-hand sides of tiH's(' formulas are :l('tually eielllen!!; of F. WI! must, as mmnl, first justify that thpsp olwrntiolls an' well-defined. Otherwise expresS(·d, w(, 11('(,.1 to show that. t II!' sum nnd produet. are independent of til(! pnrt.i.·ular I'I('nl('lIts of H uspd ill til(' dp(init iOIl. To aehicve this, let

3-3

179

FIELDS

[a, b]

=

[a', b/] and [e, d]

=

[e ' , d/l. From the equations

a· b'

=

b· a',

=

e· d'

d· e' ,

it follows that (a· d

+ e· b) . (b' · el')

- (a ' · el'

= =

+ c' · b') . (b· el)

(a· b' - b· a'l . (d· d' ) + (c· d' - d· e' ) . (b· b') O· (d· d') + 0 . (b . b') = O.

Thus, by the definition of equality of classes, [a· d

+ e· b, b· d] =

+ e' · b', b' · d/],

[a" d'

proving addition to be well-defined. In mueh t.he [a· e, b·

dl =

~mnle

way, one can show that

[a' • e' , b' . d').

The next lemma estabHs}les the algebraic nature of the triple (F, +',

./).

Lemma. The system (F, +',.') is a field, generally known as the field of quotients of the integral domain (R, +, .).

Proof. It is an entirely straightforward matter to establish that the triple (F, is a commutative ring. We leave the reader to make the necessary verifications at his leisure, and merely point out that [0, bl serves as the zero element while [-a, b] is the negative of [a, b]. That the equivalence class [a, a], where a is any nonzero element, constitutes the multiplicative identity is evidenced by the following:

+', .')

[a, a]·' [e, d]

=

[a· e, a· d]

=

[e, d],

with [e, dl arbitrary in F. To show that every nonzero clement of F has all inverse under multiplication, suppose that [a, b] is not the zero of (F, +', .'). Then a ~ 0, whence the class [b, a] is a member of F. Accordingly, [a,

bl ., [b, a] =

[a· b, b . a]

Since the product a· b is not zero, [a· b, a . [a, b]-I

=

=

bl

[a· h, a • b].

is the identity clemcnt, so that

[b, a].

We wish to show next that the field (F, +', ./) contains a subsystem isomorphic to (R, this will estnblish the requil1'd imbedding t,heorcm.

+, .);

Theorem 3-27. Thc integral domain (R,

field of quotients (F,

+/, ./).

+, .)

CIlII

be imbedded in its

180

RING THEORY

Proof. Consider the subset F' of F consisting of all elements of the form [a, 1],

where 1 is the multiplicative identity of (R, +, .):

F'- {[a,I]laeR}.

It is readily checked that the triple (F', +', .') is a subring of (F, +', .') and, in actual fact, is an integral domain. Now, let /: R -+ F' be the onto mapping defined by lea) -

[a, 1]

(or each a E R. Since the condition [a, 1] - [b, 1] implies a· 1 - 1· b or a - b, we see that f is a one-to-one function. Moreover, this function preserves addition and multiplication: f(a

+ b) = [a + b, 1] =

f(a· b)

=

Accordingly, (R, +,.)

[a· b, 1]

~

(F',

=

[a, 1]

+' [b, 1] = f(a) +' f(b),

[a, 1]·' [b, 1] == f(a) .' feb).

+', .') under f, and the proof is complete.

Several remarks are in order. First, note that any member [a, b) of F can be written in the form [a, b) = [a, 1] .' [1, b) = [a, 1] .' [b, 1]-1.

Since the systems (R, +, .) and (F', +',.') are isomorphic, one customarily identifies the element [a, 1] E F' with the element a of R. The above equation then becomes [a, b] = a·' b- 1 • The point is this: we may now regard the set F as consisting of all quotients a.' b- 1 , with a and b '" 0 in R. It should also be observed that for any a '" 0, [a, 1] .' [b, a]

=

[a· b, a]

=

[b, 1].

Again writing [a, 1] simply as a, we infer that the equation a·' s == b always has a solution in F, namely s = [b, a) = b·' a-I. A final fact of interest is that the field of quotients (F, +" .') is the smallest field in which thc integral domain (R, +, .) can be imbedded, in the sense that any field in which (R, +, .) is imbeddable contains a subfield isomorphic to (F, +', .') (Problem 14 of this section). The field of quoticnts constructed from the integral domain (Z, +, .) is, of course, the rational number field (Q, +, .). Definition 3-17. A field which does not have any proper subfields is called a prime field.

+, .),

Example 3-27. The field of rational numbers, (Q, is a prime field. To see this, suppose (F, +, .) is a subfield of (Q, +, .) and let a e F be any nonzero element. Since (F, +, .) is a subfield, it must contain the product

3-3

FIICLDS

181

= 1. In tum, n = n· 1-1 E F for any n in Z; in other words, F contains all the integers. It follows then that every rationaillumber n/m = n. m- 1, m ;o! 0, also belongs to F, so that F = Q.

a· a-I

Exampl. 3-28. For every prime p, the field (Zp, +p, .p) of integers modulo p is a prime field. The reasoning here depends on the fact that the additive group (Zp, +p) of (Zp, +p, .p) is a finite group of prime order, and therefore has no nontrivial subgroups. We r.ondudc this section by showing tha.t the rational number field and the fields (Zp, +1" .p) are, in a certain sense, the only prime fields. The proof relies heavily on earlier results. Theorem 3-28. Any prime field (F,

+, .)

+, .),

is isomorphic either to (Q, the field of rational numbers, or to one of the fields (Zp, +p, .p), where p is a prime number.

Proof. Let 1 be the identity clement of (F, f: Z-Fby

+, .)

and define the mapping

fen) = n1 "

for any integer n E Z. Then I is a homomorphism from (Z, +, .) onto the subring (f(Z), +,.) consisting of integral multiples of 1 (Example 3-22). By Theorem 3-18, we see that (Z/ker (f),

+, .) ~ (f(Z), +, .).

But the triple {ker (/h +,'.) is an ideal of (Z, +, .), a principal ideal ring. Whence, ker (f) = (n) for some nonnegative integer n. The possibility that n = 1 may be ruled out, for otherwise I would be the trivial homomorphism j that can only happen if F = {O}. Note further that if n ;o! 0, then n must in fact be a prime number. Suppose to the contrary that n =, nln2 where 1 < ni < n (i = 1,2). Since n E ker (I), (nIl) . (n21)

=

(nln2)1 = n1

=

0,

yielding the contradiction that the field (F, +, .) has divisors of zero. (This result is not entirely unexpected, because n is the characteristic of (F, and as such must be prime.) The preceding discussion indicates that two possibilities arise: either \ 1) (f(Z), +, .) ~ (Z/(p), +, .) == (Zp, +p, .p) for some prime p, or 2) (f(Z), +, .) ~ (Z/(O), = (Z, +, .).

+, .)

+, .)

+, .)

. Turning to a closer analysis of these C8.ses, suppose first that (f(Z), ~ (Zp, +p, .p), with p prime. Inasmuch as the ring of integers modulo a prime forms a field, the subring (f(Z), +, .) must itself be a field. But (F, +, .) con~ (Zp, tains no proper subfields. Accordingly, feZ) = F and (F,

+, .)

+", .,,).

182

3-3

JUNll TIfI<:OltV

Next, consider th(~ sit.uution (f(Z), +, .) ~ (Z, +, .). Under these circumstances, the subring (f(Z), +, .) is an integral domain, but not a field. Theorem 3-26, in conjunction with the hypothesis (F, +, .) is a prime field, then implies F= {a·b-1Ia,be/(Z);b;>
+, .)

+, .)

It is now 1\ JI\m'ly routinc mattcr to Khow that the fields (F, and (Q, are isomorphi(' UlI(I(~r the mapping fI{nfm) = (nl) . (ml}-l; we leave this as an exercise.

Since each field has a prime subfield, we get the following subsidiary result. Corollary. Every field contains a subfield which is isomorphic either field (Q, +,.) or to one of the fields (Zp, +p, 'p), p a prime.

to the

PROBLEMS

I. If, + and· denote ordinary addition and multiplication, for which of thp. following 1.1. field? sets It' is (F, 1.1.) F = {a - bv'21 a, b e Z} b) F = {a+ b01 a, beQ} c) F = {a b0 c~ I a, b, ceQ} 2. In a fi!'ld (F, Hhow t.hat 1IH1 !''Iuation (,2 = a impli('!! eithl'r a - 0 or a 3. Define two binary operations * and on the set Z of integers by letting

+, .)

+ + +, .),

0

a* b = a

+b -

a,bei.

1,

Prove that the triple (Z, *, 0) forms 1.1. field. 4. In the field (e, of eomplex numbers, define the mapping rule I(a, b) = (a, -b); in other words,

+, .)

/(a+

bi) =

I: e -. e by

the

a - bi.

Determine wheth!'r the function 1 is 1.1. homomorphillm. 5. A division ring is 1.1. ring with identity in which every nonzero element has a multiis a division ring, prove that (cent H, plicative inv!'rRe. As.'Juming (H, forms a field. 6. Let (H, be an integral domain and consider the set Zl of all integral multiples of the identity: Zl = {nl In e Z}.

+, .)

+, .)

+, .)

Vl'rify that (ZI, +,.) is a fi!'ld if and only if (H, +,.) has positive characteristics. 7. a) Prove that ev!'ry field is a principal ideal rinl/:. b) Consid!'r the set of numbers H = {a by'21 a, b e Z}. Show that the ring (H, ill not a field by exhibiting a nontrivial ideal of (H,

+, .)

+

+, .).

CERTAIN SPECIAL IDEALS

183

S. Derive the following rellults: a) The identity element of a subfield is the same as that of the field. h) If Wi. it! an indexed collection of 8uhfiC'ld8 of the field (F. then (n Pi. -\ •. ) is ulso a subfil'ld of W. 9. Lei. I he a homomorphism from the field W. + .. ) into itllelf and K be the set of ell'ments left fixed by I: K = {a E P I Ita) = a}.

+.. )

+..).

+.. ).

Given K # to}, verify that the triple tK. + .. ) ill a Muhfield of (l<'.

+, .).

10. a) Cnm'lider the subset S C R' defined by S = {a -I- byp I a. b E Q; p a prime}. Hhow that (8. is a sub field of (R',+ ..). b) Prove that any subfield of (R'. must contain the rational numbers.

+..)

+..)

W. +.. ) is of characteristic

II. Prove that if the field

W. +..) has characteristic p.

p

>

O. then every subfield of

+..)

+'.. ') +..

12. Let I be a homomorphillm of the ring (R. into the ring (R'. and suppOMe (R.l.·J hll.,. a !lubring W.+.·) which is a fh·ld. Hhow that either F ~ k('r (f) or "Is(' (R'. cont.ains a lIubring isomorphic to W. J. 13. If R = {a by21 a. b E Z}, then the system (H. is an integral domain, but not a field. Obtain the field of quotients of (R, 14. Suppose the integral ddmain (R.+,·) is imbedded in the field (F'.+' • .'), say (R. ~ (R', under the mapping I. Define the set K by

+'. .')

+

+..)

+..) +, .).

+', .') K

=

{a'.' (b')-I I a', b' E R'; b' # O}.

+', .')

Prove (I) (K, is a subfield of W', +',.') and (2) (K, +', .') is isomorphic to the field of quotients of (R. [Hint: For (2), consider the function g defined by gIla. b» = I(a) .' I(b) -1 where a, bE R, b ;t6 0.) 15. Show that any fi(·Jrl ill isomorphic to its field of quotiE'nts. [Hint: Make use of the previous exercio;e with I as the identity map.]

+, .).

+', .')

are illOmorphic integral domains. then their 16. Prove that if (R,+,·) and (R', fields of quotient!! arc also i!!omorphic. 17. From Problem S(h), deduce that every field W, has a unique prime subfield. Is this result still true if (p, + .. ) is assumed mefl'ly to be a division ring? IS. Estahlillh the following a.~sertion. thereby completing the proof of Theorem 3-28: If (/", is a field of eharactcristic zero and (K, t.h" prime Imbfi('ld generated by the identity element., then (Q,+.') ~ (K,+,') via the mapping I(n/m) = (nl)· (ml)-I, where n, 111 EZ, m ;t6 O.

+, .)

+, .)

+, .)

19. Use the preceding problem to prove that any finite field (i.e., a ficld with a finite number of elements) has nonzero characteristic. 3-4 CERTAIN SPECIAL IDEALS

The present sedioll il:! largely devoted to a study of eertaill special types of ideals, most notably maximal amI prime ideals. 011 UlP whole, our hypothesis will restrict us t.o eommlltative rings with identity. The requirement is moti-

184

RING THEORY

. vated to some extent by the fact that many of the standard examples of ring theory have this property. A,further reason, which is perhaps more important from the conceptual point of view, is that the moat satisfactory and complete results occur here. We begin by making the following definition. DeflnlHon 3-18. An ideal (I, +,.) of the ring (R, +,.) is a maximal ideal provided I jI6 R and whenever (J, +,.) is an ideal of (R, +,.) with I C J ~ R, then J = R. Expressed rather loosely, an ideal is maximal if it is not the whole ring and is not properly contained in any larger nontrivial ideal. The only ideal to contain a maximal ideal properly is the ring itself. Assume, (or the moment, that (1, +, .) is a proper ideal of the ring (R, +, .) and a is an element which does not belong to 1. Then the ideal «I, a), the ideal generated by the set I u {a}, is such that I C (I, a) ~ R. These inclusions imply that if (I, were a maximal ideal, the set (I, a) must be all of R, (I, a) = R. On the other hand, suppose (J. +,.) is an ideal of (R, +, .) for which I C J ~ R. If a is an element of J not in I, then I C (I, a) ~ J. The condition (I, a) = R would therefore force J = R and we could conclude that (I, + .. ) is a maximal ideal. Summing up, the ideal (I, +, .) is a maximal ideal of (R, +, .) if and only if I jI6 R and (I, a) = R for every element a ~ I. This fact will prove quite helpful a little later. To illustrate the concept, let us show that in the ring of intege.. the maximal ideals correspond to the prime numbers.

+, .),

+, .)

+, .)

be the ring of integers and n > 1. Then the principal ideal «n). +,.) is maximal if and only if n is a prime number.

Theorem 3-29. Let (Z,

Prool. First, suppose «n), +,.) is a maximal ideal of (Z,

n is not prime, then n = nln2, where 1 < nl ideals «n1), and «n2), are such that

+, .)

+, .)

~

n2

<

+, .).

n.

If the integer This implies the

contrary to the maximality of «n), +, .). / For tho opposite direction, &8Sume now that the integer n is prime. If the ideal «n), +, .) is not maximal in (Z, then either (n) = Z or elae thero exists some proper ideal «m), +, .) with (n) C (m) CZ. The first case is immediately ruled out by the fact that 1 is not a multiple of a prime number. On the other hand, the alternative po88ibility (n) C (m) means n = km for some integer k > 1 i this also is untenable, since n is prime, not composite. We therefore conclude that «n), +,.) is a maximal ideal.

+.. ),

An additional illustration may be of some interest: Let R denote the collection of all functions I: R' -+ R'. For two such functions I and g, addition and multiplication are defined by the formulas ' U

+ g)(x) = I(x) + g(x),

(f. g)(x)

= I(x)g(x),

xER'.

CERTAIN SPECIAL IDEALS

185

Then (R, +, .) is a commutative ring with identity (see Example 3-5). Consider the set M of functions in R which vanish at 0:

M = {fE R 1/(0) =

OJ.

Evidently, the triple (M, +,.) forms an ideal of (R, +, .); we observe that it is actually a maximal ideal. For, if I e M and i is the identity map on R', one may easily check that (i 2 + 12)(x) ."" 0 for each x E R'. Hence, the sum i 2 + 12 is an invertible element of R. This implies that (M, f)

;2

(i, f)

=

R,

and therefore (M, f) = R for every Ie M [here (i, f) designates the elements of the ideal generated by i and I; that is, (i, f) = {r. i + 8 • I 1 r, 8 E R} J. Our immediate goal is to obtain a general result assuring the existence of suitably many maximal ideals. As will be seen presently, the crucial step in the proof depends upon the maximal element principle, or what is commonly termed Zorn's Lemma. It would take us somewhat far afield to do much more than merely formulate thislcmma as an axiom; the reader who wishes to pursue the topic further is directed to the comprehensive discussion in [38]. For ease of reference, let us recall that by a chain is meant a collection e of sets such that A, BEe implies either As;;; B or B S;;; A. We now state Zorn's Lemma in a form best suited to our present needs. Zorn '. Lemma. Let (1 be a nonempty family of subsets of some fixed set with the property that for each chain e in (1, the union ue also belongs to (1. Then (1 contains a set which is maximal in the sense that it is not properly contained in any member of (1.

The significant point, needless to say, is that this lemma asserts the existence of a certain maximal element without actually giving a constructive process for findjng it. Theorem ~30. (Krull-Zorn).

In a commutative ring with identity, each proper ideal is contained in a maximal ideal.

+, .), a commutative ring with

Proof. Let (I, +,.) be any proper ideal of (R, identity. Ddino n family of subsets of R by taking (1

=

{J 1 IS;;; J; (J,

+,.)

is a proper ideal of (R,

+, .)}.

This family is obviously nonempty for I itself belongs to (1. Now, consider an arbitrary chain {l;} in (1. Our aim, of course, is to establish that ul; is again a member of (1. Notice first that UI; ."" R, since 1 e I, for any i. Next, let the elements a, bE UIi and r E R. Then there exist indices i ~nd j for which a Eli, bE Ii' As the collection {Ii} forms a chain, either Ii S;;; Ii or else Ii S;;; Ii; say, for definiteness, Ii S;;; Ii' But (Ij, +,.) is an ideal,

ISH

:J-4

It! Nil 'I'll EOilY

the difference a - bE I j S;;; uh For the same reason, r· a E I j • This shows the triple (uh +, .) to be a proper ideal of the ring (R, +, .). Finally, IS;;; uI i , hellce the union UI. E Cl. Thus, on the basis of Zorn's Lemma, the family Cl contains a maximal element lIf. It follows directly from the definition of Cl that the triple (M, +, .) iR a proper ideal of the ring (R, +,.) with IS;;; M. We assert (lIf, +,.) is in faet It maximal ideal. To see this, suppose (.I, is any ideal of (R, for whi('h 111 C.l S;;; R. Sinl'c AI is a mnximal I'lenH'nt of the fnmily a, the set .I cnnllot belong to a. Accordingly, the ideal (.I, +, .) must be improper, whieh implies .I = R. We therefore cOllclude (ill, +,.) is a maximal ideal of (R, +, .), ('ompleting tIl<' proof.

:-;0

+, .)

Corollary. All idl'lll.

1'11'1111'111.

is illvC'rlihll' if

1~lId

oilly if it 1)(·lolIII;M 1.0

Proof. The C'OIl('IIlSioll i:; immediatc by Theorem

+, .)

1If)

maxillllli

a-7.

One rC'nlllrk: TIII'orl'm a-30 docs not extcllcl to rings without. identity. The pre('cding argumcnt i:; no longer adequate, since the union of a chain in a ('an not be :-;hown to be a proper subset of R. Although IlIl1ximal iil('als were defirwd for arbitrary rings, we shall abandon a degree of p;enerulity and heneeforth limit our discussion to commutative rings with idellt ity. The advantage in doing so stems from the fact that each iilC'al, otlwr thall tIl<' parent ring itself, will he contained in a m~imal id{~1. '1'11111'1, unt.il furllwr notice, Wl' shall assume that

all given

rin(/.~

are commutatille with identity,

even when t.his if< 1101. (·xplic·itly ml'nlioned. To he Rum, some of the s\lb~q\lent material coulc! he prcsC'lItl'd without this additional restridion. In a striet sl'n s<, , it. i:-; inC'orrect to speak of an ideal being eontained in a maximal iilpnl. for til!' irwlusion aet.ultlly rl'fl'rs to t.he underlying sets of elenH'nts. "Nevl'rtlH'lpss, the phra.sing is eonvenient and we ('ommit this inaccuracy frcPly. As an applieation of t.lw Krull-Zorn TheorC'm, we next I!;ive an elementary proof of a somewhat slweial result; while the fad invoivpd is rather interCt!ting, tll('rl' will be no oec'asion to nlake lise of it.. Theorem 3-31. In It ring; (fl, +, .) havinl!; eXIlC't.Iy one maximal ideal (.11. ".), HIl' only idC'lIIpolC'nt. 1'1(,1II1'1I1H are 0 and I. Proof. ASSUlll(' til(' theorl'lll iH false; t.hat. is, HUppOse t.here exists an idempotent a E R with a ¢ 0, 1. The rplatioll a 2 = a implies a· (I - a) = 0, so that a and 1 -- a af(' zl'ro divisors. Hem'c, hy Problem S(d), Redion :~-l, neither thl' elemenl a nor 1 - a iH illvl'rtible in R. But. this means the prin('ipal ideals Ua), +,.) and a), 1-,·) are bot.h proppr icipalH of the ring (R, +,.). As su(·h, t1wy mllst. he ('ontained in (III, I,'), t.hc solt' maximal ideal of

«(1 -

3-4 (R,

CEH1'AIN SPECIAL WEALS

+, .); (a) ~ 1If and (1 -

a) ~ M, Accordingly, both a and 1 -

whence 1

= a + (1

-

This leads at once to the contradiction ltf

187

a lie in M.

a) E M,

= R.

While more clementltry proofs are possible, Theorem 3-31 can be used to show 1\ field has no idempotents except 0 and 1. A full justification of this statement consists of first establishing that the zero ideal is the only maximal ideal in It field .• We now come to a characterization of maximal ideals in terms of quotient rings, It fundamen tal r(·sult. IL "rOI)(~r i«II'1I1 (If I.llI' rilll( (ll, oj, .). Then (/, "', .) is a IlIIlXillllL1 ideal if and only if the quotient ring (Ill I. +.. ) is a field.

Theorem 3-32. 1.1'1. (I, , •. ) 1)(,

Proof. To hegin with, let (I, +.. ) he a maximal ideal of (R. +, .). Since (ll. +.. ) is 1\ commutative ring with identity, the quotient ring (RII, also has these properties. Thus to prove (RI I, a field. it suffices to show each nonzp!'o element of RI I has II. multiplicative inverse. Now. if the coset a + I ~ 0 + I. then a fl"I. By virtue of the fact (I. +.. ) is a maximal ideal. the ideal «I, a). +,.) generated by I and a must be the whole. ring (R, + •. ):

+.. )

+, .)

R

=

(I, a)

{i + r· a liE I, r E R}.

=

+

That is to say, every element of R is expressible in the form i r' a, where i E I and r E R. The identity element 1, in particular. may be written as 1=~+f"a

for suitable choice of i E I. f' E R. But then the difference 1 - f" a E I. This is obviously equivalent to 1+ I

+I

+ I) . (a + I), which asserts f' + I = (a + 1)-1. Hence (RI I. +, .) is a field. For the opposite direction, suppose (RI I, +, .) is It field ami (J. +, .) is any ideal of (R, +, .) sueh that I C J !; R. The argument consists of showing =

f'. a

=

(f'

thnt J = R, for then (/,1-,') will be a maximal ideal. Since I is It proper suhset of .I, thl'l'l' «'xists lUI element. a E.I with a fl I. Consequently, the co!!Ct a + 1 ¢ 01 1, 1,lle zero clement of (Rll, +,.). All (1l11, +,.) ill assumed to bc n field, a + I must havc :m illvenic under multiplicntion. (a -1- J) . (b

+ J) = 1 + J

for some coset b + IE RI I. It then follows that 1 - a· bEl C J. But the product a· b also bdongs to J (recall a is an element of the ideal (J, +.. implying the identity 1 E J. This ill turn yields J = R, as desired,

»,

188

RING THEORY

+, .), a commutative ring without identity. In this'ring, the principal ideal «4), +,.) generated by the integer 4 is a maximal ideal. The argument might be expressed as follows: If n is any element not in (4), then n is an even integer not divisible by 4; consequently, the greatest common divisor of nand 4 must he 2. By Problem 9, Section 3-2, we then have Example 3-29. Consider the ring of even integers, (Z.,

«4), n) = (2) = Z., so that the ideal generated by (4) and n coincides with the whole ring. This reasoning shows that there is no ideal of (Z., contained strictly between «4), and (Z., +, .). Now note that in the associated quotient ring (Z./(4), +, .),

+, .)

+, .)

(2 + (4». (2 + (4»

= 0 + (4).

The ring (Z./(4), +, .) therefore has divison; of zero and cannot possibly be a field. The point we willh to make is that the asllumption of an identity element is essential to Theorem 3-32. Wo now shift our attention from maximal to prime ideals. Before formally defining this notion, let us tum to the ring of integers (Z, +, .) for motivation; specifically, consider the principal ideal «p), +, .) generated by a prime p. If ab E (p), where a, be Z, then ab = np for some integer n. Since the product ab is divisible by p, either p divides a or else p divides b (corollary to Theorem 1-4). This being so, it follows that either a

=

niP E (p)

or

b = n2P E (p)

for suitable choice of n., n2 E Z. The ideal «p), +,.) thus haa the agreeable property that whenever (p) contains a product at least one of the factors must belong to (p). This observation serves to illustrate and partly to suggest the. next definition. Definition 3-19. An ideal (1,+,,) of the ring (R,+,') is a prime ideal if for all a, b E R, a· bEl implies either a E I or bEl.

+, .) are precisely the ideals «p), +, .), where p is a prime number, ,together with the trivial ideals ({O},+,') and (Z,+,·).

Example 3-30. The prime ideals of the ring (Z,

Example 3-31. A commutative ring with identity (R, +,.) is an integral is a prime ideal. domain if and only if the zero ideal ({O} ,

+, .)

The prime ideals of a ring may be characterized in the following manner. Theorem 3-33. Let (1,

+,.)

be a proper ideal of the ring (R,

+, .).

Then

(I, +,.) is a prime ideal if and only if the quotient ring (RIl, +,.) is an integral domain.

CERTAIN SPECIAL IDEALS

189

Proof. First, take (1, +, .) to be a prime ideal of (R, +, .). Since (R, +, .) is a commutative ring with identity, so is the quotient ring (R/1, +, .). It remains therefore only to verify (R/1, +, .) is free of zero divisors. For this, assume that (a

+ 1) . (b + 1) =

I.

In other words, thll product of these (lOsets equals the zero element of the ring (R/ I, +, .). The foregoing equation is plainly equivalent to a· b I = 1, or what amounts to the same thing, a· bel. Since (1, +, .) is a prime ideal, one of the factors a or b must lie in I. But this means either the coset a + 1 = 1 or else b + I = I, hencc (R/ I, is without zero divisors. To prove the convcl'!lC, we just reverse the argument. Accordingly, suppose (R/ I, is an integral domain and the product a· bel. We then have

+

+, .)

+, .)

+ J) . (b + I) = a· b + I = I. By hypotiwsis, (R/ I, +, .) contains 110 divisors of.zero, 80 that either a + I = 1 or b + I = I. In finy event, one of a or b belongs to I, forcing (I, +,.) to be (a

n prime idolLl. There is an important 'class of ideals which are always prime, namely the maximal ideals. From the several ways of proving this result, we choose the argument given below; another approach is indicated in the problems at the end of this section. Theorem 3-34. In a commutative ring with identity, every maximal ideal

if!

0.

prime ideal. ..

Proof. Assume (I, +, .) is a maximal ideal of the ring (R, +, .) and that a· bel with a fl I. We propose to show bel. The maximality of (1, +,.) implies that the ideal generated by I and a must be the whole ririg: R = (I, a). Hence there exist elements i e I, r e R for which

l=i+r·a. Since both a· band i are in I, we conclude

+ r· a)· b ~ i· b + r· (a· b) e 1, from which it is clear that (I, +, .) is a prime ideal. b = (i

We should point out that in rings without an identity element this ~t does not remain valid; a specific illustration is the ring (Z., +, .), where «4), +,.) forms a maximal ideal which is not prime. One more definition is required: a principal ideal domain is a principal ideal ring which is also an integral domain.. Otherwise expressed, a principal ideal domain is an integral domain in which every ideal is a principal ideal.

190

3-4

RING THEORY

Let us remark that the converse of Theorem 3-34 need not hold; there exist examples of prime ideals which are not maximal ideals. The special properties of the principal ideal domains, however, guarantee that the notions of primeness and maximality are equivalent for this important class of rings. Sinee every integral domnin contains the two trivial prime ideals, the use of the term prime ideal in a principnl ideal domnin customarily excludes these from consideration.

+, .)

Theorem 3-35. Let (R, be a principal ideal domain. A (nontrivial) ideal of (R, is prime if and only if it is a maximal ideal.

+, .)

Proof. In vi(!w of Theorem :J-34, it iR Ruffieient to Khow that if «a), +, .) is a prime ideal of (U, 1-, .), UJ('n «a), 1-,') iK also maximal. To this end, SUPPORC (1,1-,.) iK !lny idpal with «(l) C I (.~ U. Hinee (ll, I,') it! a "rill(~iplLl ideal ring, t.here exist!> an clement bE R for which I = (b). Now a E I = (b), hence a = r· b for Rome choice of r in R. But «a), +, .) is a prime ideal, so either r E (a) or b E (a). The pORsibility that b E (a) immediately leads to the contradiction (b) ~ (a)_ Therefore r E (a), which implies r = 8' a for suitable 8 E R, or a = r· I, = (.~. a) . II. Since a ¢ 0 alld (Il, +, .) it! all integral domain, we then have 1 = 8' b. This meanK the identity 1 E (b) = I, or equivalently, 1= R. Since no ideal lies between «a), +, .) and the whole ring, we conclude that «a), is a maximal ideal.

+, .)

Corollary. A nontrivial ideal of the ring (Z, is maximal.

+, .) is

prime if IMld only if it

Note that in asserting the equivalence of prime and maximal ideals, Theorem 3-35 fails to actually identify these ideals. The situation is easily remedied though by introducing the idea of a prime element. Definition 3-20. A nonzero ('Il'l1Ient a of the ring (R,

+, .)

is called a

prime element of R if a is not invertible and in every factorization a = b . c with b, C E R, cit.hcr b or c is invertible.

The nonpriml's thus ('(Insist. of zero, clements having inverses, and all elements which can be written as the product of two factors neither of which iR invertible. In rings 11\1eh IUl diviKion rillj!;s and fields, where each nonzero element pORRCsses a multiplicative inv('I'/!{', the eOlwppt of a prime clement is of 110 sigllificanee. The theorem providing the bridge between prime elements and prime ideals may now be stated as follows:

+, .)

Theorem 3-36. Let (R, be a principal ideal domain. The ideal «a), +,.) is a prime (maximal) ideal of (R, +,.) if and only if a is a prime plcmcnt. of R. Proof. The Ht.l'\let,ure of the first. part of the proof i~ similar to that of Theorem 3-35. In other words, suppose a is a prime element of Rand (1, is any ideal for which (a) c I ~ R. By hypothesis, (R, +, .) is a principal ideal ring,

+, .)

:i-4

CEHTAI:-.I SPECIAL mEAL!'!

HH

so there is an c1PlJlent b in R with I = (b). As a E (b), a = r· b for some r E R. It followli from the fact that a is a prime clement that either r or b is invertible. Were l' to have a multiplicative inverse, then b = r- I . a E (a), which implies I = (b) ~ Cal, an obvious contradiction. Accordingly, the clement b must he invertible, so thut (b) = R. This ar~ull1ent shows thnt ((a), +,.) is a maximal ideal of (R, +,.) and consequently prime, by Theorem 3-34. On the other hand, let (a), +, .) be !\ prime ideal of (R, +, .). For a proof by eontmdietion, aHsume thnt a iH not. n prime element of U. Theil a admits the fad.orizat.ion a = Ii· c, where h, C E fl, and ncith(!r h nor c is invertihle. (The alt,('I'lIlltivn pOHHibilit.y t.hat. a hUH lUI illv"!'",, implics (a) = fl, HO it may hI! 1'lI1('«I oul..) Now if till' d(,III1'IIt. b w('rn ill (Il), t.h(,11 b· ,.. a for SOIlW r E R, and a = b· c = (r· a) . c. From the cancellation law, we could infer 1" C = 1. But this f('Hulis in the contradiction that c is invertible. By the same reasoning, if c IiI'S in (a), t.hell b possesses an invef/o;e, eontrnry t.o assumption. We then have b· C E (a), with b fl (a) and c fl (a), whi('h denies that (a), is a prillle id"al. Helice our o!'il(inal supposition is fal,,1' alld a must be a prime clemen L of H.

+, .)

As might be expected, prime clements in a prilwip:d ideal domain playa role analogous to the prime numbers in the ring of intl~gers. To give one illustration, the following genernlization of the Fundamental Theorem of Arithmetic ('lUI be obtained: every nonzero element of a prineipal ideal domain is either invertible or can be written as a produet of prime clements in an essentially unique way (up to invertible clements as fai:tors and the order of factors); the proof of this fact is omitted. It. would s('cm irmppropriat,e to eoneiudc this scetion without some brief nU'lIt.ioll of t.he radil'nl of n rilll/;. The notion ('lUI he ehamderize
II

+, .), denoted by rad R, is the set llIaximal ideal of (R, +, .)}.

+, .)

is a ring wilhout radical or is a semi-

Deftnition 3-21. The mdical of a ring (R,

md U If rad R = simple riny.

to},

=

niM

I (AI,

/,.) is

then we say (R,

Th(' rluli('al always ('xist.s, sinee we know by Tlwofl'm 3-30 that any ring contains at, h'ast one maximal ideal. It is abo imnwdiale from the definition that the t.riple (rad Il, f-,.) forms :tn i
192

BING THEORY

principal ideals «p),

+, .), where p is a prime; that is, radZ = n{(p) I p a prime number}.

Since no nonzero integer is divisible by every prime, rad Z - {O}.

First, let us establish a connection between the radical and invertibility of ring elements. At the risk of being repetitious, we recall our convention that "ring" always means commutative ring with identity. Let (1, +,.) be an ideal of the ring (R, +, .). Then the set I s;;; rad R if and only if each element of the coset 1 + I has an inverse in R.

Theorem 3-37.

Proof. We begin by assuming that Is;;; rad R and that there is some element a E I for which 1 + a is not invertible. Our object, of course, is to derive a contradiction. By the corollary to Theorem 3-30, the element 1 + a must belong to some maximal ideal (M, +, .) of the ring (R, +, .). Since a E r&d R, a is also contained in M, and therefore 1 = (1 + a) - a is in M. But this means M = R, which is clearly impossible. To prove the converse, suppose each member of 1 + I has a. multiplicative inverse, but 1 ~ rad R. By definition of the radical, there will exist a nWtimal ideal (M, +,.) of (R, +,.) with 1 ~ M. If a is any element of 1 which is not in M, the maximality of (M, +, .) implies that (M, a) = R. 'Fhe identity element 1 can therefore be expressed in the form l=m+r·a for: suitable choice of m E M and r E R. But then m = 1 - r • a E 1 + I, so that m possesses an inverse. The conclusion is untenable, since no proper ideal contains an invertible element. The form this result takes when (1, +, .) is the principal idelif generated by an element a E rad R deserves special attention. While actually a corollary to the theorem just proved, it is impOrtant enough to be singled out as a theorem. Theorem 3-31. In any ring (R, +, .), an element a E rad R if and only if 1 + r· a has an inverse for each r E R.

This theorem adapts itself to many uses. Three fairly short and instructive applications are presented below. Corollary 1. An element a is invertible in the ring (R, +,.) if and only if the coset a + rad R is invertible in the quotient ring (Rlrad R,

+, .).

Proof. Assume the coset a + rad R has an inverse in (Rlrad R, +, .), so that (a

+ rad R) . (b + rad R) = 1 + rad R

for some b E R. Then a· b - 1 lies in rad R. We next appeal to Theorem 3-38,

3-4

CERTAIN SPECIAL IDEALS

193

with r = 1, to conclude that the product a· b = 1 + 1· (a· b - 1) is invertible; this, in tum, forces the element a to have an inverse. The other direction is el!8entially trivial. Corollary 2. The only idempotent in the radical of the ring (R,

+, .) is O.

Proof. Let the element a E R with = a. Taking r = -1 in the preceding theorem, we see that 1 - a has an inverse in R; say, for purposes of argument, that (1 - a)· b = 1, where b E R. This leads to

a2

a

=

a2

+ a· b -

a· b

=

a· (a + a· b - b)

=

a· (a - 1)

=

0,

which completes the proof. Corollary 3. Let N denote the set of all noninvertible elements of R. Then the triple (N, +, .) is an ideal of the ring (R, +, .) if and only if N = rad R. Proof. The inclusion rad R ~ N clearly holds. Suppose, therefore, that the element a E N. If the system (N, +,.) forms an ideal of (R, +, .), then r· a E N for any r E R. Moreover, the element 1 + r· a E N, for otherwise 1 = (1 + r· a) - (r· a) ~ould be in N. From the definition of N, it follows that 1 + r . a must be invertible, implying a E rad R. This shows N s; rad R, whence the equality N = rad R. The other direction of the corollary is immediate.

We close this section with a result which provides a convenient method for manufacturing semisimple rings. Its proof utilizes both implications of the last theorem. Theorem 3-39. For any ring (H, semisimple.

+, .), the quotient ring (Hlrad H, +,.)

is

Proof. Before beco~ing involved in details, let us remark that since (rad H, +, .) is an ideal of (R, +, .), we Pl&Y certainly form the quotient ring (H/rad R, +, .). To simplify notation somewhat, we will denote rad H by 1. Supposo the coset a + 1 E rad (HI 1). Our strategy is to show that a e 1, for then a + f = I, which would imply that rad (HI J) consists of only the zero coset. Since a + flies in rad (RI f), Theorem 3-38 asserts (1

+ f) + (r + J) . (a + J) = 1 + r· a + 1

is invertible in Hl1 for each r E H. Accordingly, there exists a coset b + f (which, of course, depends on both r and a) such that (1

+ a· r + 1) . (b + 1) = 1 + 1.

This is clearly equivalent to b + a . r· b - 1 E 1

=

rad H.

194

3-4

Again appealing to Theorem 3-38, we conclude that the element b

hUH un

illVl'rfW!

c

ill

+ a· r· b =

1 -I- 1· (b

+ a· r· b -

1)

fl. But then

(I

+ r· a) . (b· c)

= (b

+ a· r· b) . c =

1,

that 1 -I- r· a iii inwrtible in R. Bt'Cl\use this argument holdH for every r E R, it follows that a E rad R, as desired.

IiO

PROBLEMS

In thl' fullowill~ idl'ntity.

~t

of prohll'm>!, all rings arll aSflllmlld to bl'

(~ommutativll

with

J. Show that the only maximal ideal in a field
+', .'),

+', .'),

+', .'),

+, .).

+", ',,)

+, .)

+, .)

+, .)

\1'1 = {rElllrnE/fnrRomenEZ+}. Estahlish I.hl' following assl'rtion!l: a) The triplt· (\1'1, -1-,') is an i,II'al of (R, +,.) such that I r;; Vi.' b} If (J, +,'J i" an ill('al uf (Ii, -1-,') fur which I r;; J r;; vI, then c) If (II, 0.011 (12, -1-•. ) arl' both ideals of (R, then

+, .),

+, .)

vII

n

12 =

VYin

+, .)

vJ

=

Vi.

vI2'

7. If f is IL homomorphism from the ring (R, onto the ring (R', -1-', .'), show t.he following: a) If (/,-1-,') is a prime ideal of (R,+,') with 12 ker(j), then the triple (fU), J) is a prime idl'al or (R',

+',

+', .').

:l-4

195

CJ<:HTAIN SPI<;CIAL IDEALS

+', .')

b) If (I', is a prime ideal of (R', prime ideal of (ll,

+, .).

+, .) is a prime illl'lIls of 1.11(' ring (R, +, .). If

-+', .I), then the triple

(f-'(I'),

H. Lf't (Ii, I,') 1m lin indf'xf'l\ eollectinn of tlH' fllIllily of st!t>! {I,: for illS admin, proVt' t.hat bot.h (U/;, I,') alit \ (n/;, arc prime icleal~ of (R,+, .).

+, .)

9. Prove that if (1,,+,,) and (12,+,') are two ideals of the ring (R,+,') such that neither the Het I, nor the set 12 is contained in the other, then the idea! (I, n I:l,·t,·) is lint prime. [llint: Piek aE II - 12, bE 12 - II, and considl'r a . b.] )(1. lTtilize Tlll'nrems 3-32 and a-33 to give an altt'fnativ(' pronf of the fact that every maximal idl'nl is IL prime id(·nl. II. An idl'ul (I, 1 , .) of IL ring (fl, 1 , .) i" ,,"id to I,,· u ",':/Il11rl/ idm/ if a . II E 1 with It fl. I illlplit'" /," E I for SO III" positivI' intl'gl'r 11. Jo:"tllhlish tlw following faett!

('OIlCl!rJlillJ!: prilllllry ideals: a) Every prim!' idl'1I.1 is a prirnnry idnal. b) If (I, i>j a primnry ideal of (ll,I', .), l.hl'lI thl' i,leal (0, is prime. c) H t.he illeal (I, +,.) is such that (VI, 1 ,.) for illS a maximal id('al of (Ii, tI}(,1l (I, 1-,') is a pri llIary ill(·al.

+, .)

+, .)

+, .) is a primary +, .) is nilpotent.

+, .).

+, .)

+, .),

12. Let (I, be an ideal of the ring (R, Prove that (I, ideal if and only if every zero divisor of the quotient ring (RI I,

I a. An ideal is called a nil ideal if 1'lIeh of it.t! f'leml'nt.s is nilpot.(·nt.

Prove t.hat if (1,·1,') iH a nil idt'ILI of (.III' ring (ll, I, .), tlll'lI 1 c;:: rad U. [llint: If rEI with rn = 0, tll('l1 (I - r) . (I -I- r -I- r2 j ... ··1 rn-I) = 1 - r" = I.]

14. Describe the radiea! of (Z.,

+., ',,).

II/int: Considl'r the prime factorization of n.)

15. Given (I, +,.) is all ideal of the ring (R, +,.) such that the quotient ring (RI I, is semisimp!e, prove that rad R ~ I. [Him: If a E rad R, show that a IE rad (Ill I) by usc of Theorem 3-:J8.) 16. Prove that if (I, is an ideal of t.he ring (R, thell rad I = I n rad R. 17. a) Hhow that the annihilator of a semisimple ring (R, +,.) is zero; in other words, ann R = {O}. b) Let (R,+,') h(, a ring with the property that I = aIm (ann 1) for every ideal (/, +,.) IIf (/i, I, .). (In gl'lIl'rlLl, WI' IInly IlILvl' I.h.· illl'itl"ioll I!; unn (ILnn 1).) Pro v,' I,hat. if (.II, I,·) is a maximal ideal of tl", ring (Ii, I, .), then (ann .11, +,.)

+

+, .)

+, .)

+, .),

iH a minimal idl!al. IH. Let (1,+,,) hi' an ideal of ti\(' ring (R,+,') 8.nd

S~

Il he a Het that is closed

under multiplication and is disjoint from I. a) Using Zorn's Lemma, show that t111're t'xiHts all icleal (/', which is maximal with respect to the,;e prop('rtiell: (i) I

~

P,

(ii) P

nS

+,.)

of (R,

+, .)

= ~.

b) Prove that (/', +,.) is n('cl'Hsarily a prime ideal. [/lint: If a fl. P, b fl. 1', then (I', a) and (I', b) intersect S, whieh implies a . b fl. P.) 19. Prove: 8.11 ideal (I, +,.) of (R' +,.) i~ the intersection of prime ideals if and only if a 2 E I implies a E 1. [/lint: Foreadl a fl. I, there is a prirneideal (P,+,')

196

3-5

RINO THEORY

of (R, +,.) which is maximal with respect to"disjointedness from the set ," \a,

20.

21.

22.

23.

2 ... , a" } a, , ...

and such that P"2 I.] Verify that an ideal (1, +,.) of (R, +,.) contains a »rime ideal if and only if, for every n, al . a2 ... a" ... 0 implies that some a. E I. [Hint: Use Problem 18 with S - {bl' b2' .. b.. 1 b. E -I, n ~ I}.] A ring is said to be a local ring if it has a unique maximal ideal. If (R, is a local ring and (M,+,·) is its maximal ideal, prove that any element a EM is invertible. If I: R -+ R' is a homomorphism of the ring (R,+,·) onto the ring (R',+', .'), prove that I(rad R) ~ rad R' and, whenever ker
+, .)

3-5 POLYNOMIAL RINGS

The next step in our program is to apply some of the previousty developed theory to a certain class of rings, the so-called polynomial rings. For the moment, we shall merely remark that these are" rings whose elements consist of polynomials with coefficients from a fixed, but otherwise ar\>itrary, ring. (The most interesting results occur when the coefficients are specialized to a field.) The matter of formalizing the intuitive idea of what is meant by a polynomial always presents a serious problem. We propose first to put this notion on a sound basis, then to discuss the properties of polynomials and polynomial rings. Our investigation will culminate with a survey of the basic facts relating. roots of polynomials and field extensions. En route, the reader will see that many of the familiar results of elementary algebra are but special cases of general theorems to be obtained. For an arbitrary ring (R, +, .), let poly R designate the set of all infinite sequences of elements of R in which at most a finite number of the ak ~ 0; in other words, (ao, at. a2, . .. ) E poly R if and only if there is some nonnegative integer N such that a" = 0 for all k ~ N. The elements of poly R are called polynomials over the ring (R, +, .), or just polynomials over R. At times, it will be cOllvenient to use the notation

(ao, a" ... , all, 0, 0, ... )

3-5

POLYNOMIAL RINGS

197

for the polynomial with last nonzero term an; when n = 0, we permit ao = 0 in order to include the zero polynamial (0, 0, 0, ... ) each of whose terms is zero. With this agreement, the set of polynomials over R may be regarded as the IKlt poly R

=

{(ao, ai, ...

,an, 0, 0, ... ) I at E

R, n

~

O}.

Thus, the sequence (1, 1, 1,0,0, ... ) would be a polynomial over (Z2, +2, '2), but (0, 1,0,1, ... ,0,1, ... ) would not. We next introduce operations of addition and multiplication in the set poly R, 80 that the resulting system is a ring containing (R, +, .) as a subring. Let us first, however, make clear that two polynomials and arc equal if and only if their corresponding terms are equal: / = g if and only if ak = bk for every integer k ~ O. Now, define the sum / + (J by the rule

/ +g =

(ao

+ bo, al + bl, a2 + b2, ... ).

The reader may easily verify that the pair (poly R, +) forms a commutative group. We state only that the zero element is the zero polynomial (0,0,0, ... ) and that the additive inverse of (ao, al, a2, ... ) is (-ao, -ai, -a2, ... ). Finally, we specify an operation of multiplication in poly R by taking /. g = (ao' bo, ao . b 1

+ al . bo, ao . b2 + al . b + a2 . bo, ... ) 1

= (co, -CI, c~:·:-:-.), where

C" =

L

iH-t

ai' bi = ao . bt

+ al . b"-l + ... + 'at . boo

°

Note that if ai = 0 for i> n and bi = 0 for j > m, then c" = for all > n + m (i j = k > n + m implies either i > n or j > m), whence /. g is once again a polynomial. A routine computation, which we omit, establishes that this multiplication is associative as well as distributive with respect to addition; in fact, these properties follow almost at once from the corresponding properties in (R, +, .). All this may be summarized in a theorem.

k

+

Theorem 3-40.

0/ polynomial8

The triple (poly R, +, .) forms a ring, known as the ring

+, .)

over R. Furthermore, the ring (poly R, is commutative with identity if and only if (R, +, .) is a commutative ring with identity.

At the risk of belaboring the obvious, let us point out that while the operations in (poly R, +, .) are defined in terms of those of (R, +,.), these are entirely new operationR; we have used the same symbols only to avoid unnecessary notational complications.

198

3-5

If 8 rl!prcHcmtH Ule HPj, of all constant TJolyrtomials,' that is, the set

=

8

{(a, 0, 0, ... ) I a Ell},

+, .)

then it is not particularly difficult to show that (S, constitutes a subring of (poly R, +, .) whieh is isomorphic to (R, +, .); one need simply consider the mapping that scnds the eonstant. polynomial (a, 0, 0, ... ) to a. In this sense, (poly R, +,.) eont.ains (R, +,.) as a suhring. As 1\ result of the uforementioned imbedding, we shall no longer distinguish between an clement a E R and the constant polynomial (a, 0, 0, ... ) of poly R. By developing IlOme additional notation, it is possible to express polynomials in Hw fl1ll1ilillr form usually PIII'Oulltl'rpci ill f'lf'mnntary COllrHHS in algebra. As a firHl. HI.I·JI ill this din·d.iulI, wC' Id a.c del:liI(IIILf.c~ til(! polynomial (0, a, 0, 0, ... ).

That is, ax is the spcdfie nu'mber of poly R which has the element a for. its 8('cond term nnel 0 for nil other terms. :\[ore genemlly, the symbol ax", n ~ 1, will dC'lIot.e tlw polynomial (0, ... , 0, a, 0, ... ), wll(~re

t.he ('Iem('nt a iH t.Iw (n

have

+ I )Ht. tc'rm in this polynomial; for

ax 2

=

example, we

(0, 0, a, 0, ... )

and

ax:!

=

(0,0,0, a, 0, ... ).

Utilizing these conventions, ench polynomial

mny be uniquely represcnted in the form

f = (an, 0, 0, ... ) -I- (0, alt 0, 0, ... ) -I- ... = an

-I- alx -1 a2x2

+ (0, ... ,0, a", 0, ... )

+ ... + a"x",

with, of course, ao repl:wing (an, 0, 0, ... ). Thus, there is no harm in regarding t.he polynomial rillj!; (poly n,l, .) IlS consisting of all formal expressions

where the ('1('nI('IIts all, aI, ... , an (t.he cl)('.jJidcnts of f) lic in R. WC' Hhould pllll'liaHizp Ihal It('c'ordilll( to our dl'finil.ion, x is Himply 1\ new symbol, or indctcrmina7l', totally 1I11Tf'lated to the rillg (R, +, .), and in no Hellse r('pn'spllts all ('\('nwlIl of H. To indimte the indeterminnnt x, it is com mOil prac~tic'e to write Rf.rJ for \.Iw 1'11'1. poly R, ulld f(.c) for any member of the Rl1mC'. From 1I0W 011, W(' Hlmlllllalw exc·lllsivl' use of this IIotation.

3-5

POLYNOMIAL RINGS

199

Remark. If t.IlC rillg (U, +, .) has a llIult,ip\i('utive identity 1, many authors will identify the polynmnilll Ix with the ill(let<~rminltnt. x, thereby treating x itself as It sp('pilie IlIl'mber of Rlxj, namely, t.he polynomial x = (0,1,0,0, ... ). From this view, ax becomes an actual product of clements in Rlx]:

ax = (a, 0, 0, ... ) . (0,1,0,0, ... ). To simplify the writing of 1\ polynominl, it is custolllary to omit terms with zero coefficients and to replace (-ak)x" by -akx". Thus, for example, 1 - 23;2 would stand for 1 Ox (_2)X2. An important. definition in connection with polynomials is that of degree, givl'll Iu·low.

+ +

Deftnition 3-22. If

is a nonzero polynomial in Rlx], we call the coefficient an the le(J(iing coefficient of f(x) and tlH' inl.('gl'r n, t.he de(II'ec of the polynomial. Thl' d(~grt,1' or ILIly nOlY.!ero polynomilll is therefore a nonnegative integer; no degf'(~e is assigned to the zero polynomial. It will be observed that the polynomials of degree 0 are precitICly the nonzero constant polynomials. If the ring (R, has an identity clement, a nonzero polynomial of degree n whose leading eoeffieient an = 1 is said to be a monic polynomial; ill this case, we take the liberty of dropping the 1 and writing/(x) = ao ajX + ... xn. As a matter of notation, we shall hereafter write deg/(x) for the degree of any nonzero polynomial I(x) E Rlx]. SUPl>OtIC I(x), g(x) E Rlx] with deg/(x) = n and
+, .)

+

+

+ ajX + ... + anxn, bo + hr + ... + bmx m ,

I(x) = ao

an ~ 0,

g(x) =

b",

~

O.

From the definition of multipliention, the reader mny ensily check t.hat all coefficients of I(x) . g(x) beyond the (n + m)th arc zero, whence I(;c) . g(x) = ao· bo

+ (ao' b + a • . 1Jo)x + ... + (an' bm)xn+m • j

If we 1l.,'i,'IIUne that at least olle of the leading eoeffi(~iellt.i'! an or bm is not a divisor of zero ill (R, I, .), thell a tl • bm ~ 0; lLl'eol'dillgly,/(x), g(;r) ~ 0 and deg (/(;r) . !I(.,.» = n

+ lit =


+ dcg !I(x).

+, .)

This ('ertainly holds if (R, +, .) is an integral domain, or again if (R, has IlII id('II!ily "IPIIlI'II! and 0111' of till' polynomials I(x) or fI(.r) hili'! all illvertihle lead ing l'oeHil' iell t.

200

BING THEORY

The foregoing argument serves to establish the first part of the next theorem; the proof of the second auertion is left as an exercise.

+,.) be nonzero elements of (R[x), +, .). I} deg (f(x) . g(x» = deg/(x) 2) either I(x) g(x) = 0 or deg

Theorem 3-41. Let (R,

an integral domain and I(x), g(x) be two Then deg g(x), and (f(x) g(x» ~ max {deg/(x), deg g(x)}.

+

+

+

The notion of degree may be used to prove the following corollary. Corollary. If the ring (R, ring (Rlx), +, .).

+, .) is an integral domain, 80 also is its polynomial

Proo/. We observed earlier that whenever (R, +, .) is a commutative ring with identity, these properties carry over to (Rlx], To see that (R[x], +,.) has no zero divi80rs, choose/(x) p4 0, g(x) p4 0 in R[x]. Then deg (f(x) . g(x» = deg!(x) deg g(x) ~ 0, hence the product I(x), g(x) cannot be the zero polynomial.

+, .).

+

+, .)

Example 3-33. As an illU8tration of what might happen if (R, baa aero divi80rs, considcr (Zs, +s, 's), the ring of integers modulo 8. Taking

+ 2%, 4 + x + 4x 2 ,

I(x) = 1 g(x) =

+ x + &;2,80 that deg (f(x) . g(x» = 2 < 1 + 2 = deg/(x} + deg g(x). While many of the properties of the ring (R, +, .) carry over to the polynomial ring (B[x], +, .), it should be observed that for no ring (R, +, .) is (B[x], +, .) a field. In fact, no element of B[x] which has positive degree can

we then have/(x), g(x)

=

4

have a multiplicative iuverse. For suppose I(x) E B[x] with deg I(x) I(x} . g(x) = 1 for 80me o(x) E B[x], we would obtain thc contradiction

o = deg 1 =

deg (j(:f) . ,(x»

= deg/(x) + deg g(x}

p4

> 0;

if

O.

For polynomials, we have .. MUtt analogous to the division "algorithm for integers.

+, .)

Theorem 3-42. (Diviaitm Algorithm). Let (R, be a commutative ring with identity and I(x}, g(x) p4 0 be polynomials in Blx], with the leading coefficient of g(x) an invertible clcmcnt. Thcn there exist unique polynomials q(x), rex) E Rlx] such that I(x)

where either rex)

=

= q(x) . g(x) + rex),

0 or dcg rex)

< de,; (1(x).

3-5

POLYNOMIAL RINGS

201

Proof. The proof proceeds by induction on the degree of f(x). First note that if f(x) = 0 or degf(x) < deg g(x), then a representation meeting the requirements of the theorem exists with q(x) = 0, rex) = f(x). Furthermore, if degf(x) = dcg g(x) = 0, f(x) and g(x) are both elements of R, and it su1ficea to take q(x) = f(x) . g(x)-t, rex) = O. So, suppose the theorem is true for polynomials f(x) of degree less than n (the induction hypothesis), and let deg f(x) = n, deg g(x) = m, where n ;;:: mj

that is, f(x)

= ao + alX + ... + a..x",

g(x)

=

bo + b1x

a.. 'F 0,

+ ... + b".x"',

b", 'F 0,

(n;;:: m).

The polynomial

lies in R[x] and, since the coefficient of x" is a.. - (a..' b;l) . b", = 0, has degree less than n. By induction, there are polynomials ql(X), rex) E R[x] such that

. !I (x) =

where rex)

ql(X) . g(x)

= 0 or deg rex) < deg g(x). f(x)

+

+ rex),

Substituting, we obtain the equation

= (91(X) (a..' b;I)X"-IR) • g(x) = q(x) . g(x) rex),

+

+ rex)

which shows that the desired representation also exists when deg f(x) = n. As for uniqueness~ suppoSe f(x) = q(x) . g(x)

+ rex) =' q'(x) • g(x) + r(x),

where rex) and rex) satisfy the requirements of the theorem. Then rex) - r'(x)

=

(q'(x) - q(x» • g(x).

If rex) - r'(x) 'F 0, we would have

deg (r(x) - r'(x» = deg «((x) - q(x» ;;:: degg(x).

+ deg g(x)

On the other hand, since the degrees of rex) and r'(x) are both less than that of g(x), it follows that deg (r(x) - r'(x» This contradiction implies rex)

=

< deg g(x).

r'(x), which in tum yields q(x)

= (x).

202

:J-5

ItINO TIU;OIlY

The polynomialll q(x) and rex) appearing in the represcntation f(x) =" q(x) . (I(x) f- rex) givl'n hy the Divillioll Algorithm arc (~alled, rcllpectively, the quotient and remainder on dividing f(x) hy g(x). Obscrve also that if g(x) is a monic polynomial, or if (R, is a field, it is not necessary to assume the leading cocffident of fI(X) t.o he invl'rt.ihlE'. Bl'forc~ dilll!lIll~illg HII' l'OII~I'qllmwclI of Theorem :~-42, wr~ paulii' to intl'odlle(l I"\CIIl\(' nddit.illlllli (·IIII1"I'ptS. To this I'lId, II!L (Ii, I,·) h(~ n eonullut.ntivc ring with idellt ity allll,' he nil nrhitrnry d!'llJ('lIt of N. For e:wh polYllominl

+, .)

ill R[x),

WI'

llIay dpfint' fer) E R hy fer)

=

ao

+

(II •

r

+ ... + an' rn.

The clement J(r) is said to he the result of su/,slituling l' for x in J(x). Suffice it to Ray, the addition and multiplieation used in defining fer) arc those of the ring (R, I, .), 1I0\. t.holl/! or (N[x), 1-, .). If J(r) -~ 0, WI! ('nil the e1l'1ll1mt r It "oot or zero of t.he giv('11 polynomial/(x). Hllpposc J(.r), g(r) arl' in R[r) and I' E R. The reader should prove that if It(x)

J(x)

+ g(x),

k(x)

=

f(x) . g(x),

= J(r)

+ (I(r),

k(r)

=

fer) • f/(r).

=

t.hen

her)

In partirmlar, it. llIay hI' C'OIlI"llIded t.hat t.he mapping whieh III'nds f(x) to J(r) is a hOlllomorphisllI from (lllr), +, .) into (Ii, the range of this homomorphism will he (It'noted hy R(r). Some diffi('uIt.y arisl'M whell t.h(' ring (R, fails t.o he (!ommutative; in this case, the above result lIeed not hold. Observe that if we let

+, .);

+, .)

hex)

=

(x -

a) . (x -

b)

=

x2

-

(a

+ b)x + a· b,

UlI'n "(I') .,.-

,.2

«(I I

I,)·

1'·1

a . II.

Without the hypothesis of ("ommlltativity, it ennnot be conduded that (r - a) . (I" .- /1) =

1'2 -

a·

1" -

1" /, -I- a· b

will (~qual her); ill ot.hpr wordR, hex) = J(x)· fI(X) docs not always imply her) = J(r) . (1(1'). 0111' IIIC1fI' r1 .. finit iOIl Hhollirl I", illfl"llrilll'l'ri: H J(r) mill I/e,·) rtf () Ilre ill Rlx), W(~ Hay that fI(X) iH a Jactor of J(x) [or !ICc) dil'irle.~ f(x)] provid(~d there ('xists some IlOlynominl h(.r) E ll[x) for whi('h J(x) = hex) . g(x).

:J-5

J'Ol,vNOMIAL RINGS

With this terminololO' in mind, we cOllie to tlw fal'forizllt.ioll prop!'r!.i!';; (If N[x/.

IL

sCI'iell of t.heorems conl:crninl(

Theorem 3-43. (Il(:mairuler Theorem). Let (R, -t , .) he a eomlllutative ring with identity. If f(x) E R[x] and a E then there is a unique poly-

n,

nomial q(x) in nIx] such that f(x) = (x - a) . q(x)

-I- I(a).

I'roof. Applying UII' c1ivi"ioll algOl'it hIll t.(I f(.r) nlllt x f(x) = (x -- a) . q(x)

(I, W('

o!ltnin

+ I'(X),

\\,1\('1'1' r(.r) ~ 0 or d('1!; r(.r) < del( (x a) = 1. It. followli in either caMe that rex) is a eonstant polynomial r E Substituting a for x, we have the desired result: f(a) = (a - a) . q(a) rea) = 0 r = r.

n.

+

+

The polynomial f(x) E nIx] ill divisible if anel ()nly ir a ill n root of J(x).

Corollary. (Factorization Theorem).

hy x -

fl

ProoJ. The ('orollary follows immediately from the theorem, since

J(x) = (x -

if and only if J(a)

a) . q(x)

= O.

Let us next show t.hat t.he number of root.s of a polynomial over an integral domai II call not ('xcccd i ts d~gf(~e. Theorem 3-44. Let (fl, 1-,') be

lUI integrnl elomnin and J(x) E nIx] be a nOllzero polynomial of degl'ee n. Theil I(x) has at mO!:lt n dilltinet roots in

n.

ProoJ. We proC!l!Cd by indu(~t.ioll on til(! degree of f(J:). When degJ(x) = 0, the result is trivial, silll'e J(x) eallllot have any root.s. If degf(x) = 1, say J(x) = ax + b, a ~ 0, t.hen J(x) Ims at most olle 1'00t; indeed, if a is invertible, _a-I. b ill the only root, of J(x). Now, suppose the theorem is t.rue for all polYllolllinls of dl'grcl' n - 1 ~ 1, and let dcgJ(x) = n. If f(x) has a root r, the prl!eeding ('())'olhu'y I!;iv!'s J(x)

=

(x -

I') . q(x),

where the polynomial q(x) haR degree n - 1. Any root r must be It root. of q(x), for, by substitut.ion,

1'1

of J(x) distinct from

allll, sill(!(~ (ll, I,') has 110 zero divisors, Q(I'I) =- n. FI'OIII our illlilwtioll hypothesis, q(x) has at. most. n - 1 distinet roots. AI! the only root!! of f(x) are r anu those of q(x), J(x) cannot have more than n distinct roots in R.

204

3-5

RING THEORY

Corollary. Letf(x) and g(x) be nonzero polynomials of degrees ~n over the integral domain (R, If there exist n 1 distinct elements a. E R (Ie = 1, 2, ... , n 1) for which f(ak) = g(a.), then f(x) = g(x).

+

+, .).

+

Proof. The polynomial hex) = f(x) - g(x) is such that deg hex) ~ n and has at least n 1 distinct roots in R. By Theorem 3-44, this is impossible unless hex) = f(x) - g(x) = 0, or f(x) = g(x).

+

Example 3-34. Consider the polynomial x" -

x E Z,,[x), where p is a prime number. Since the nonzero elements of (Z", +", ',,) form a cyclic group, under multiplication, of order p - 1, we must have a,,-l = 1 or a" = a for every a ¢ O. But the last equation clearly holds when a = 0, so that every element of Z" is a root of the polynomial x" - x. This furnishes an illustration of a nonzero polynomial which "vanishes identically. »

Using the terminology of the present section, we now state the result known as the Fundamental Theorem of Algebra.

+, .) be the field of complex numbers. If f(x) E Crx) is & polynomial of positive degree, thenf(x) has at least one root in C.

Theorem 3-45. Let (C,

While many proofs of this theorem are available, none is strictly algebraic in nature; thus, we shall assume the validity of Theorem 3-45 without proof. The reader may however establish the following corollary. Corollary. If f(x) E C[x] is a polynomial of degree n > 0, then f(x) can be expressed in C[x) as a product of n (not necessarily distinct) linear factors. Throughout the remainder of the section, we shall focus our attention on polynomials with coefficients from a field (F, +, .). In this important and interesting case, the associated polynomial ring (F[x], +, .) is an integral domain (but not a field I); in fact, (F[x), +, .) is a principal ideal domain. Theore~ 3-46. If (F,

+, .) is a field, then the ring (F[x), +, .) is a principal .

ideal domain. Proof. By Theorem 3-41, it is already known that (F[x1, +,.) is an integral domain. To see that any ideal (I, of (F[x], is principal, we need .

+, .)

+, .)

oiuy mimic the argument of Theorem 2-24: If 1= {O}, the result is trivially true, since I = (0). Otherwise, there is some nonzero polynomial p(x) of lowest degree in I. For each polynomial f(x) E 1, we may use the Division Algorithm to writef(x) = q(x) . p(x) rex), where either rex) = 0 or deg rex) < deg p(x). Now, rex) = f(x) - q(x) . p(x) lies in 1; if the degree of rex) were less than that of p(x), a contradiction to the choice of p(x) would result. With this possibility ruled out, rex) = 0 and f(x) = q(x). p(x) E (p(x»; hence, I ~ (p(x». But the opposite inclusion clearly holds, so that I = (p(x».

+

Corollary. A nontrivial ideal of (F[x), +, .) is maximal if and only if it is a prime ideal.

3-5

POLYNOMIAL RINGS

205

+, .)

By custom, the prime elements of the principal ideal domain (F[:c], are referred to as irreducible polynomial8. Translating Definition 3-20 into the language of the present section, we see that f(x) is an irreducible polynomial if and only if f(x) is of positive degree and in any factorizationf(:c) = g(x) • hex), with g(x), hex) E F[x], either g(x) or hex) must be a constant polynomial (recall that the invertible elements of F[x] are precisely the nonzero constant polynomials); the constant polynomials are neither reducible nor irreducible. Let us record this observation as a formal definition. Definition 3-23. A nonconstant polynomial f(x) E F[x] is said to be irreducible in F[xl if and only if f(x) cannot be expressed as the product of two polynomials of positive degree. Otherwise, f(x) is reducible in F[x].

+, .)

The dependence of this definition upon the domain (F[x], is essential. It may well happen that a given polynomial is irreducible when viewed as an element of one domain, yet reducible in another. One such example is x 2 1; it is irreducible in (R'[x], but reducible in both (C[x], where x2 1 = (x + i)· (x - i), and (Z2[X], wherex 2 1 = (:c 1)· (x+ 1). Thus, to ask merely whether a polynomial f(x) is reducible is incomplete and meaningless. For the ques~ion to make sense, one must indicate what c0efficients are to be allowed in the factorization. It is often quite difficult to decide whether a particular polynomial is irreducis a finite field, say one of the fields ible relative to a specific field. If (F, of integers modulo a prime, we may actually examine all of the possible roots. To cite an illustration, x 3 x 1 is irreducible in Z2[X); in this case, the only possible roots for a polynomial are 0 and 1, but 0 +2 0 +2 1 pi! 0, 1 +2 1 +2 1 = 1 ;oil O.

+, .),

+

+, .),

+

+, .), +

+

+, .)

+ +

+

Example 3-35. Any linear polynomialf(x) = ax b, a;oll 0, is irreducible in F[x]. Indeed, since the degree of a product of two nonzero polynomials is the sum of the degrees of the factors, it follows that a representation

ax

+b=

with 0 < deg g(x) < 1,0 < deg hex) polynomial has degree at least 2.

g(x) . hex),

<

1 is impossible. Thus, every reducible

Example 3-36. The polynomial x 2 - 2 is irreducible in Q[xl, where (Q, is the field of rational numbers. Otherwise, we would have

x2

-

2

= =

(ax

+ b) . (cx + d) + (ad + be)x + bd,

(ac)X2

where the coefficients a, b, c, d E Q. Accordingly, ac

=

1,

ad

+ be = 0,

bd

=

-2,

+, .)

:.!fl(j

whl'llee c obtain

:l-5

=

I/a, Ii =

o=

Rubst.ituting in thc relation ad I-- be

--2/b. -2a/b

+- b/a =

(-2a 2

= 0, wc

+- b2 )/ab.

Thus, - :!a 2 -1- b2 = 0, or (b/a)2 = 2, which is impossible bec~ause V2 is not a ratiorllli number. While irreducible in Q[x], the polynomial x 2 - 2 is nonetheIl's." r<'dueihle in R'[x]; in this case, x 2 - :! = (x - V2)(x +- V2), where both fal'lors arl' in R'[X). For easl' of rdcren(:l', let us summarize in the next theorem some of the rl'sult.s of till' pn'vious s('(d,ion (sp('(~ili('ally, TIH'ormns :i-:i2 IUld :i-ali) in Uw (~nlll' of (P[x], I,')' Theorem 3-47. If (F, I,,) is IL lipid, t.he following st.af,ements are (·ljuivalcnt.: 1) f(x) is an irreducible polynomittl in F[x). :!) TIH' prin(~i"al idl'al «f(x», +,.) ill:t maximal (prime) ideal of (F[x], +,.). a) The quot.ient ring (F[xl/(f(x», is It field.

+.. )

A further faet whieh we Hhall require shortly is that every nonconstallt polynomial ('1111 hl' faetor('d into irredueible polynomials. Theorem 3-48. (Unique Factorization Theorem). Eaeh polynomial f(x) E F[x] of positive degree is the produet of a nonzero clement of F and irreducible monic' polynomials of F[x]. Apart from the order of the factors; this faetorizat ion is unique.

Proof. We prove our thoorem by induction on the degree n of f(x). If n = 1, Ihl'n f(x) = ax +- b = a' (x +- a-I. b), where by Example 3-35, x +- a-I. b is_ a monic irrl'dw·ihle polynomial. Next, suppose f(x) has degree n > 1 and that the thoorem holds for all polynomials of degree less than n. If f(x) is irreducible in F[x], WI' are through Ilfh'r factoring out. its IC'uding (·oeffi(·ient. Otherwise, .r(x) ill rt'.hll'ihl(~ and it. ill pORllihl(l t.o writ(~ f(x) = !I(x), hex) with y(x), hex) E F[x], 0 < d('g y(x) < 11, 0 < d('g hex) < n. Therefore, by the induction' hypothesis, (/(.r:)

=

flJ •

OI(X)' 02(X)' .. O,(;!:),

hex) = a2 . h, (.1') • h2(.r) ... h.(x),

whC're a" a2 are nonzero elemen!.s of F (in fact, the leading (:oefficients of o(x) and h(x») al\ll (I.(:c), ".(x) art' irr('(hwihle monic polynomial!:!. Thus, /(./,)

(11'1'

(I:.d . f/.(./,)· .. Yr(./') . h,(.c) .. · 1I.{.r)

i!:! a faelorizllt ion satisfyinl!; the condit.ions of the t.heorC'lll. Tht' IIniqll('n('~s of th(' flldors still rt'lll11inl< t.o he shown. W(, again proceed hy indudion on 1/ = th·gf(x}. Th(' r('slllt is lrivial for n = 1: if

POLYNOMIAL RINGS

207

then a, = a2 and a, . b, = a2' b 2, whellC~e b, = b2 by the cancellation law for multiplieation. Now, let n > 1 and f(;l') = c· p,(.r) . P2(X) ... Pr(x) = d· q,(x) . Q2(X) •.• q,(x)

be any two faetorizations of f(x) into irreducible monic polynomials. Surely = d, for euch is the leading coefficient of f(x). Furthermore, Pl(X) divides the produc!t q,(x) . q2(X) ..• q.(x) and must thercfof
Hi'lI'e dt'gf,(x) < n, its fadol'ization iM unique by the induction ILHSumptioll, so that the seqlleJwe q2(X), ... ,q.(x) iM simply 11 rearrn.ngem(mt of P2(X), ... ,1Ir(X). Together with the observation HlIlt c· p,(.x) = d· ql(X), this completes the proof. Needless to lillY, Theorem 3-48 can b(' made more explicit when one knows, for n given fi(·ld (ft', +, .), exactly what polynomials are irreducible. If F = C, for instance, the Fundamentnl Theorem of Algebra implies that the only irreducible polynomials in C[xl arc the linear polynomials. In the event (F, is the real field, we have the following corolInry.

+, .)

Corollary. If f(x) E n'[xl is of positive degree, then f(x) can be factored into

linear and irreducible quadratic factors.

Proof. Since f(x) also helongs to linear polynomials x - CIc, Cic E C. Ck = a -/- bi, where a, b E n' and nomials O('I~ur iu I!oujugate pairM a root of f(x). Thu8 (;1: -

Ck) •

C[x], f(x) factors in C[xl into a product of If Cic E n', then x - Cic E n'[x]. Otherwise, b ~ O. But the complex roots of real poly[problem H(b) I, so t.hat ~k = a -- bi is also

(x - ~k) = x 2

-

2ax

+ (a 2 -/- b2 ) E

n'[.r)

iM Il fad-or of f(x). The 1IIlIldmti!! polyuomial x 2 :!ax 1 (a 2 1 02 ) is im~dudulc in R'[xJ, sirll:e any factorization in Rt[xl is also valid in C[xl and (x - Ck) . (x - ~k) is it.s unique factorization in C[xJ. . An interest.ing remark, to be rec!orded without proof, is that if (F, +, .) is lipId, 1.111' polynolllial rinl( (FIx), I,') I'ontains irrpl!w·ihlp polynolllials of I!VI!"Y tll·gr·!!I'. By an extension (F', +,.) of a field (F, +,.) wc :;imply mcltn any field which eontains (ft', I,') ml It sublil'ld. For inst.:uwl', the field of J'l'lLl numbers is lUi l!xt.l"usion of «(J, I,')' t.lw lil'ld of rational UllIlllll'rs. In vipw of Theor('1l\ a-2R, it Illay hi! ),I'marketi that (!very lield (F, 1-,') i:; an extcn~i()n of a field i:;olllorphic to (Q,I,') or to (ZI" h,,' p), aCI'ortiiug as thl" dl:lradl"'i~tie of (F, +, .) is zero 0)' a ).lriulI· ]I. I~ linit.l~

208

RING THEORY

Given a field (F, +, .) and an irreducible polynomial J(z) e FIz], we may ask whether one can construct an 'extension field (F' , +,.) of (F, +,.) in which J(z), thought of as a member of F/[z), has a root. (If degJ(z) = 1, then, in a trivial sense, (F, +,.) is itself the required extension.) Our n~t theorem anawen this question in the aftirmative. Theorem 1-49. (Kronecker). If J(z) is an irreducible polynomial in F[z], then there is an extension'field of (F, in which J(z) has a root.

+, .)

ProoJ. Let (I, +,.) denote the principal ideal of (F[z), +, .) generated by the polynomial/(z); that is to say, I = (f(z». SinceJ(z) is assumed to be irreducible, the corresponding quotient ring (FIz]/ I, +, .) is a field. To see that (FIz]/ I, +, .) constitutes an extension of (F, +, .), consider the natural mapping natl: F[z] - F[x)/l. According to Theorem 3-25, either the restriction nat, I F is the trivial homomorphism or else the triple (natl (F), +, .) is a field isomorphic to (F, +, .), where nat, (F)

=

{a

+ I I a E F}.

But the fint possibility is immediately excluded by the fact that natl (1)

= 1+ I

¢

I,

the zero element of F[z]/ I. Therefore, (F, +, .) is imbeddable in the quotient field (F[z]/ I, +, .) and, in this sense, (F[z]/ I, +, .) becomes an extension of (F,+,')' Next, we need to show that the polynomiaIJ(x) actually has a root in F[x1/ I. If J(z) = ao + alx + ... + a,.x", then, from the definitions of coset addition and mUltiplication, (ao

+ 1) + (al'+ 1)· (x +

1)

+ ... + (a.. + I)· (x + 1)" = ao + alx + ... + a,.x" + 1 . = J(x) + I = 0 + I.

Using the identification, justified above, of the coset a",:+ I with the corresponding element a", E F, we obtain

+ al (x + I) + ... + a.. (z + 1)" = 0 In other words, the coset x + 1 = Ix + I

ao

or J(x + 1) = O. root of J(x).

is the desired

Since each polynomial of positive degree has an irreducible factor (Theorem 3-48), we may drop the restriction that J(z) be irreducible. Corollary. If the polynomial J(x) E F[x] is of positive degree, then there exists an extension field of (F, containing a root of J(x).

+, .)

3-5

POLYNOMIAL RIN08

209

Before illustrating this theorem, let us take a closer look at the nature of the cosets of I = (f(x») in F[x], with the view of expressing F[x]/ I in a more CODvenient way. As usual, these cosets are of the form g(x) 1, with g(z) E F(z). By the Division Algorithm, for each g(x) there is a unique polynomial r(z) such that g(x) = q(z) . fez) + r(z), where r(z) = 0 or deg r(z) < degf(x). Evidently O(x) - r(z) = q(x) . f(x) E I, which implies g(x) and rex) must belong to the same coset, g(x) + I = rex) I.

+

+

From this, we draw the following conclusion: each coset of I in F[x] contains exactly one polynomial which either is the zero polynomial or has degree less than that of f(x). In fact, the cosets of 1 are uniquely determined by remainders upon division by f(z), in the sense that g(x) + I = hex) + I if and only if o(x) and h(x} leave the same remainder when divided by f(z}. If degf(x) = n > 1, say f(x) = ao alX + c..x", the quotient field (F[x)/ I, may therefore be described by

+

+, .)

F[x]/I = {b o

+ ...

+ blx + ... + b.. _Ix..- + I 1

I b" E

F}.

Identifying b" + I with the element b", we see that a typical coset can be (uniquely) represented in the form

+ bl(x + I) + ... + b.. _l(x + 1)"-1. As a final simplification, let us replace x + I by some new symbol X, bo

so that

the elements of F[z)/l become polynomials in X:

= {b o + blX + ... + b.. _IX.. - I I bl: E F}. Observe that since X = x + I is a root of f(x) in F[x)/I, calculations are carried out with the aid of the relation Co + alX + ' .. + a,. X.. = o. F[xJ/ I

We pause now to examine two concrete examples of the ideas just introduced. Example 3-37. Consider (Z2, +2, '2), the field of integers modulo 2, and the polynomial f(x) = x 3 X 1 E Z2[X). Since neither of the elements 0 or 1 is a root of Z3 x 1, f(x) is irreducible in Z2[X]. Theorem 3-49 thus guarantees the existence of an extension of (Z2, +2, '2), specifically the field

+ +

+ +

(Z2[X]/(J(X»),

+, .),

in which the given polynomial has a root. Denoting this root by X, the discussion above tells us that

Z2[X]/(J(X»)

= =

where, of course, X3

{a + bX + eX21 a, b, e EZ2} {O, 1, X, 1 + X, X2, 1 + X2, X + X2, 1 + X + X2},

+ X+ 1 =

'0.

210

3-5

lUNG THEORY

As an exnmple of opcrnting in thilJ field, let UK calculate the inverse of 1 t-)" + ),.2. Before starting, observe that by using the relations ),.:1

= -(),. + 1) = ),. -I-

1,

(our ('o('fIi('i('nts ("ollie from Z2, whence -1 = 1), the degf(~ of any product can he kept h-ss than:t Xow, the problem at hand is to determine elen\('lIt.s a, /1, f E :x ~ f(II' whi(~h

C:U'ryill~ out. t.he lJIult.ipli('atioll I\nd I'mhsLituting for ),. 3 , allll ),. 2 , we ohtain (II 1 /1 1 c) 1 a),. 1 (a 1 11»,.2 =, I.

Rinec 1 is Ilniqudy f(~pn-liCnted by 1 = 1 + 0)" of linear equnl ions (£

1 /1 1- e = 1,

a= 0,

+ 0),.2,

),. 4

in terms of 1, ),.

thilJ yields the system

a+b=O

with solution a = b = 0, e = 1; therefore, (1 + ),. + ),.2)-1 = ),.2. Finally, note that x 3 + x + 1 factors completely into linear factors in Z2[xll(j(x») and has the three roots),., ),.2, and),. + X2: x3

+x +1 =

X) . (x -

(x -

X2) . (x - (X

+ X2)).

Example 3-38. Tlw quadratic polynomial x 2 + 1 is irreducible in H'[x]. For, if x 2 + 1 wert' r('(itll'ihll', it wOII)d be of the form . •'1: 2

-I- 1 =

(a.r

+ I»~ . (ex + d) =

acx 2

1- (arl

where a, b, e, d E R'. It follows at once that ae Therefore be = -(ad), and 1

=

(ae)(bd)

=

(ad)(be)

= -

or rather, (ad)2 = -1. which is impossible. In this instance, the extension field (R'[x]/(x 2 R'[x]/(x 2 1 I) =

(a

=

+ be)x + bd,

bd

=

1 and ad

+ be = O.

(ad) 2

-f 1), +, .) is described by

+ b)" I a, b E R' i),.2 + 1 =

OJ.

Performing the usual operations for polynomials, we see that (a

+ bX) + (e + d),.) =

(a

+ e) + (b + d)X

and (a

+ b),.)· (e + clX)

= (ae -

= (ae -

+ (ad + be)X + bd(X 2 + 1) bd) + (ad -I btl)X. bd)

:3-5

POLYNOMIAL RINGS

211

TIll' Milllilarity of theMe formulas to the rules for addition and multiplication of eompl('x numbers should be obvious, As a mattcr of fact, the field (R'[x]!(x 2 + 1), +,') is adual\y i";olllorphi(, to (C, under the mapping 4>: Jl'[.cI/(x 2 1 1) --> (,' givell hy

+, ,)

4>(a

-1- bA)

= a

+ bi,

H(·forl· pro('('pding, two ('0111111('111.,.; 1t"P in OI'd("', l'it"Ht, EXlllllpl(~ :J-:17 HhoW8 that there ('xist finite fields other than the fields (ZI" -h, ',,) of integers modulo It prime 71, TIll' flu,t that. til(' ficlt! of this example' IUlS 2 3 = 8 e'1e'ments is typical of til(: 1!;(,II('ral ,.;iflmtion: givl'll II prime Ilullllwr 7' Ilnd It positive int.eger n, HIP"p is PXI""t Iy Oil(' (lip t.o isolllorphi";Jll) fipld with [I" pic:nwnt.s. Jlldcml, if j(.r) i,.; /lily irrPlhwiIJl1' polYllolllial of dl'gl"l'l' It ill /'plJt t.llt· tjuot.ilmt lidll (/'p[x]!(J(:r», -1-, ,) eonHiHts of all polynomials bo bRA b,,_lA,,-I, where bk E Zp; Hillee there arc only 7' choice'S for eaeh coefficient bk , we thus obtain Il finit(: fi(~ld with p" members, HI'I'0I1d, til(' I·OIlHt.l"lldioll of TllI'orclII a-4!l yiddi'l lUI extelllSioll of a field (P, 1 , .) ill whil,h a giv('n polynomial f(x) E PIx] splitH off olle lillt'llr fll(,tor. By repeated application cU this procedure, we can build up an extension (F', of (F, in whichf(x), thought of as a member of F'[x], factors into a product of linear factors; that is, the field (F', is large enough to contain all the roots of f(x) (tedmieally srll'aking, the polynomial splits completely in F'[x]). We phrase this result ill the form of an existence theorem.

+

+ .. , +

+, .)

+, .)

+, .)

Theorem 3-50. If f(x) E "'[x] iH a polynomial of po::;itive degree, then there exi,.;t.s UII cxt,ellHion lield W', +,.) of (p, +,.) ill which f(x) factors completl'ly int.o lil1l'llr polynomial::;,

Proof. The proof iH by iuduction on n = deg/(x). If n = 1, f(x) is already linC'ur and (F, +, .) is itself th(l required field. Therefore, assume that n > 1 and t.hat t.iw tlworem is t.ruC' for all polynomials of degree less than n. Now, the polynomial f(x) IlIlIst have some irreducible factor (I(x). By Theorem 3-49, there is lUI ('xtension liPId (/"1, +,.) of (F, +,.) in which y(x), and hence f(x), has a root al; specifieally, 1'\ = P[x]/(g(x»). Thus, f(x) can be written ill F l[X] as f(x) = (x - al) , fl (x). Since degfl (x) = n - 1, there exists, by our induetioll hypothesiH, an (·xt.ension field (F', -+ , .) of (F 1, in which fl (x) = ao(x - a~)(.r. - aa) ... (x - a,,), with ak E F', ao ¢ O. From this, we see thatf(x) call he completely factored into linear factors in F'[x].

+, ,)

Corollary. Lt't, f(x) E F[x] with deg f(x) = n tension of (F, in which f(x) has n roots.

+, .)

>

O. Then there exists an ex-

Example 3-39. To illustratl: this situation, let us consider the polynomial f(x.) = X4 - !ix 2 t- (j = (x:.! - 2) . (x 2 - 3) over the field (Q, of rational lIumbers. F"om EXatllpll':3- ali, x 2 - 2 (mid similarly x 2 - 3) is already known

+, .)

212

3-5

BING THEORY

to be irreducible in Q[~]. So we first extend (Q,

Fl

+, .) to the field (F 1, +, .), where

2)'== {a + bX I a, b eQ; X' - 2 == O},

= Q[z]/(x' -

and obtain the factorization I(z)

= =

(z (x -

+ X) • (x' - 3) V2) . (x + V2) . (x 2 X) • (x

(Aa X' .... 2, one customarily identifies X with

3).

V2.)

However,l(x) does not factor completely, since the polynomial Xl - 3 is irreducible in F I[X], For, suppose to the contrary that X2 - 3 has a root in F 11 say e + dV2, with e, d E Q. Substituting, we find that (el

+U 2 -

3)

+ 2cdV2 == 0,

hence el

+U'

- 3

== 0,

cd== O.

This latter equation implies that either e = 0 or d = 0; but neither c nor d can be zero, since otherwise we would have d ' = 3/2 or cl == 3, which is impossible. Thus X2 - 3 remains irreducible in Fl[X]. In order to factor I(x) into linear factors, it is necessary to extend the c0efficient field further. We therefore construct the extension (F" where

+, .),

The elements of F I may alternatively be expressed in the form (a

+ bV2) + (c + dV2)V3 == a + bV2 + cv'3 + dV6.

It follows at once that the original polynomial factors in FI[x] as I(x)

=

+

+

(x - X) • (x X) • (x - 1') . (x 1') == (x - 0) . (x + 0) . (x - v'!) . (x + v'!).

Observe that the four roots all lie in F ••

+, .)

+, .)

Let I(x) E F[x). An extension (F', of (F, is said to be a 'Plitting field for I(x) over F provided I (x) can be factored completely into linear factors in F'[x], but not 80 factored over any proper subfield of (F', containing (F, Loosely speaking, a splitting field is the smallest extension field in which a given polynomial decomposes as a product of linear factors. To obtain a splitting field for any polynomiall(x) E F[x] of positive degree, we need only return to Theorem 3-50 and consider the family (Fi, of all subfields of

+, .)

+, .).

+, .)

3-5

POLYNOMIAL RINGS

+, .)

213

(F', in which f(x) factors completely (the theorem guarantees the existence of such extensions); then (nF" serves as a splitting field for /(x) over F. Having thus indicated the existence of splitting fields, it is natural to inquire about their uniqueness. As a final topic for this section, we shall prove that any two splitting ficldll of the I!&mc (nonconstant) polynomial are isomorphic; this being 110, one ill justified ill using the definite article and speaking of the splitting field of a given polynomial. Before presenting the main theorem, two preparatory results of a somewhat technical nature are needed. As previously noted, if r is a fixed element of a field (K, +,.) and F!',;;; K, we write F(r) for the set of finite sums:

+, .)

F(r)

=

{ao

+ al . r + ... + ~. r"l aAo E F, n

~

I}.

Lemma. Let f(x) be an irreducible polynomial in F[x] and r be a root of /(x) in some extension field (K, of (F, Then (F(r), ~

+,.)

+, .).

+,.)

(F[x]/(/(x», +,.) under an isomorphism whereby the element r corresponds (j(x». to the coset x

+

Proo/. The mapping II; F[x] - K defined by setting IIf(x) = /(r) is easily checked to be a homomorphism of (F[x], +, .) onto the ring (F(r), +, .). It follows at once that f(x) E ker (II) = {g(x) E F[x] I g(r) = O},

whence (/(x» !; ker (II). One observation is quite pertinent: the possibility that ker (II) = F(x].does not arise, since the identity element of (F[x], +,.) is not mapped onto zero. Asf(x) is assumed to be irreducible in F[x], «(f(x» , +, .) is a maximal ideal of (F[x], +, .), so that (f(x» = ker (II); in other words, (f(x» simply consists of all polynomials in F[x] having the element r as a root. Thus, by the fundamental homomorphism theorem for rings (Theorem 3-18), there exists an isomorphism 8 of (F[x]/(I(x», +,.) onto (F(r), +,.) such that II = 8 nat(/(:<)l: As regards the last assertion of the theorem. we evidently have (I(x»). r = II(X) = (8 nat(/(:<»)(x) = e(x 0

+

0

Although of some interest in its own right. the value of this lemma is that it leads almost immediately to the following theorem: Theorem 3-51. Let t/> be an isomorphism from the field (F,

+..)

onto the field (F', +', .'). Also let f(x) = al + alX + ... + ~x" be an irreducible polynomial in F[x] and 1'(y) = t/>(ao) + t/>(al)Y + ... + t/>(a..)y" be the corresponding polynomial in F'[y]. Then1'(Y) is irreducible in F'[y]. Furthermore, if r is a root of f(x) in some extension field of (F, +.. ) and r'is a root of f'(y) in some extension field of (F', +', .'), then t/> can be extended to an isomorphism 4> of (F(r), onto (F'(r'), with 4>(r) = r'.

+, .)

+'.. ')

214

UlNH '1'ln;OTty

Proof. Let us first extend 4> t.o a mapping f, between the polynomial rings (F[x), and (F'[y], by taking

+, .)

f,g(x) =

+', .') f,(b o -I- blx + ... + II"x") =

4>(b o)

+ I/>(bdy + ... + 4>(b,,)y"

for every polynomial g(x) = bo + blx + ... + b"x" E F[x). Using the fact that I/> ill an isomorphism, it is an easy matter to verify t.hat f, is an isomorphism of (F[x), +,.) onto (F'[y], .')j we leave the render to supply t.he necessary dctni\s. Not i('(! that. for !lny polynomilll Vex) ill FIx}, an clement 0 E F is a root. of g(x) if and only if 4>(0) iN a root of 4iu(x). Indeed, if (I(x) = bo + blx + ... + b..x", we hnvc

+',

(f,(J(;rJ)(r/>(fl»

(/'.)·r/>(ll) 1 .,. 14>(I'")'4>(tt)" "-c

1/>(11 0 -I b l • a -I ••• 1- b,.· a")

~: 4>(u(a»,

from which the nr-;s('rtion follows. In pnrti('ular, the givl'n polynomial f(x) is irrcdueiblc in FIx} if and only if f'(y) = f,f(x) is irredueible in F'[y]. Now, by !.Ill' fort'going lemma, we know that there exist isomorphisms

a: F(I')

-+

F[.rl/(J(x»

and

(3: F'(r') -+ F'[y]/ (J'(y».

Furtlwrmof(', it. is not. diffi('ult to 8how t,hat, thl'1'<' is also an iso1llorphism (F[x]/(J(r», +,.) onto (F'lyl/(J'(y» , defined by T ((I(x)

+ (f(x»)

=

+', .,) f,{,(X) + (J'(y»

,

T

of

fl(X) E FIx].

OblS('rV<' , parti('ularly, that T ('luries the ('Of;(~t, x + (j(x» onto 11 -I- (j'(1J». WI! (!Ollt(,(ld that (F(I'), I,·) ~ (F'(r'), I', .') vin tIl(! ('ompoHition

where «1>: F(l')

-+ F'(l") j

':

thiR situation is portrayed in the diagram below: ."

-r----~----.

Flx]/(I(x))---..;T---_. F'r,]/(r(y))

+', .'),

O'rtninly «I> is an i8()morphism of (F(r), -I,,) onto (F'(r'), individual mnppinjl;s a, T, and (3-1 nrc themselves isomorphisms. nrhitrnry ('I('nlt'lI!. of F, then «I>(a) =

uri

0

T) (a(a)) = (rioT) (a

= fj-I(r/>(a)

-I (f'(1J»)

= 4>(a) ,

+ (j(x»)

for th(l If a is an

I'OI.YNOMIAL IUNGR

215

when('e is actually an extension of tJ> to F(r), Finally, we point out that (r) = =

({r l oT)(a(I'» = urI OT)(X+ (J(x») rl(y+ (J'(y») = 1"

as requin'd, amI tlw t.h"on'lll i:-; 11I'ovl'd in its entirety. For :l silllpl(' illll:-;tl'lltiOlI of thi:; ..p:-;ult, Ipt both (F,I,') and (F', +', .') 1)(' Ute ..pal number fidd (U', +, .); take f(x) E R'[x] to he the irrcdueihlc polynomial /(.1') = ;1'2 I I, >10 t.h II t. f'(y) = y2 -+- 1 [l'l'('nll that. tllIl identity mil» ii'l thl' only illomorphi>lm of (U', +,.) onto it.III'If.1 Finally, we let. ,. = i Ilnd r' i. Tlworl'lll:\ :>1 tlu'n :LKS"rt s that.

(U'U), ,1, .)

~ (lif( - i),

I,')

IIlull'r llll isonlo .. phislll whidl eu .... il's i OIl!.O i. Hillel! (.' -- H'(i) = R'( --i), this mapping is just the eorresp0Jl(knee hetwel'lI a eomplex lIumber and its I'olljllgat P. \Ve arc now ill a positioll to :;how the uniquenl's:;, to within isomorphislIl, of splitting fields; actuall~, we shall prove a somewhat more general r('sult:

+, .)

Theorem 3-52. IA't tJ> be an iiSomorphism of the field (fl, onto the field (F', +', .'). Let f(x) = ao + a.x + ... + anx n E FIx] Ilnd f'(y) = tJ>(ao) tJ>(a 1)11 -I ... -I- tJ>(u,,)yn hI' the eOrr('sl)Onding 1)OIynomiai in F'ly]. If (K, is II splitting field for f(x) and (K', is It splitting field for f'(y), then .p eall be extended to an isomorphism of (K, +,.) onto (K', +', .').

+', .')

+ +, .)

Prollf. Ou .. argllllll'lIt. pl·o(~p(·ds by ilHltu·tion on t.Iw nllmbl'r n of roots of f(x) that IiI' outside of P, hilt, lI(,p(l\e:;s /.0 :-;ay, in K. When n = 0, all the roots of I(x) belong to F alllI (F, is itself the splitting field of f (.r); that is, K = F.

+, .)

This in turn induces !l splitting of the polynomial 1'(11) into a product of linear fnctors in F'[lI], so that K' = F'. Thus, in the ease n = 0, the isomorphism .p itself is tIll' dl': I which iH irredlH'ihll' ill F[.d. LI't. !I'(y) d(,llote the (~OITl'HpOlldillg irreduciblp fador of f'(y). SineI' (K, iH a splitting field of f(x) ()VPl' 1", !I(:r) ill pal'ticulm' must have It root in K, ('all it f. Himilnrly, one of the motH of thl' polynomial I'(y), Hay,.', is It roo/. of (/'(y) in K'. By Theorpm :I-til, '" (':tn hI' I'xtl'nlil'd to all isomorphism .p'I)('tw{'(·n I.Iw liPId!! (F(r), +,.) and (F'(r'), j ',.'). :\ow (K, -I,') is a splitting field of /(x) viewed llS t\ ))Oly-

+', -'),

+, .)

216

3-.

RING THEORY

+, .)

nomial with coefficients from the field (F(r), i in a like manner, (K', +', .' can be regarded as a splitting field of J'(y) over F'(r/). Because the numbc' of roots of f(x) outside of F(r) is less than n, the induction hypothesis pennit us to extend the isomorphism .;' (itself an extension of .;) to an isomorphil;u ~ of (K, +,.) onto (K', +', ./). This completes the induction step and thll the proof of the theorem. Corollary. Any two splitting fields of a nonconstant polynomial f(x) E Fl.!" are isomorphic by an isomorphism ~ such that ~ I F is the identity map.

Proof. This is an immediate consequence of the theorem on taking (F, (F',

+', ./) and.; to be the identity isomorphism

+, .)

o·

iF.

Let us complete the picture by giving a brief application of these ideas. Theorem 3-53. Any two fields having p" elements, p a prime, are isomorphic'

Proof. In view of the preceding corollary, it is enough to establish that all,\ field (F, +,.) with p" elements is a splitting field of the polynomia xl''' - x E Zp[x]. [We remind the reader that (F, being of prime character istic p, has a subfield isomorphic to (Zp, +1" '1').] Now, the multiplicativ( group (F*, . ) of nonzero elements of F has order p" - 1. By Lagrange's Theorem one sees that rp"-l = 1 for all r E F* and consequently r P " - r = 0 for ever~ member of F, including O. In other words, the polynomial xl'· - x E Zp[)' possesses p" distinct roots in F. Since the degrec of xl''' is p", this poly nomial must split completely into linear factors in F[x]. Needless to say, il cannot split in any proper subfield of (F, for no proper subfield contaill' p" elements. Hence (F, is a splitting field of xl''' - x over Zp, as asserted

+, .),

z

+, .),

+, .)

A related result, which 'we shall not stop to prove here, is that any finit( field has p" elements for some n > 0, where the prime number 11 is the character· istic of the field. Granting this, Theorem 3-53 may be interpreted as assertilll! that any two finite fields having the same number of elements are isomorphi(', PROBLEMS

+, .)

1. For an arbitrary ring (R, with identity, prove that a) the polynomial 12:" E cent R[x], b) if (1, is an ideal of (R, then (l(x], is an ideal of the polynomial

+, .)

+, .), +', .').

+, .),

+, .)

ring (R(x], c) if (R, +,.) and (R/, +/,.') are isomorphic rings, then (Rlx], +,.) is isomorph ii' to (R/(x],

+, .)

2. Given (R, is an integral domain, show that a} the only invertible elements of R(x] are the constant polynomials determined by the invertible elements of R, b) the characteristic of (R(x],+,·) is equal to the characteristic of (R,+, .). 3. Prove that no monic polynomial can be a zero divisor in (R[x],

+, .).

3-5

POLYNOMIAL RINGS

217

4. Show that the relation - defined by taking f(x) - g(x) if and only if deg fex) = deg g(x) ill an equivalence relation in the set of nonzero polynomials of R(x]. 5. a) Let P be the set of all polynomials in Z(x] with constant term 0: P = {a\x

+ a2x2 + ... + a"x" I a. E Zj ft ~

+, .)

I}.

+, .).

Establillh that the triple (P, is a prime ideal of (Z[x1, b) Show the principal ideal «12:),+,') is a prime ideal of the polynomial ring (Z[x], but not a maximal ideal. 6. If (Il, +,.) iM a commutative ring with identity, prove that the polynomial I ax is invertible in R[x] if and only if the element a is nilpotent. 7. Let (R, +,.) be a commutative ring with identity and the element r E R be fixed. a) If R(r) denotes the set

+, .),

+

R(r) = U
+, .)

+, .).

prove that the tripJe (R(r), forms a subrlng of (R, b) Show that the mapping.: R[x) ..... R(r) defined by.(f{x» = f(r) is a homomorphism of (R[x), onto (R{r), 8. a) Given that fex) - x~ + 22:3 - U + 1 and g(2:) - 22:2 u + 1, find polynomials q(x), rex) E Z5[2:] for which f(x) - q(x) • g(2:) rex). b) Returning to the ring (R,+,·) of Example 3-8, show that the polynomial (a, 0)x2 E R(x] has infinitely many roots in R. 9. For f(x) = Go 412: ~-.- a"x" E 0[2:], define the polynomial lex) by

+, .)

+, .).

+

+

+

+ +

+ 41X + ... + 4"x", where 4. indicates the complex conjugate of CI•. Prove that lex) ... c10

a) rEO is a root of f{x) if and only if , is a root of lex). [Hint: f{rj - 1(1').] b) if f(x) E R'[x] !; C[x] and rEO is a complex root of f{x), then I' is also a root of f{2:). 10. Assume that p/q (where l' and q are relatively prime) is a rational root of the polynomial f(x) .. Clo + CltX + ... + CI"x" E Z[x). Verify that p I Go and q I a,.. 11. Let (R,+,·) be a commutative ring with identity and f(x), g(x) be two polynomials in R[x] which are not both zero. Then d(x) E R[x] is said to be a greate8t common difli_ (gcd) of f(2:) and g(x) provided i) d(x) is a divisor of both f(x) and g(x), and ii) every polynomial which is a divisor of fex) and g(x) also divides lI(x). Prove that any two polynomials fex), g(x) E R[x], not both aero, have a unique monic gcd dex) in R[x) which can be expressed in the form d(x) = f(x) • p(x)

+ g(x) . q(x)

for some p(x), q(x) E R[x]. [/lint: Follow the pattern of Theorem 1-12.1

218

a-5

UINe; TIIEOlty

12. Let (R, +,.) 1)(' a commutative ring with identity, and let f(x) E Rlx). The function ]: R -+ R defined by taking ](r) = fer) for all r E R is called the polynomial function associated with f(x). Assuming that S denotes the set of all polynomial functions alisoeiated with elements of Rlx), prove that

+, .)

a) the triple (S, forms a ring, b) the> mapping a: R[xl --> S given by af(x) = ] is a homomorphism of thC' rinl( (RlII, I.') 0111.0 (S, I, .), e) if rE U i>! fixed aud Ir = {]ES!](r) = 01, then (lr,l,') ill an idC'al of

(8,+,')' that if the> integral domain (R, +,.) ha.'! an infinite number of elC'mentH, tlwn distinct polynomial>! in Rlx) ineluce distinct polynomial functions. h) Givll an "XILlIlph~ or two di",tirwt polynomialH whil:h induce the flame Jlo\ynowild fundioll.

13. a)

~how

14. J.d. (Il, I,') I"'IL ,·olllll\ut.atiVl' rillg with idl'"t.it.y. IA·t. Ow flllwtioll 6: Ulx) -+ t 11l' 1l0-"ILIII·.1 dt'Til'III1:,'c fllIlrtiOll, ht, dl'filll'd IL>! follows: if f(x)

= ao + alx +

... + anx" E

HlrJ.

Rlx),

then !Jf(x) = al

+ 2a2X + ... + na"x,,-I.

For any f(x) , g(x) E Rlx) and any a E R, verify that a) /J(f(x) h) fI(af(x»

I g(x»

+ !Jg(x), of(x) . g(x) + f(x) . og(x).

= flf(r)

= a!Jf(x) ,

c) fI(f(x)' g(x» = nonz('fO t('rms of f(x).)

[/lint: Induct on the number of

15. ),e·t W,I, -) he a commutativ(' ring with identity, and let r E R be a root of the nonz('ro polynomial f(x) E R[x). We call r 0. multiple root of f(x) if f(x)

=

(x -

r)" . g(x),

n>

I,

wh('re g(x) E Rlx) is a polynomial such that g(r) ~ O. Prove that an element r E R i.. 0. multiple root of f(x) if and only if r iii 0. root of both f(x) and of(x). 16. Determine a) 0.1\ irreducibl{' Jlolynomials of degree 2 in Za[x], and b) all irreducible polynomials of degree 3 in Z2[X).

+, .)

17. Let (f', b(' a fie>l.l anll/(x) E F[x) be a Jlolynomial of degree 2 or 3. Establish that f(x) is irre>ducihle in FIx) if and only if f(x) ha.'i no root in F. Give an l'xample which >!howH that this fI'sult IU'C't\ not he true if t\e·g fIx) ~ 4. IS. Prov!' tluLt if f(x) is all irn'dllcible polynomial in Z[xl. then f(x) i>! also irredllcibl(' whf'n rf'gardf'd 1\." all 1'1C'lIll'lIt of Qlx). 19. Df'riw Iht· following analog of Euclid's LI'llIma: L.. t (R, +,.) be a commutative ring with iell'nlhy and f(x) be all irrC'ducihlt' polynomial ill Rlx). If f(x) divides the prulhH't g}(x) . U2(:r) .•• g,,(x), then !(x) divi.l(·s gk(X) for some k, 1 ::; k ::; n.

+, .)

20. In regan 1 to ProblC'm 7, show that the ring (R(r), is a field if and only if ker (I/» = (f(x» where I(x) is an irreducible polynomial in R[x).

219 21. a) Prove thl' Eisl'nstein Criterion for irreducibilil,y: If f(x) = ao -I- alx

+ ... -I- anxn E Z[X)

and p is a prime number such that pi ak (k = 0, \, ... , n - I). pt an. p2 tao. then fix) is irrl'ducible in Z[x). [!lint: Proof is by contradiction.) h) 1l1<1' pllrt. (11) t.o Rhow I.hat. t.he polynomii~1 2xa -- fi.r. 3 9.r. 2 - 15 ill irreducible ill XI.d.

+

22. Civl'n fer) = .c 2 j .c -1- 2, an irredudhll' polynomial in Za[x), conMtruct the multiplieation tahle for the field (Za(xl/(j(x» ,

+, .).

-I-

x 2 -I- 1 E Z2[X) fad,ofl~ into linl'ar factors

2:1. Vt'fify that the polynomial f(x) = ill X21r.1/(j(;r».

x!!

:.!-l.

2), I,,) forlll'" b E: (J:.

I'rov," thnt. t.lll' tripl" «(JI.I'I/Ir. 2

W, I, .), wlll'ro o I,'

~ [II

I !Jv'21

-

II,

IL

li .. ld j,mlllorphi,·. to tl ... fi"lo!

25. DeRcrihe the splitting fields of the following polynomials: a) x 3 c) x4

-

+ +

3 E Q(x),

+ 2X2 + 1 E

b) x 2 X I E Zs[x), d) (x 2 - 2)(x 2 E Q[x).

R'[x),

+ \)

26. Let f(x) E FIx) be irreducible and r. 8 be twu roots of f(x) in some splitting field. Show that (F(r). -1-, .) ~ (P'(8). -1-, .) by a unique isomorphism that leaves every clement of F fixl'd. 27. Hhow that if g(x) i,; a factor of the polynomial f(x) E FIx), then any splitting field of f(x) over F contains a subfield which i!l a Hplitting field of g(x).

+..)

+, .)

2H. Prove: An extl'nsion field (K, of (Ti'. is a splitting field of the polynomial fix) E F[x) if and only if both i) fix) factors eompletely into lim'ar faetors in K[x), and also ii) (K, -I,') ill gem'mh',1 by F and all til(' roots of fix) in K. 29. Let hex), hex) be two irreducibl!> polynomials of degree n in Z,,[x) , p a prime nlllllbl'r. If ri is a root of ''(x) in 80m!> splittin~ fi('ld (i = I, 2), establish that

(Zp(rll,+,') ~ (X,,(r2).-I-•. ). 30. Derive Fermat's Little Theorem: If p is a prime number and a ¢ 0 (mod pl. then a,,-l == 1 (mod pl.

3-6 BOOLEAN RINGS AND BOOLEAN ALGEBRAS

The thcory of Boolean algebras is IlII algehmie eOlillt.erpart to thc logical theory of the I'l\lelllllll of propm;itiolU;. It.s origius lit' in the work of the English mat.lwnmti('ian Gcorge Boole (ISI!i-1Sfl4), who first attcmpted 1.0 give a systematic t.reatnH'nt of lop;i(~ by abstract methods. Siu('(' su(~h n strueture ma.y be viewed as t,hc postulational ahHt.met ion of the rules gov!'rning the algebra of sets. it providcs a suitable topie with whi('h to eonelude our discussion of twoopcrat.ional systemll. Indecd, as will be evidenced shortly, Boolean algebras may be subsumcd undcr thc gcncral theory of rings.

220

3-6

RING THEORY

We begin by studying the properties of a special class of rings, which we shall designate as Boolean rings. '. Definition 3-24. A Boolean ring (R, +, .) is a ring with identity every element of which is idempotent; that is, a 2 = a for every a E R.

It should be pointed out that the existence of an identity is frequently omitted in the definition of a Boolean ring. (One can show that if the number of elements in a Boolean ring is finite, then an identity element always exists.) The definition given here, however, will be more convenient for the applications we have in mind. Let U8 pause to give several examples of the concept just introduced. Example 3-40. The ring of .integers modulo 2, (Z",

ring, since 0·" 0

= 0 and 1 ." 1 =

+:h .,,),

forms a Boolean

1.

Example 3-41. The ring (P(X),.1, n) of subsets of a nonempty set X is easily verified to be a Boolean ring. In this case, we have A n A = A for every subset A ~X. Example 3-42. For It. 1~1!H obvious example of a Boolean ring, let R consist of all (unctioll8 from lUI arbitrary noncmpty set X to Z" with the operations defined pointwise; specifically, if I and g are in R, then

(J

+ g)(x) =

(f· g)(x)

=

I(x)

+" g(x),

f(x) ., g(x),

(x EX).

Under these operations, the triple (R, +,.) is a commutative ring with identity; the proof is straightforward and will not be given in detail. To establish the idempotency condition, we proceed as follows: If the function fER is such that f(x) = 0, then (J2)(X) = I(x) '"/(x) = 0. 2 0 = O. While if lex)

=

1, then (J")(x) =

In any event, (J')(x)

= f(x)

lex) ·,/(x)

= 1·,1 = 1.

for every x in X, whence'"

= f.

Our ultimate purpose is to prove the celebrated theorem of Stone which asserts that each Boolean ring may be represented by a ring of sets. Looking forward to this result, we first develop a number of the fundamental properties of Boolean rings necessary to the proof. While the conclusions obtained are somewhat restrictive (Boolean rings have an almost embarrassingly rich structure), they bring together much of the material developed earlier. 3-54. Every Boolean ring (R, +, .) is a commutative ring of characteristic 2.

Theorem

3-6

221

BOOLEAN RINGS AND BOOLEAN ALGEBRAS

Prooi If a and h are arbitrary elements of H, then

+ h)2 = a2 + a· h + h· a + h2 = a + a· h + b· a + b, and hence a . b + b . a = O. In particular, setting a = h, we obtain 2a = a + a = a 2 + a 2 = 0 a

+h =

(a

+, .)

for every a E H. This shows that (H, is of characteristic 2. But then by adding a . b to both sides of the equation a . b + b • a = 0, we obtain

We proved earlier that, in any commutative ring with identity, maximal ideals are automatically prime ideals. For Boolean rings, the converse is also true.

+, .) be a Boolean ring. A proper ideal (1, +, .) +, .) is prime if and only if it is a maximal ideal. Proof. It is sufficient t.6 show that if the ideal (I, +, .) is primo, then (1, +, .) is also maximal. To see this, suppose (J, +, .) is an ideal of (H, +, .) with the property 1 e J s; H; what we must prove is that J == H. If G is any element of J not in 1, then a· (1 - a) = 0 E 1. Using the fact that (1, +, .) is a prime Th.....m 3-55. Let (H,

of (H,

ideal with a

e 1, we conclude

1- aE IeJ.

As both the elements a and 1 -

(J

1= a

lie in J, it follows that

+ (1 -

a) E J.

The ideal (J, +,.) thus contains the identity, and consequently J == H. A natural undertaking is to determine which Boolean rings are also fields. We may dispose of this question rather easily: up to isomorphism, the only Boolean field is the ring of integers modulo 2. Theorem

3-56. A Boolean ring (H, +, .) is a field if and only if (H, +, .)

~

(Z2' +2, '2)'

Proof. Let (H, then have

+, .) be a

Boolean field. For any nonzero element a

E

H, we

This argument shows that the only nonzero element of H is the identity; in other WOrdR, R = {O, I}. But any two-element field is isomorphic to (ZI' +1, '1)' The opposite direction of the theorem is fairly obvious.

222

3-6

lUNG TIIF..olty

+, .)

Corollary. A proper ideal (I, of the Boolean ring (R, ideal if nnd only if (RI I, +, .) ~ (Z2' +2, '2).

+, .) +I

Proof. Fil'8t, note t.hut the quotient ring (HI I, since (a + 1)2 = a 2 + I = a

+, .) is a maximal

is itself a Boolean ring,

for each clement a in R. By Theorem 3-32, (I, +,.) is a maximal ideal if and only if (HI I, is a (Boolean) field. An appeal to the above theorem now completes the proof.

+, .)

The next theorem is a major olle and re1luires a preliminary lemma of some difficulty. Lemma. Let (H, I,') he a Boolean ring.

there exiHtH a homomorphiHm f from (H, such thatf(a) = 1.

Proof. Let (I, that is, The set I

+, .)

For cnch nonzero clement a E H, +, .) onto the field (Z2, +2, . 2)

be the principal ideal generated by the clement 1 + a, 1= {r·(l+a)lrER}.

R, sinee the identity is not a member of I. Indeed, if 1 E I, then

~

1

=

r· (1

+ a)

for some choiee of r in H; this means

+ a)2 = (r· (1 + a») . (1 + a) = 1· it folloWH that a = 0, contrary to hypothesis.

1

=

r· (I

(1

+ a),

from which BCCIUlHe (/, ~I •. ) is a proppr iil(,lLl, Theorem a-ao aHHurefl the ()xistence of 1\ mnximal ideal (111, +,.) of (fl. + .. ) sud, that I ~ M. In light of the result just proved, the associated quotient ring (RI 1II, will be isomorphic to (Z2, +2. '2) vin KOIlW homomorphism g. W(~ may thl'refom define I~ funet.ion f: U --+ Z2 hy t.aking f = !I natM, where nlLtM ill Himply the lIatural mapping of U Ollto Illlll. The situation ill conveniently depicted by the following diagram of maps:

+, .)

0

'~/Z',~

._

RIM

The rcmllinil('r of till' proof amounts t.o showing that t.he function I, flO i1efillC'i1, hUH t hI' 11I'/)pI'rlil's ILH.'!I'rted in till! statement of the tlwormn. Plainly, f is hoth 1\1\ unto mup and a homomorphism I)('illg the composition of two such

223

:~-6

functions. Rillce 1 + a E 1 ~ Af, the ('oset 1 1

+a+

Af

=

Af, so that

+2 f(a) = f(1) +2 f(a) = f(l + a) = g(1

+ a + 111) =

But, 1 -h f(a) = 0 if and only if f(a)

=

(!(III) = O.

1, which finisheR the proof.

An immediate consequence of this lemma is the following corollary. Corollary. Every Boolean ring (R, +,.) iR a semisimple ring; that is to say, rad R = {O}.

I'roof. In order to n.rrivl! at a cOIlt.rll.didion, w(~ aliHulnc t.hltt a E rad Il with a "., O. Thrn t.here exi8t.8 a homomorphism f from (R, -1--, .) onto (Z2, +2, '2) for whi<·h f(a) = 1. It follows that the ideal (ker (f), +, .) must be a proper ideal of the ring (R, +, .). Henee there is some maximal ideal (111, +,.) of (R, +,.) with ker (f) ~ M. In particular, the element 1 - a E ker (f) ~ M. But also a E rad R ~ 111, which implies

1

= a + (1

- a) EM.

This leads at· once to M = R, the desired contradiction. Having a!i.'iembled the necessary machinery, we now Ret ourselves the principal taHk, that of showing that each Boolean ring is (~8sentially a ring of sets. This theorem, now considered a landmark, was first proved in 1936 by the American mathematician, l'larshall Stone. Theorem 3-57. (Stone Representation Theorem). Every Boolean ring (R, ill isomor(lhie t.o a ring of Ilubllets of !rome fixed 8
+, .)

Proof. To begin the attack, let H denote the collection of all homomorphisms from (R, +, .) ont.o the field (Z2, +2, '2)' Next define It flllWtioll h: R -. P(H) hy assigning to caeb element a E R t.hose members of II which assume the value 1 at a; in other wordll, h(a) = {fEHlf(a) = I}. While the notation is perfectly clear, let us emphasize thll.t h is a set-valued function in the sense that its functional values are certain subsets of H. By means of this function, we shall establish the isomorphism mentioned in the theorem. Let us now give some details: FOl' any fEll, the productf(a) '2 feb) = 1 if and only if both f(a) = 1 and feb) = 1. This being so, we eon('lude h(a· b) =

UE

H I f(a . b) = I}

U E II I f(a)

UE

'2

II I f(a) =

feb)

I:

11 n U E

II I feb) =

Il

= h(a)

n h(lI) ,

224

RINO THEORY

showing that the function h preserves multiplication. The verification that h(a

+ b) =

h(a) 11 h(b)

is equally straightforward, depending chiefly upon the observation that the sum/(a) +2/(b) = 1 if and only if one of /(a) or /(b) is 1, while the other is 0; the reader may easily fill in the steps for himself. These remarks demonstrate the fact that h is a homomorphism from (R, +,.) into the ring of sets (P(H), 11, n). All that is needed to complete the proof is to show that h is a one-to-one function or, what amounts to the same thing, that ker (h) = {O}. But this follows immediately from the preceding lemma which asserts that the set h(a) is empty if and only jf a = 0, whence,

=

ker (h)

{a E R I h(a)

= 0} =

{O}.

The pieces all fall into place, and we see that the ring (R, +, .) is isomorphic to a subring of (P(H), 11, n). The definition of a Boolean algebra which we are about to present is based on a structure introduced by E. V. Huntington in 1904. A variety of other sets of postulates could be chosen that would define the algebra equally well; indeed, few areas of mathematics have received more diverse postulational treatment. Aesthetically speaking, it seems desirable to build our theory on is few assum~ tions as possible. The axiom system quoted below was therefore selected with the intention that no axiom could be derived from the others. Definition 3-25. A Boolean algebra is a mathematical system (B, V, 1\) consisting of a Donempty set B and two binary operations V and 1\ defined on B such that

(PI) Each of the operations V and 1\ is commutative; that is,

aVb=bVa,

al\b=bl\a

for all a, b E B.

(P 2 ) Each operation is distributive over the other; that is,

a V (b 1\ c)

=

(a V b) 1\ (a V c),

a 1\ (b V c)

=

(a " b) V (a 1\ c)

for all a, b, c

e B.

(P.) There exist distinct identity elements 0 and 1 relative to the operations V and 1\, respectively; that is,

aVO=a,

al\l=a

for all a E B.

(P,) For each element a E B, there exists an element a' E B, called the complement of a, such that

a V a' = 1,

a 1\ a' = O.

BOOLEAN RINGS AND BOOLEAN ALGEBRAS

225

Example 3-43. An obvious example of a Boolean algebra, but nonethe1eas an

important one, is the system (P(X), u, n), where X is a nonempty set. It is apparent that we should take 0 = 0, 1 = X, and whenever A!;;;; X, A' = X-A. More generally, if B is any family of subsets of X, including 0, which is closed under unions and complements, then (B, U, n) will be a Boolean algebra, in fact, a Boolean subalgebra of (P(X), n, U). Example 3-44. For an illustration quite removed from the algebra of sets, consider the set B of positive integral divisors of 10, that is, B == {I, 2, 5, IO}. Given elements a, bE B, we define a V b to be the least common multiple (lcm) of a and b, a A b to be the greatest common divisor (gcd) of a and b:

a V b

=

lcm (a, b),

a A b

= gcd (a, b).

The tables for these operations are given below.

1

V

1 2 5 10

5 10

A

1

1 2 5 10 2. 2 10 10 5 10 5 10

1 2 5

10 10 10 10

10

1 1 1 1

2

2

5 10

1

1 1 1 2 5 5 5 10

2

1 2

The formal verification that the triple (B, V, A) constitutes a Boolean algebra is left as an exercise. (We should caution the reader that in this example the integer 1 plays the role of the identity element for the operation V, while the integer 10 serves as the identity element for the operation A.) A quick inspection of the foregoing tables will reveal the various complements to be l'

=

10,

2'

=

5,

5'

=

2,

10'

=

1.

We ca.ll attention to the fact that a' is simply the quotient when 10 is divided bya. The first thing one notices on inspection of the axiom system for a Boolean algebra is the perfect symmetry or duality between the properties of the two operations V and A. That is to say, if V and A are interchanged in the axioms and if, at the same time, 0 and 1 are also interchanged, then the properties are merely permuted amongst themselves. This principle of duality permits us to state all theoreIns in dual pairs (unless, of course, a statement happens to be its own dual) and guarantees that the proof of one of the pair of statements will be sufficient for the establishment of both; the proof of the dual theorem is obtained by maki~g the appropriate interchange of symbols in the proof of the original theorem. We now proceed to deduce from the postulated properties of the operations in a Boolean algebra a series of further properties, including, for instance, the

226

3-fi

lUXG THEORY

aSHOciative laws for V and 1\. Dual statements are placed side by side; ill view of the principle of duality, only one statement from each dual pair need be proved. In order to condense the demonstrations, we shall arrange the steps 80 far as possible one under another, citing to the right those I>ropositions used in passing a('ross succcssive equality signs. Theorem 3-58. III !LilY Boolc!L1I ILlgchra (8, V, 1\), the following prolK!rtil'H

hold: 1) The c1cmellts 0 and 1 are unique. 2) For each clement a E B, a V a

= a,

al\a=a.

=

a 1\ 0

:J) For l'll.{'h dement a E B, a V 1 4) For eaeh pair of elements

1,

= o.

a, b E B,

a V (a 1\ b)

=

a,

a 1\ (a V b) = a.

Proof. To establish (1), we need only appeal to Theorem 2-1. The proof of (2) is indicated below:

a=aVO = a V (a 1\ a / )

= =

(a V a) 1\ (a Va')

(a V a) 1\ 1

=aVa

(by P a) (by 1>4) (by P2) (by P 4 ) (by Pal.

We obtain (3) as foIlows: l=aVa'

= a V (a' 1\ 1) = (a Va') 1\ (a V 1) = 1 1\ (a V 1) =aVI

(by (by (by (by (by

P4) )la) P2) P4)

Pal.

The proof of (4) requires the use of (3): a=al\l = a 1\ (b V 1) = (a 1\ b) V (a 1\ 1)

= (a 1\ b) V a = a V (a 1\ b)

(by Pa) (by 3) (by P 2 ) (by P a) (by PI)'

3-6

BOOLEAN RINGS AND BOOLEAN ALGEBRAS

227

We did not include the 8II8Ocint.ive laws for V and 1\ among our axioms for a Boolean algebra, as is frequently done, since they arc logical consequences of the properties listed. This is demonstrated in our next theorem. Theorem 3-59. In any Boolean algebra (B, V, 1\), each of the operations V IUlIl 1\ is I\IIHOdat.iv(·; thl~t. is, for pVl'ry t.ripll· of I'll'tnc'nts a, b, c e B, a V (b V c)

=

(a V b) V c,

a 1\ (b 1\ c)

=

(a 1\ b) 1\ c.

Proof. First, set x = a V (b V c) and y = (a V b) V c. We wish, of course,

to prove that x = y. Note that

la

a 1\ x = (a 1\ a) V

1\ (b V c»)

= a V fa 1\ (b V c») =a

(by 1'2) [by Theorem 3-58(2») [by Th(.'Orem 3-58(4)1

and also a 1\ Y = [a 1\ (a V b)] V (a 1\ c)

a V (a 1\ c)

=

=a

(by P 2 ) [by Theorem 3-58(4») [by Theorem 3-58(4)].

Therefore a 1\ x = a 1\ y. Now, a' 1\ x

=

(a' 1\ a) V [a' 1\ (b V c)]

=

0 V [a' 1\ (b V c)]

= a'

1\ (b V c)

(by P 2) (by Ph P,,) (by P a)

and also

a' 1\ y

=

[a' 1\ (a V b») V (a' 1\ c) = (a' 1\ a) V (a' 1\ b») V (a' 1\ c) = [0 V (a' 1\ b)] V (a' 1\ c)

= Therefore a' 1\ x

(a' 1\ b) V (a' 1\ c)

= a'

(by (by (by (by

P 2) P2) Ph p.) Pa).

1\ y. From these observations, we conelude that

(a 1\ x) V (a' 1\ x) (a 1\ a') V x

= =

(a 1\ y) V (a' 1\ y) (a 1\ a') V y

IVx=IVy

x=y

(by Ph P 2) (by P,,) (by Pa),

proving the associative law for the operation V; that 1\ is also associative follows by a dual argument.

228

3-6

BING TBJlORY

As yet nothing has been said about the properties of complementation. In the next group of results, we sIuill prove, among other things, that each elem6nt has & unique complement; thus, ' may be viewed as & function from B into itself (as & matter of fact, onto the set B).

Theorem 3-60. In any a E B Boolean algebra (B, V, 1\), the following hold: 1) Each element a E B has a unique complement. 2) For each element a E B, a" = a. 3) 0' = 1 and l' = o. 4) For all a, b E B, (a V b)'

=

(a 1\ b)'

a' 1\ b',

=

a' Vb'.

Proof. For (1), asaume there are two elements x and y of B such that

= 0,

a V x = 1,

a 1\ x

a V 11 = 1,

al\y=O.

We then have (by Pa) (by hypothesis) (by P,) (by p.) (by hypothesis) (by Pa).

x=xl\1

= x 1\ (a V y) = (x 1\ a) V (x = (a 1\ x) V (x = 0 V (x 1\ y)

1\ y) 1\ y)

=xl\y

In the 8&11le manner, 11 = 11 1\ x = x 1\ y, 80 that x = 11. Accordingly, any two elements asaociated with a as specified by axiom p. must be equal; in other words, the complement a' is uniquely determined by a. From the definition of the compleinent of a, a V a' = 1 and a 1\ a' = O. Hence, by Pit and a' 1\ a = O. a' V a= 1 From this, we conclude that the element a is the complement of a': (I," = (a')'

=

a.

Using the uniqueneaa of the complement and the relatioDB

o VI = it follows that 0'

=

1. But then 0

1,

01\1=0,

= 0" =

1'.

3-6

BOOLEAN RINGS AND BOOLEAN ALGEBRAS

229

Finally, we prove the first statement of (4). Note that, since complements are unique, it is enough to establish (a V b) V (a' A b') = 1,

=0

(a V b) A (a' A b')

Now, (a V b) V (a' A b')

=

[(a V b) Va'] A [(a V b) Vb']

= [(a Va') V b) A [a V (b Vb')] = (1 V b) A (a V 1) =lAI

=

1

(by P 2 ) [by Ph Theorem 3-59] (by p.) [by Theorem 3-58(3)] (by Pa).

Furthermore, (a V b) A (a' A b')

=

(a' A b') A (a V b)

= [(a' A- b') A a] A [(a' A b') A b)

=

b'l

A [a' A (b' A b)] = (0 A b') A (a' A 0) =OAO [(a' A a) A

=0

(by PI) (by P 2) [by Ph Theorem 3-59] (by Pl. p.) [by Theorem 3-58(3») (by Pa).

These considerations imply that (a V b)' = a' A b'. The last three theorems do not, in any sense, exhaust the theory of Boolean algebras; we could continue to deduce a large number of results. This brief outline should serve, however, to give some idea of the structure of the system, as well 8.8 to I>repare the way for our final objective-that of showing that every Boolean algebra can be transformed into a Boolean ring and vice versa. As a first step in this direction, we indicate how, by the introduction of suitable definitions of addition and multiplication, a Boolean algebra may be converted into a Boolean ring. The argument relies heavily on the results of the previous three theorems. Theorem 3-61. Every Boolean algebra (B, V, A) becomes a Boolean ring

(B,

+, .) on defining addition and mUltiplication by the formulas a

+b=

(a A b') V (a' A b),

a· b = a A b,

(a, b E B).

Proof. It is obvious that addition as defined above is commutative, for

a

+b=

(a A b') V (a' A b) = (b' A a) V (b A a')

=

(b A a') V (b' A /I) = b

+ a.

230

3-6

RING THJo:OllY

Furthl!rmore, a

+0 =

(a A 0') V (a' A 0)

= (a A 1) V (a' A 0) =aVO=a and a

+a =

«(,I

A a') V (a' A a)

=

0 V 0

=

O.

This shows that the element 0 acts as the identity for the system (B, +), while each element is its own (additive) inverse. To establish the associativity of addition, let us first perform a preparatory calculation: (a

+

b)'

=

[(a A b') V (a' A b)]'

=

(a A b')' A (a' A

bY

= (a' V b) A (a Vb') = [(a' V b) A a) V [(a' = (a A b) V (a' A h').

V h) A b']

Utilizing this relation, we then have (a

+ b) + e =

[(a

+ b)

A e') V [(a

= [«a A b') = (a A b' A

(a'

V

+ b)'

A b») A

A e)

e]

V

_

[«a

A b) V

(a'

A

b'»

A

e)

e) V (a' A b A e) V (a A b A e) V (a' A b' A e).

Note, however, that the foregoing expression is symmetric in a, band e; that is to say, it is unaltered by permuting these clements. Thus, after interchanging a and c, we obtain (a

+ b) + c =

(c

= a

+ b) + a + (b + e).

From all this, one may infer that the pair (B, +) is a commutative group. Turning next to a discussion of multiplication, it is evident that both the commutative and associative laws hold, while 1 serves as the mUltiplicative identity. Because a2

=

a· a

=

a A a

=

a,

each element a in B is also idempotent. Finally, to establish that the triple (B, +,.) is a Boolean ring, it remains only to verify that multiplication is distributive over addition. We may dispose of this rather easily, since a· (b

+ e) = =

a A [(b A c') V (b' A c)]

(a A hAc') V (a A b' A c),

a-Ii

JlOOLEAN IUNUR ANI> 1I01ll••:AN ALUEDUAS

231

wher('IUI

a· b

+ a· e =

(a /\ b) + (a /\ e) /\ b) /\ (a /\ e)'] V [(a /\ b)' /\ (a /\ e)] = [(a /\ b) /\ (a' V c')l V [(a' Vb') /\ (a /\ e)] = (a /\ b /\ a') V (a /\ b /\ e') V (a' /\ a /\ e) V (b' /\ a /\ e) = (a /\ b /\ e') V (a /\ b' /\ e).

= [(a

The proof of the theorem is therefore complete. We now reverse this process; in other words, we start with a Boolean ring and transform it into a Boolean algebra by suitably defining the opera.tions V and /\. Theorem 3-62. Every Boolean ring (B, (B, V, /\) on defining

a V b

= a + b + a· b,

+,.) becomes a Boolean algebra

a /\ b

= a' b,

(a, b E B).

Proof. That V and /\ are both commutative followli immediately from the commutativity of the operations in (B, A simple calculation will show that /\ is distributive over V :

+, .).

a /\ (b V e)

= = =

a . (b + e + b . e) a· b + a . e + (a· b) . (a· e) (a· b) V (a· e)

=

(a /\ b) V (a /\ e).

The verification of the other distributive law relics on the fact tha.t, since 2x = 0 for every clement of the ring (R, +, .), it is ullnecessary to distinguish betwccn addition and subtraction:

(a V b) /\ (a V e)

(a + b + a ' b) . (a + e + a . e) = a+a·b+a·b+a·e+b·e+a·b·c+a·c +a·b·c+a·b·c = a+b·c+a·b·c

=

=a

V (b· c)

=

a V (b /\ e).

If 0 and 1 are the additive and mUltiplicative identities of (B,

a V 0= a+O+a'O= a,

+, .), then

a/\l=a·l=a

for every a E B. Filllllly, we hllve

+ a) = a + (1 + a) + a . (1 + a) = 1 + 4a = (1 + a) = a' (1 + a) = a + a 2 = 2a = 0,

a V (1 a /\

1,

232

3-6

RING THEORY

which implies 1

+ a is the complement of a in (B, V, 1\), that is a'=l+a.

Theee computations show that the postulates of Definition 3-25 are all 88.tisfied and, consequently, the triple (B, V, 1\) is a Boolean algebra. Taken together, Theorems 3-61 and 3-62 indicate that the theory of Boolean algebras is equivalent to the theory of Boolean rings. What is to be considered remarkable is the identification of a. notion a.rising out of questions of logic and set theory with a. system amenable to the powerful techniques of modem algebra.

PROBLEMS

+, .), every triple of elements a, h, c E R satisfies (a + h) • (h + c) • (c + a) = O.

1. Prove that in a Boolean ring (R, the identity

2. If a Boolean ring (R,+,·) has at least three elements, show that every nonzero element except the identity is a divisor of zero. [Hint: For a, hER, consider the product (a h) • a • h.] 3. Prove that any ring (R,+,·) in which each element is idempotent can be imbedded in a Boolean ring. [Hint: Let R' - R X Z2 and mimic the argument of Theorem 3-16.] 4. a) Let (R, +,.) be a commutative ring with identity and S the set of idempotents of R. For a, h E S, define the operation * by taking

+

a * h .. a

+h-

2(a . h).

Prove that the triple (8, *, .) forms a Boolean ring, known &8 the idempoUnt. Boolean rifl(J of (R, b) In particular, obtain the idempotent Boolean ring of (Z12,+12, '12)' 5. Suppose (1,+,·) is an ideal of the Boolean ring (R,+, .). Show that (1,+,.) is a proper prime (maximal) ideal if and only if for each element a in R, either a E I or 1 - a E I, but not both. 6. Given (R,+,·) is a Boolean ring. For an element a E R, define the set I(a) by

+, .).

I(a)

>=

{I 1(1,+,,) is a maximal ideal of (R,+, .); a E I}.

Verify that the sets I(a) have the following properties: a) I (a) ,. ., whenever a ,. O. b) l(a h) - l(a) ~ l(h). c) l(a' h) - I(a) n I(h). d) l(a) - I(b) if and only if a-h. [Hint: a E 1 if and only if 1 - a E 1.]

+

BOOLEAN RINGS AND BOOLEAN ALGEBRAS

233

7. In reference to Problem 6. prove that if M

=

{I 1(1.+.') is a maximal ideal of (H.+.

·n.

+..)

then the ring (H. is isomorphic to a subring of (P(M), A, n). [Him: Consider the mapping I: H -+ P(M) given by I(a) - I(a).] 8. Establish that there is no Boolean ring having exactly three elements. 9. In any Boolean ring (R. an order relation ~ may be defined by taking a ~ b if and only if a . b = a. If the elements a. b. c. d E H, establish the following order-properties : a) a ~ a, 0 ~ a:S 1 for every a E H. b) a ~ band b ~ c imply a ~ c. c) a:S b and b ~ a imply a-b. d) a :S c and b ~ d imply a . b ~ c· d. e) b· c = 0 implies a • c = 0 if and only if a :S b. 10. a) Let (H. be a Boolean ring and I be a nonempty subset of H. Show that (1,+,') is an ideal of (H,+,') if and only if i) a, bel imply a bel, ii) a E I and r E H with r :S a imply r . a E I. b) If the set la is defin.ed by I • ... {r E HI r ~ a}. verify that the triple (1., forms an ideal of (H,

+, .).

+..)

+

+, .)

+, .).

11. Suppose that (8,+.,) is a subring of a Boolean ring (R,+, .). Prove that any homomorphism I from (8, + •. ) onto the field (Z2, +2, '2) can be extended to all of (H. + .. ). [Him: The ideal (ker (f), +,.) is contained in maximalideal (M,+, .), where (H/M,+.·) ~ (Z2, +2, '2).] 12. For elements a. _~. and__c_~t a Boolean algebra (B. V, A), prove that a) (a A b) V (b A c) V (c A a) - (a V b) A (b V c) A (c Va), b) a A c = b A c and a A c' == b A c' imply a - b, c) a = b if and only if (a A b') V (a' A b) - 0, d) a ... 0 if and only if (a A b') V (a' A b) - b, e) a A b = a if and only if a V b = b. 13. Let the set B consist of the positive integral divisors of 30, that is, B

=

{I, 2, 3, 5, 6.10,15, 3O}.

If V and A are defined by a Vb,. lcm (a, b),

a A b - gcd (a, b),

show that the triple (B, V, A) is a Boolean algebra. 14. Given that X is an infinite set. Let B be the family of all subsets A !;;; X such that either A or X - A is finite, plus ~ and X. Determine whether the triple (B, U, n) forms a Boolean algebra. 15. Prove that if (B, V, A) is a Boolean algebra having identity element 1 for the operation A, then every Boolean subalgebra must contain 1. Contrast this with the case of rings.

234

RING THEORY

16. By means of Theorem 3-61, convert the Boolean algebra (B, V, A), as defined below, into a Boolean ring. d

V

abc

a b c d

abc d b b b b c b c b d b b d

A

abc

d

a a a a b abc cae c dad a

a d a d

B == {a, b, c, d}

17. Suppose a Boolean algebra (B, V, A) is made into a Boolean ring (B, +,.) via Th('orem 3-61, and then (B, +,.) is convertt"d baek to a Boolean algebra (R, V \,1\ \) via Tlworem 3-62. Verify that (R, V, 1\) ... (B, VI, 1\ .). 18. Suppose (B, V, 1\) is a Boolean ring and 0 ;o!i I\; B. The triple (I, V, 1\) is said to be a (Boolean) ideal of (B, V, 1\) if and only if i) a, bEl implies a V bEl, ii) aE I and bE B imply a 1\ bE I. a) Prove that every Boolean algebra (B, V, 1\) has two trivial ideals, namely, ({O}, V, I\) and (B, V, 1\). b) If (Ii, V, 1\) is a collection of ideals of (B, V, 1\), show that (nIi, V, 1\) is also an ideal. e) Prove that if (I, V, 1\) ill an ideal of (B, V, 1\) and 1 e 1, then 1 - B. 19. Let (P(X), U, n) be the Boolean algebra of subsets of a nonemptlt set X and Xo be any element of X. Prove that a) if I is the family of all subsets A \; X such that Xo ~ A, then the triple (1, U, n) is an ideal of (P(X), U, n), b) if X is infinite and I is the family of all finite subsets A \; X, then the triple (1, U, n) is an ideal of (P(X), U, n). 20. Let (B, V, 1\) and (B\, V \,1\ \) he two Boolean algebras and I a mapping from B into B.. Then I is Haid to be a Boolean homomorphiam from (B, V, 1\) into (Bl' VI, 1\.) provided I(a V b) = I(a) V. I(b), I(a 1\ b) = I(a) 1\. I(b) , I(a') "" I(a)'

for all elements a, bE B. (The formation of complements may be regarded as a unitary operation.) Show that such a function has the following properties: a) 1(0) ... 0.,/(1) = 1\. b) (f(B), V., 1\.) is a suhalgebra of (B., VI, 1\ .). c) If a :S b is taken to mClI.n a 1\ b .. a, then I(a) :S I(b) whenever a d) The triple (ker(!), V, 1\) is an ideal of (B, V, 1\), where

k('r (f)

=

{a E B I/(a)

:S

b.

= Od.

e) If (It, VI, 1\ 1) is an ideal of (Bl, VI, 1\ .), then (f-l(It), V J 1\) is an ideal of (B, V, 1\).

CHAPTER 4

VECTOR SPACES

4-1 THE ALGEBRA OF MATRICES

The theory of matrices has long occupied a strategic position in various branches of mathematics, physics, and enginccring. Only in reccnt years has its importance in the social and biological sciences as well hecome apparent. Thc subject t.oday has become an indispensable tool ill :mcll new field!! as game theory, linear programming, and statistical decision theory. Part of the reason for the widening applicability of matrix theory is no douht the role it plays in the analysis of di!!eretc'obHcrvations and the case with whieh matric operations may bl~ progmmmed for modern highapeed computen;. We do not intend to give a complete nnd systcmatie aeeount of the problems of matrix theory and its diverse applications. Rather, the operations and the basie properties of veetors and matrices arc approached from an algebraic point of view with the aim of illustrating some of the conccpts of the previous chapters. One result of sueh a study will be t.hc formulat.ion of a mathematical sy!!tcm, somewhat more complicated than tho!IC st.udied earlier, known as a v~tor spa(!e. The basic definition whieh atl1rts U!! ofT ill that of II. vector, the fundamental object in our study. Deftnitlon 4-1. An n-componcllt or n-dimensional vector over a field (F, is un ordered n-tuple (aI, a2, ..• ,an) of clements ak E F.

+,.)

The clements ak E F ure called the comTllInents of the vector and we say n is its dimension. Clearly, the set of all one-dimensional veetors over (F, can be identified with F itself. To have to write out the whole vector is somewhat awkward; hereafter, we will condense our notation and designate the vector with (~OmpollentR ak by (a,,). It is hardly necessary to point out t.hat two n-component V(,(,t.()rtI (ak) and (Ilk) Iln~ (~qlllll, ill whidl ellS(. w/' writ./' (nk) ~ (lJd, if Illld only if t.hcir eorrcspolldillg eompOltelltll arc equal: (ak) = (I)k) if and only if ak = bk for k = 1, 2, ... , n.

+, .)

Definition 4-2. n) The sum of two n-component vectors (ak) and (b k) , denoted by (at) + (b,,), is the veetor obtained by adding their corresponding components. Thus, (at) + (b,,) = (a" + bk). 235

236

.4-1

VECTOR SPACES

b) The product of a vector (a,,) and an element r of F, denoted by rea,,), is the vector obtained by multiplying each component of (a,,) by r. Thus, real)

=

(r· a,,).

Here, we conform with the standard practice of using the plus sign in two different contexts, for vectors and for their components. It should be perfectly clear in any given situation whether we are adding vectors or elements of F. Note, incidentally, that the difference of two vectors may be expressed in terms of the operations already given:

For a simple illustration of these ideas, consider vectors over (R', +, .); in this case, we have (1,2,3) - 2(1,0, -1)

=

(1,2,3)

+ (-2,0,2) =

(-1,2,5).

Definition 4-3. Any vector whose components are all zero is called a

zero

vector and is represented by the symbol O. Let V,,(F) denote the set of all n-component vectors over an arbitrary field +, .). Inasmuch as vector addition enjoys the basic additive properties of its components, the following theorem concerning the algebraic nature of the pair (V,,(F), +) is obvious. (F,

Theorem 4-1. The system (V,,(F), +) is a commutative group, having the zero vector of dimension n as its identity element and (-a,,) as the inverse of a vector (a,,) E V,,(F).

The operation of multiplication of vectors by elements of F, as defined above, has the following properties: if r, • E F and (a,,), (b,,) are vectors in V,,(F), then

+

+

1) (r .)(a,,) - r(a.) .(aAl), 2) r[(a,,) (b,,)] = rea,,) r(b.), 3) r[.(a,,)] = (r· 8) (a.,) ; l(a.) = (a,,); O(a.) =

+

+

o.

Verification of these facts is not particularly difficult and is left to the reader. Vectors may also be combined under a rule of composition known as inner product multiplication. Definition 4-4. The inner product of two vectors (a,,), (b,,) E V,,(F), denoted

by (a,,)

0

(b.), is defined to be (aAl)

0

(b,,)

=

.

L al . bl;. 1-1

According to this definition, inner product multiplication • may be regarded as a function from V,,(F) X V,,(F) onto F; that is, the inner product of two vectors is an element of F. Note also that the product of two nonzero vectors

4-1

THE ALGEBRA OF MATRICES

237

may be zero, as in Va(R'), where (1, 2, -3)

0

(3, 6, 5)

=

1 . 3 + 2· 6 + (-3) . 5

=

O.

However, one should not jump to hasty conclusions concerning divisors of zero, for on the right-hand side we have the real number zero and not a 3-eomponent zero vector. While failing to be even a binary operation, inner product multiplication nonetheless enjoys some interesting properties, several of which are listed in the next theorem. Theorem 4-2. If rEF and (at), (bt ), (Ct) are vectors in V .. (F), then 1) (a.,) 0 (bAo) = (bAo) 0 (at), 2) 0 (a,,) = 0 = (a,,) • 0, 3) r[(a.,) (b t )] = (r· at) • (b t ) = (a,,) • (r· bAo), 4) (a.,). [(b,,) (Ct)] = (aAo) • (bAo) (al;) • (CI;). 0

0

+

+

Proof. Let us illustrate the type of argument involved by establishing the last statement; the remainjng parts of the theorem are left as an exercise. We proceed as follows: (a.,)

0

[(b.,)

+ (CI;)] =

(al;) • (bl;

+ CI;) =

..

L aAo' [bl; + Ct]

"-1 .. .. = E a., . b., + E a., . c.,

"-1

"-1

= (a.,) • (b.,) + (a.,) • (c.,). Definition 4-5. By an m X n matriz (plural: matrice.) over the field (F, we mean a function from {I, 2, ... ,m} X {I, 2, ... I n} into F.

+,.)

In the case of matrices, one customarily arranges the functional values block fashion in a table made up of m rows and n columns. S~cifically, if the value of the matrix at the ordered pair (i,j) is denoted by aij, where 1 ~ i ~ m, 1 ~ j ~ n, then the matrix is indicated by the following rectangular array: (

all

a12

~21

~22

a... 1

a...2

... a l .. ) ...

~2. . .

... a.....

Abusing terminology, we shall hereafter refer to the above display of mn elements of F as the matrix itself (in the strict sense, this display is simply a representation of the matrix). Further, we will call 4;, the ijth entry or element of the matrix, and we shall speak of the integers m and n, the number of rows and columns, as its (limenBiona.

238

4-1

VECTOR SPACES

Note that clements are located ill the matrix by the use of double subscripts. the first subscript indicating ttHl row. and the second subscript the column in which the element is found. For instance. the element a23 is in the second row and third column. To avoid cumbersome notations. it is convenient to abbreviate a matrix aH (ai;)",X", to be read "the matrix of dimension m X n whose elements are the ai/s." When the numbers of rows and columns are clearly understood. we may instead simply write (aij). If m = n. the mntrix is snid to be square of order n. Definition 4-6. Two m X n matrices (ai;) and (b i ;) are equal, for which we write (ai;) = (b i ;) , if and only if their corresponding elements are equal; Lhllt iH, flij • l)ij fill' ILII i ILIIII j.

Since the rowl'! of all m X n mlttrix may be regarded as clements of the vector space V,,(F) , it iH 1I0t HlIl'priHing that the operations defined in V,,(F) have natural generalizations to matrix operations. Definition 4-1. a) The Hum of two m X n matriceH (ai;) and (bij). denoted

by (aii) + (b i ,), is the matrix obtained by addillg their corresponding elements. Thus, (aij) + (b i;) = (ai; + bi ;). b) The product of the matrix (ai;) and the element rEF. denoted by r(ai;), is the matrix obtaitwd by mult.iplying each element of (aij) by r. Thus, r(ai;)

=

(r· ail)'

Observe that by its definition, addition is a binary operation on the set of all matrices of a given size; that is, the sum of two n X m matrices is again an n X m matrix. Example 4-1. Taking (ll',

+, .)

as the base field, let

~) ,

-6

A=e

3

0

4

B= (:

-1

:).

Then

2A

+ = (:

-12

B

0

-8 = (:

-1

:)+(: :).

4 -1

~)

A matrix each of whose elements is zero is called a zero matrix and is denoted by O. Accordingly, a zero matrix need not be square. For the zero matrix whose dimensions are those of (ai;), we have (ai;) (ai;)

+0

=

-I- (-I)(aij)

= 0 + (a;j). = 0 = (-I)(a;j) + (aij). (ai;)

4-1

THE ALGEBRA OF MATRICES

239

+, .)

Let. us denote the set of all m X n matrices over the field (F, by the symbol M ",n(F). The following theorem establisheH the algebraic properties of (M",,,(F), +) under matrix addition.

Theorem 4-3. The system (JIf ",,,(F), +) is a commutative group, with the m X n zero matrix as the identity element and (-aij) as the inverse of a matrix (aij) E M ",,,(F). Proof. Definition 4-7 indicates that each property of matrix addition is derived from the correHpondillg ndditive property in the field (F, +, .). For instanoe, to establish the commutative law, let (aij), (bij) EM",,,(F). Then

The rest of t.he proof proceeds along similar lines and is left to the reader. Although multiplieation of matrices by a field element is not a binary operation on M ".,,(F) lunless, of couI"s(~, m = n = 1), this operation has several int.eresting features. Specifieally, if (aij), (b ii ) E JIf ".n(£<') and r, 8 E /I', then r[(aij)

+ (bij)] =

(r· 8)(aij) (r

+ 8)(ai;)

l(a,;)

=

=

r(aij)

r[8(aij)],

= rea,;)

(a,j),

+ r(bij), + 8(a,;),

O(ai;)

= O.

Our main purpose for introducing inner product multiplication for veetors becomes apparent with the following definition. Definition 4-8. If (aij) is an m X n mntrix and (b ij) is an n X r matrix, then their product (Cii) = (aii)' (bi') is an m X r matrix whose elements are given hy the formula

.

Cij = ~ ail, . bk ;

for i

=

1,2, ... ,mi j

= 1,2, ... , r.

k-1

As the SUbscripts indicate, the ijth entry c'l in the product matrix (ai;) . (b i ;) is obtained by taking the inner product of the ith row of (aii) and the jth column of (h ij): eii

=

(ail, ai2, ... , ain)· [;::].

b"J As we observed previously, the inner product of two vectors is defined only if the vectors involved have the same number of components. Thus for the matrix produet (aij) . (bij) to exist, the number of columns in the matrix (ai;)

240

4-1

VECTOR SPACES

[which determines the number of elements in a row of (1Ji,)) must be equal to the number of rows in the matrix (b tJ ) [which determines the number of elements in a column of This means that we could not, for example, multiply a 4 X 3 matrix and a 2 X 3 matrix. Restricted to the set of square matricea of order n, matrix multiplication is a binary operation. For if (IJiJ) and (b'/) are both n X n matricea, then 80 is their product (a.ii) • (bi/)' Before going on, it would be worthwhile to consider an example in detail.

(b,,».

Exam.... 4-2. Again taking (B', +, .)

A

°2 -11)

= (:

88

the field, let

and

Then

2X3 (

=

3X2

2, 3 + O· (-1)

+ 1· 0

- (6 4

2·1+0·0+1·2 ) 3·1 + 2· + (-1)·2 -

°

3·3+2·(-1)+(-1)·0

7

2X2 On the other hand,

° 1) 2

3X2

=

~1

2X3

3.2+1.3

3·0+1-2

0·2+2·3

0-0+2-2

3.1+1.(-1>] -1·2+0-3 -1·0+0·2 -1 . 1 + 0· (-1)

1

-[-: :-~l· 6

4-2

3X3

0·1 +2· (-1)

)1

4-1

THE ALGEBRA OF MATRICES

241

We next dispose of one natural question that arises here, namely, the question of commutativity of matrix mUltiplication. First of all, given an m X n matrix (ao/), the matrix products (a'/)' (bu) and (b'/)' (O(/) are both defined if and only if (b ii ) is an n X m matrix. When the latter condition holds and it is a.t least possible to form these two products, (aii) • (bi/) and (bii) • (4;i) will be of different dimensions unless m = n. Even if this is the case, where it is meaningful to ask whether (ail) . (bi i ) a.nd (b ii ) . (aii) are equal, matrix mUltiplication will not as a rule be commutative. One need only consider the computation

Due to the asymmetric way in which two matrices combine in a product, such an outeome is not totally unsuspected. It is quite possible, of course, that a particular pair of matrices may commute. I·'or the zero .matrices of appropriate dimensions, (aii)' 0 = 0 and O· (ail) = O. In particular, if both (aii) and 0 are square matrices of the same order, then (aii) . 0 = O· (aii) = O. If (a;j) is any square matrix, then that part of the matrix consisting of the the elements ali is called the (main) diagonal of the matrix. DefInition 4-9. The identity matrix oj order n, designated by I .. , or simply I when there is no chance of confusion, is the square n X n matrix having ones down its diagonal and zeros elsewhere.

It is helpful to have some notation for the elements of the identity matrix. Consequently, we will denote the element in the ith row and jth column of I .. by the symbol o;/t where 'il

and thus write Ifl

={ =

I

for i

= j,

o

for i

~

(the Kronecker delta)

i,

(O;i)' To illustrate,

For each positive integer n, the set of all square matrices of order n over the field (P, will be represented by M fI(F), rather than M .... (F). The identity matrix I .. serves as an identity element for the operation of matrix multiplica-

+, .)

242

4-1

VI
tion in the set M,.(F). Indeed, if (a;i) E M,.(F), then (aii) . I,. = (aii) . (8;i) =

(t

aik . 8ki) = (aii),

k-l

and similarly I,. . (a'j) = (a;i)' We have just proved part of the following theorem. Theorem 4-4. The system (M,.(F),

+,.) is a ring with identity.

Proof. It has already been observed that (M,. (F), +) is a commuta.tive group and that mat.rix multiplication is a bin&~ operation on M ,.(F). What remains 1M to verify tlw RHMOdl\tlvity of multiplication and tho distributive laws. To establish thnt muItiplieation is left distributive over addition, let (a;;) , (bij ), and (e;i) E Mn(F). Then (a;i) . [(b.;)

+

(e;i)]

=

(a;i) . (b;;

(t

+ eli)

a;k . [liki

+ eki])

k-l

The rest of the proof offers

110

diffilmlty and is omitted.

For our next theorem we need the following notation: define Eii to be the n X n matrix having 1 as its ijth entry and zeros everywhere else. Thus for. n

=

2,

The reader may readily prove that In

= Ell + E22 + ... + En,.

and

E..

IJ •

L'

.l~.,

_

{Eit o

=

8,

if j -F

8.

if j

4-1

THE AWEBIlA OF MATRICES

243

+, .)

This last relation shows, incidentally, that the ring (Mn(F), has divisors of zero. The proof of TheorC'm 3-24 indicates· that nny commutative ring with ident.ity whieh iH not a field ILlwu.YI! POSHCHllCI! nontrivial ideals. There is no real!on to u.SHume thnt in the nbHence of (!ommlltat.ivity the Harne conclUl!ion followH. Indeed, as we Hhall sec, the Hystcm (Mn(F), +,.) provides an example of a lIoncommutative ring without nontriviu.l ideu.ls. Theorem 4-5. The ring of matrices (Mn(F),

that is, (M,.(F),

+,.) has no nontrivial ideals;

+,.) is IL simple ring.

Pro,,!. Ruppofl(! t.hlLt (l'
where til!' nllltrix (lJij) IIml the value a;jl down itt! llIain diltgonnl lUll I zeros Due to t.he presnrwc of nil the 1:ero elll.rieH in the variout! fnetors, thit! produet is equal t.o H. t • l\Iorcover, Hilwe (S,·I-, .) is an ideal, the matrix belongH t.o S. The relation

ell«lwhN(~.

g.,

i,j

=

1,2, ... , n,

shows further that all n 2 of the matrices Eij arc contained in S. The set S, being closed under addition, then has the identity mlLtrix In as a member, from which we conelude that S = 111n(F), In other words, if S ~ {O}, then S = M,.(F). In passing, let. us remark that while the ring (llf,,(p), +,.) fails to have nontrivial (two-sided) ideals, it may very well possess one-sided ideals. For example, the set of all matrices of the form

a, b EF, compriHCs t.he elcmcnt.s of Il left ideal of (111 2(P), .1-, .). Matrices which have a multiplicative inverse are said to be nonsinyular; otherwise they are ealled sinyular. Sinee a nonsingular matrix (aij) eommutes with its inverse [by definition (aij)· (aij)-I = (aij)-I . (aij) = I], it follows that both the matrix and its inverse must be square and of the same order. While only square matrices can possess an inverse, not every square matrix is nonsingu\ar. For instance, eonsider the 2 X 2 matrix

244

4-1

VECTOR SPACES

If this matrix were nonsingular, we would then ha.ve

On the other hand, the associative law yields

which leads to an obvious contradiction. This argument shows that, without further restriction, the system (M,,(F),') does not constitute a group. The obvious thing to do is to consider only those matrices having mUltiplicative inverses, so that the object of interest becomes the group of invertible elements of (M ..(F), +, .). Theorem 4-6. If M:(F) denotes the set of nonsingumr matrices of order

n,

then the pair (M:(F),') forms 8: group. By further limiting the set of matrices under consideration, one can obviously obtain more specialized results, as is evidenced by the next two examples. Example 4-3. Consider the set S of all matrices in M 2(F) of theJorm

a,beF.

We propose to show that the system (S, +,.) is a field. If the matrices

are arbitrary elements of S, then

b- d) eS, d)c = (a-c -(b - d) d) = e

(a. c-

a-c

b· d -(a·d+b·c)

c) eS.

a· d + b· a·e-d·d

Consequently, S is closed under both differences and products. This makes (S, +, .) a subring of (M 2 (F) , +, .), the ring of square matrices of order 2. It is easily checked that the elements of S commute under matrix multiplication. Since the 2 X 2 identity matrix 12 is plainly a memoor of S, the triple (S, +, .) thus forms a commutative ring with identity. All that remains is to establish that each nonzero element of S has a multiplicative inverse in S.

4-1

THE ALQ,EBRA OF MATRICBS

Now if

(-: :) ~ (~~), either a ~ 0 or b ~ 0, 80 that a 2 + b2 ~ and, as a direct computation will show, (

a b)-l = (a2 -b a

o.

Accordingly, (as

+ b')-l exists

-b) S.

+ bS)-l(a

e

b

a

An interesting observation is that the additive groups (S, +) and (V s(F), +) are isomorphic under the mapping f: S - V.(F) given by

f( b) a

-b

= (a, b).

a

This function is obviously a homomorphism, lor

f[( b) + ( a -b a

Cd)] = -d

e

j(

•

a+

C

+ d) = (a+ e, b + d) = (a, b) + (e, d) -(b

=f(-b b)+1"'\-d d). I( b) .f( Cd), a

e

a

Moreover, if

e

a

-d e , -b a then from the definition of equality of the vectors (a, b) and (e, d), we must have a = e, b = d. This implies that

8Ojisaone-to-onemapping. It is clearly onto Vs{F),hence (S,+) ~ (V.{F),+). Example 4-4. Let the set T consist of all real matrices of order two having equal integral entries. A routine argument, which we omit, shows the triple (T, +,.) to be a subring of the ring (M 2(R'), +, .). Hereafter, a matrix

(: :),

neZ,

246

4-1

VECTOR Sl'ACES

in T will hc~ cll'llot{~d Hilllilly hy (n); while thit! iH 11I1 incorll'c~t notation, it ill much leHl! ullwiddly than the I~orrect olle. WllIlt we willh to show here is that the mapping J: T - Z. defined by J«n» = 2n yield!:! an isomorphism between (T, +,.) arid (Z., +, .), the ring of even integers. The demonstration that J preserves addition is straightforward, so we shall consider only multiplication: for n, m E Z,

J«n) . (m» = J«2nm» = 4nm = 2n· 2m = !«n» . J«m». Thull, t.he funl:tioll f is a (ring) hOlllomorphi!:!m, ILnd is evidently one-to-one. Next, let m 1)(' lUI arhitrary evcm illt.egl'r, NO tlmt m = 2n for some n E Z; thCII, hy UI(' 1I1111111C'r ill whil~h f WII~ dc·lilll·II, f( (n») =, m. Thill HhoWA t.hat f(T) = Ze ILnd illdc~(~d ('1', I,') ..... (Z., I,') vin f.

PROBLEMS 1. Determine the values of a and b for which the following matrix equation holds:

[30] 1 I

-3] [20 15]

[a

. (4 7) + 2 0

5 2

6 8

4

5

10 25

b

40 25

2. C'omputt' .t . B, .1 2 , B· 0, O· A, and O· B for the matrices

..J ...

[~ I

0

-:]

B- [ : : ] ,

0-1

3. Show that ('ach of the following matrices from J\'h(R') is a solution of the matrix t'qllation .\'2 - 5.\' 41 = 0:

+

and

( 3-2). -I

4. a) l~illcl a matrix in .I/2(R') whoH(, Hqllare iH tht' matrix

(3-4). 1 -I

b) Obtain all matrict's t.hat. commut.e with

2

,-1

THE

.u.(a~mtA

OF MATIUCJ<;S

247

a) Show that the pair (G,·) is a group, where· indicatcs matrix multiplication. b) Ruppose II and K denote the sets of matrices of the type nnd n·"IH·{·!i\,!·ly. I'ru\'t'I.hnl (II,·J i.. "l
«(I, .J,

wllil!' (1\,,)

6. Uivt'n til(' matrix (a,j) E .Hm.(ft') and r, 8 E 10', prove t.hat a) (r+8)1 .. = rl .. +81., b) (r·8)/. = (rl.)·(81 .. ), c) I",· (ao) - (a;j)' I .. "" (aiS). 7. Let thl' Ret 0 conKiHt. of the six 3 X 3 matrices 0

[:

[~

0

~]

0

0

[:

~]

0

[:

0

;]

[:

0

:]

0

0 0

0

:]

[;

0

:]

EHtahliKh t.hat thl' pair (G, .) formll a group il«lmorl.hic to (Sa, .), the symmetric grllu p on a HymholM.

:

8. ConHidt'r setH Ganci /I conHiHting of all matrices rrom M a(/") of the form

[: ::]

and

[~ ;]

reHIK'(:tivl!iy. Vt'rify that I.htl pair (G,') is a group and that (II,') formH a normal I!ubgroup of (0, .). 9. In quantum mechanics, the Pauli theory of electron spin utilizes the following complex matrices:

248

4-1

VECTOlt SPACES

where, of course, i 2 = -1. Prove that the Pauli matrices, together with matrix multiplication, consititute a group. 10. A square matrix (aii) is said to be diagonal if au - 0 for i pi. j; in other words, (41/) - (a,8i/). Assuming diag M ,,(F) denotes the set of all diagonal matrices of order n, prove that the triple (diag M ..(F),+,·) forms a commutative subring of (M ..(F),+, .). 11. By a ICalar matrix is meant any diagonal matrix (4i/) having equal diagonal elements: (aii) - (diJ). If S ..(F) is the set of scalar matrices of order n over the

field (F,+, .), show that a) the triple (S.(F),+,·) is a field isomorphic to (F,+, .), b) S .. (F) - cent M .(F). [Hint: Consider the products (41/) •

E., - E.,· (41/)']

12. If M:(F) designates the set of nonsingular matrices of order n, verify that the triple (M:CF), +, .) does not form a subring of the ring (M ..(F), +,.). 13. A square matrix (aii) is upper triangular if ail - 0 for i > j and atricll" upper triangulm- if ail = 0 for i ~ j. Let T,,(F) and n(F) denote the sets of all upper triangular and strictly upper triangular matrices of order n, respectively. Prove the following: a) A matrix (ail) e T .. (F) is nonsingular if and only if 4i1 '" 0 for i - I , 2, ... , n. b) The triple (T,,(F),+,.) isasubringofthering (M ..(F),+,·). c) Each matrix (aii) e T:(F) is nilpotent; in particular, (41/)" - o. d) (T:(F),+,') is an ideal of the ring (T ..(F),+, .).

14. The trampoae of a matrix (41/), designated by (ail)', is the matrix whose ijth entry is the jith entry of (a,;), that is, (aii)' .. (alf)' Given matrices (ail), (h#) e M ..(F), verify that a) (41/)" - (41/), b) [r(ai/) .(hi/)]' ., r(ai/)' .(hi/)', r,. e F, c) [(4i/)' (bi /»'

-

+

+

(bi/)'· (4ii)',

d) whenever (a#) is nonsingular,

80

also ill (a#)', with [(44/)'1- 1

-

(4#)-11'.

15. Show that the field (S, +,.) of Example 4-3 is isomorphic to the complex numbers (C, under the mapping

+, .)

j I\_b 4

h) =

(a, b).

a

16. In the ring (M2(C),+, .), let D be the set of all matrices having the form

+, .)

is a division where 4 is the complex conjugate of a. Prove that the triple (D, ring, but not a field. 17. Show that for any element a e F, the following matrices are both idempotents in (Al2(F),+,'): and

4-2

ELEMENTARY PROPERTIES OF VECTOR SPACES

249

18. A matrix (a.;) E M,,(F') is said to be aymmetric if (a.;)' - (atj) and ,kew-.ymmetric if (ai;)' ... -(ail)' Establish the following assertions: a) If (a.;) and (bij) are symmetric matrices, so also is r(ai;) .(bo;). b) The products (ai;) . (ail)' and (ai;)' . (aij) are both symmetrio. c) If (aij) and (b ij) are symmetric, then (aij) . (bij) is a symmetrio matrix if and only if (ail) . (bii) - (bi;) . (ail)' d) The diagonal elements of a skew-llymmetric matrix are all sero. e) Every (square) matrix can be written as the sum of a symmetrio and skewsymmetrio matrix.

+

19. Let the set 0 be comprised of the following matrices:

I -

U

=

(~

:),

8 - (_: :),

e-:), e_:), V

=

y

=

T - (-: _:),

w - (: :),

(0 -1). -1

0

a) Prove that the pair (0, .) forms a group. b) If II = {I, 8, T, U}, show that (H, .) is a normal subgroup of (0,·) and find the cosets of H in O. 4-2 ELEMENTARY PROPERnES OF VECTOR SPACES

-

We saw in the last section that the collection Mn(F) of square matrices of order n over a field (F, +, .), together with the operations of matrix addition and multiplication, constitutes a ring. At the time, our third matrix operation, mUltiplication of a matrix by an element of F, seemed relatively unimportantparticularly, since it failed to be even a binary operation on M n(F). However,' by abstracting the essential features of this operation, we now define a mathematical structure having the set M .. (F) [more generally, the set M ..... (F)] under matrix addition and multiplication by a field element as a model. Basically, this is a matter of combining two different algebraic systems into a single entity known as a vector space. Due to the availability of a number of excellent texts on the subject, there is no need for us to develop the theory of vector spaces in any great detail. Instead, our goal shall be to give a survey of some, but by no means all, of the main ideas. The pace will frequently be brisk and much is left to the reader. Definition 4-10. A vector space (or linear space) over a field is an ordered triple «V, +), (F, -t-,.), .) consisting of 1) a commutative group (V, +) whose elements are called vector., 2) a field (F, whose elements are called acalar.,

+, .)

250

4-2

VECTOH SPACES

3) an operation of scalar multiplication conneeting the group and field which satisfies the properties: 0

a) for each c E F and x E V, there is ctefinpd an element cox E V; that

is, V is closed under left multiplication by scalars;

+

+ C2)

h) (CI c,) (el'

C2)

+

0 x = (CI 0 x) (C2 0 x); x = CI 0 (C2 0 x); y) (c x) (c y);

0

+

d) co (x = e) 1 o:r = x, wh('re I is the fi('ld id('utity element. 0

0

While the addition symhol has been used in two contexts in the above definition, to d!'signate the operation of the group and one of the operations of the fi!'ld, no eOllfllsioll should arise from thiN pmdil'e. It will alwaYII he dear in any giv('n :-;it.uatioll wlwtlwr v('(·t.ors or sealars are being added. When both veetor additioll and sealar multipliention are involved in an expression, we follow our usual understanding in omitting parentheses: multiplication takes precedence over addition. In the Sl'qu!'l, a vector RP:WC over the fi!'ld (F, +,.) will be denoted merely by l"(F), rather than the corred but cumbersome notation (V, +), (F, .),0). The ('onveniellC'e rmmlt.ing from this convention more than outweighs its lack of pl'('('i8ion. For fu.-Owr simplicity, we shall hel'('aftcr drop the and write cx for the procluet r. x. It should be apparent that a veetOl' space is markedly diffel"Cnt from the previous systems we have discussed in that the products of scalar multiplication employ clements from both F and V. Part (3) of the definition relates the possible wayR th('se prorlu('tR ('ombine t.he operation + of (V, +) with + and· of (P, 1-, .). l'\ot.1' all'lo tlmt t.!IC' hYIHIt.llPsil'l Ix = X il-l quite el!RCntinl; without it, every field and c'ommutative jl;roUJl would yield a vector space under the trivial scalar multiplicat.ion ex = 0 for all C E P, x E V. Before discussing the implications of t.he axioms of a veetor space, let us give It sel('ction of ('xamples. The formal verification that each example de-. sl'ribed aetually I'ollstitutes a vee\.or SPIlC'!! iH left as all exercise.

+,

0

0

Example 4-5. Let t.he commutative grollp be (Vn(P), +), where Vn(F) is the IlCt of all n-('ompOII<'llt row vedorH over an arbitrary field (F, and is the usual compon('nt.wise addition of veet.ors. For C E F and (ak) E Vn(P), define s('alar multiplication by

+, .)

In view of th(' r('sults of the lnst. seetion, we thus obtain will hl'nc~pfOl'f.h hI' denoted simply hy 1" "(F).

Il

+

vector space, which

Example 4-6. If /tf", .. (F) rcpre'spnts t.hl' c'oll('dioll of m X n matriccI-I over (P, and I is t.he op('rntion of matrix additioll, thl'n (At mn(P), +) is a

+, .)

4-2

ELEMENTAItV PROPERTIES OF VECTOR SPACES

251

commutative group. A vector space rcsulUi on defining scalar multiplication as in the previous example: ceF.

The particulur v('(:tor lipaee whieh arilles when m indicated by !Ifn(F).

=

7~

will, in the future, be

Example 4-7. Given a field (F, +, .), take V to he all functions from an arbitrary llet X into F. For f, g e V and c e F, define the functions f + g and cf by spe(!ifying their vnlues at each point of X: (J -I- g)(x) = f(x)

+ ,,(x),

(cJ)(x) = c· f(x),

xeX.

With vector addition and scalar multiplication so defined, we obtain a vector space V(F). Example 4-8. Let (F, +, .) be a field and W[x), +, .) be the ring of polynomials in the indcterminant x with
and where m

~

n, then

+ bo) + (a1 + b 1)x + ... + (a... + b...}x"', cp(x) = (c· ao) + (c· a1)x + ... + (c· a,.)x". (Needless to say, ill the sum p(x) + q(x), we sct ak = 0 for n + 1 :S k :S p(x)

+ q(x) =

(ao

m.)

Let us retain the symbol F[x) for this vector space, in preference to the correct but awkward (F[x])(F). Example 4-9. For any field (F, +, .), Example 4-5 may be generalized by using infinitdy many elenumts of F: jUtit. take V .. (ft') to be the totality of all illfiniw 1IC
having infiniwly IlIlLny (!omponents. In this

('ILIIC,

if and only if ak

= bk for all k e Z+.

252

4-2

VECTOR SPACES

Vector addition and multiplication by a scalar c e F are perfonned componentwise: (ah a2, aa, ... )

+ (b., b1h b., .•.) =

(at

+ bt, a2 + b2, a. + ba, .•• ),

c(a., a2, Ga, ••• ) = (c· a.. c . a2, c· aa, ... ). Using these operations, V.(F) becomes a vector space over (F,

+, .).

Example 4-10. As a final, and not quite 80 simple, example of a vector space, consider a commutative group (V, +) in which every nonzero element baa order p (p a prime); that is to say, px = 0 for all x e V. If In] e Z. and x e V, we take the product [n)x to mean [n]x = x

+ x + .. : + x

(n summands).

With scalar multiplication defined in this way, V(Z.) may be regarded as a vector space over the field (Z., +., .•). A further comment on notation: To avoid a proliferation of symbols, 0 will be UIIed to designate the zero element both of (V, +) and of (F, +, .). The additive inverae of a scalar c e P is denoted by -c, while the inverse of a vector x e V is also represented by its negative, -x. These conventions should lead to no ambiguity if the reader attends closely to the contexHn which the notation is employed. For the sake of brevity, we shall frequently speak of a vector space over a field F when, in aCtual fact, we mean over a field (F, Some immediate consequences of Definition 4-10 are embodied in our first theorem.

+, .).

Theorem 4-7. If V(F) is a vector space and x 1) Oz = 0, 2) cO = 0, 3) -(CI:) = (-c):t = c( -:t).

e

V, c e P, then

Proof. To establish (1), we use the field result 0 + 1 = 1. Then Oz + x = Ox + Ix = (0 + l)x = l:t = :t

= 0 + x. Since (V, +) is a group, the cancellation law yields Ox = o. The proof of the second part of the theorem follows from the group result x. We have

o+ x =

cO

+ CI: =

c(O

+ x) =

CI:

= 0

+ CI:.

Again the cancellation law gives the desired conclusion. Finally, to obtain (3), observe that

o=

Oz = [c + (-c)]x =

CI:

+ (-c)x.

4-2

ELEMENTARY PROPERTIES 01' VECTOR SPACES

This means that (-c)z

=

253

-(ex). Similarly,

0= cO = c[z

+ (-z)] =

which proves c(-z)

=

ex

+ c(-z),

-(ex).

As the reader should expect by now, a formal investigation of vector spaces involves consideration of such notions as subsystems, operation-preserving functions, quotient structures, etc. Following our standard pattern of presentation, we begin the study with the question of subsystems. In the case of vector spacc!!, the subvcctor spaces are customarily referred to as subspacee. Definition 4-11. Let V(F) be a vector space over the field F and W!; V '¢ 0. Then W(F) is a BUbBpacB of V(F) if, under the operations of V(F), W(F) is itself a vector space.

with W

Since W!; V, much of the algebraic structure of W(F) is inherited from V(F). The minimuJD conditions that W(F) must satisfy to be subspace are: 1) (W, +) is a subgroup of (V, +); 2) W is closed under scalar multiplication.

The usual criterion for deciding whether (W, +) is a subgroup of (V, +) is to see if W is closed under differences. The second of the above conditions implies that -z = (-1) z will belong to W whenever z is an element of W. Because z - y = z + (-y), condition (2), together with the closure of W under addition, is sufficient to guarantee that W be closed under differences. This observation allows U8 to recast Definition 4-11 as follows: Definition 4-12. W(F) is a subspace of the vector space V(F) if W is a nonempty subset of V such that 1) z, yEW implies z YEW, 2) z E Wand c E F imply ex E W.

+

Example 4-11. Every vector space V(F) has two trivial subspaces, namely V(F) itself and the zero subspace {O} (F). Subspaces distinct from V(F) are

said to be proper. Example 4-12. Consider the set

W of vectors in Va(F) whose components add

up to zero:

If (alt a2, aa) and (b l , b2, ba) are arbitrary elements of W, then their sum (al + bit a2 + b2, aa + ba) is such that (al

+ bl) + (a2 + b2) + (aa + ba) =

(al

+ a2 + aa) + (bl + b2 + hi)

= 0+0 = O.

254

.

4-2

VECTOR SPACES

This estnhlishcH the eJosure of W under addition. It is equally clear that W is closed under scalar multiplication, hence W(F) is a subspace of V(F). Example 4-13. Let W denote the collection of all elements from the space

M 2(F) of the form

It follows immediately from the definition of the matrix operations in M 2(F) that W(/") is a subspaee, for

k( a b)_( k'a k.b)EW -b a -(k·b) k'a . The two conditions of Definition 4-12 may be combined into a single easily applied criterion. Theorem 4-8. W(F) is a subspace of the vector space V(F) if and only if W £;;; V and ex dy E W whenever x, YEW, e, d E F.

o¢

+

Proof. If W(F) is a suhspace, t.hen hy definition W is nonempty and contains ex + fly for all x, YEW, e, dE F. Conversely, if thi!! condition holds, W must contain Ix +- ty = x + y and ex + Oy = ex for every x, YEW, C E F. Accordingly, W is c1osl'd wit.h rl"slled In t.lm vl'dor HIl'U~!l ClI)(!rUUoIIH.

We next consider operations on the subspaces of a vector space that produce other subspaccs, the mOlolt important of which are sum and intersection. If U and W are nonempt.y suhsets of thc veetor space V(F), their (linear) sum is . defined to be the set IT

+W =

{u

+ W I U Ell, W E

W}.

The following simpl<', hut quite uS<'ful, fact. will be needed on several occasions. Theorem 4-9. If P(F) and W(F) arc suhspace!! of the vector space V(F), then (If W)(Ji') is also a subspace.

+

+ W is plainly not empty, for U and W each contain the zero vector, hence 0 = 0 + 0 E U + W. Suppose x and y are arbitrary vectors in U + W. Th!'n x = UI + WI and y = U2 + 102, where Ui E U, Wi E W. For scalars c, d E F, Proof. The sum l!

ex + dy

= =

+ WI) + d(U2 + W2) (CUi + dU2) + (ewJ + dW2). c(?lJ

4-2

ELEMENTARY PROPERTIES OF VECTOR SPACES

255

alHl WW) are hoth to!UhHPIlCI!H of V(ft'), it followH Umt CUI + dU2 E l! Thus cx + dy is again in U + W, implying that W)(F) is a subspace.

Hj'l!'l~ (l(ft')

and (U

CWI

+

+ dW2 E W.

From our earlier work with other subsystems, one would expect the intersection of subspaces to be also a subspace and, indeed, this is the content of the next theorem. As it would be repetitious to present the details again, tpe proof is left to the reader. Theorem 4-10. If Wi(F) ito! an indexell eollcction of to!uhsplwes of the vector Kpuce V(I<'), then (nWi)(I<') is a :mbspace of V (1<').

In analogy with the corresponding ideas for groups and rings, this result provides an effective means for generating subspaces: Given a nonempty subset 8 of the vector space V(F), we define [8]

=

n{W I 8 ~ Wi W(F) is a subspace of V(F)}.

There is at least one subspace containing 8, namely the whole space V(F), 80 that [8] c(~rtainly exists and is unique. We infer at once from Theorem 4-10 that [S](F) is a subspace of V(F), called the subspace generatell or spanned by the sct, S. More important still, [SJ(F) is the smalIl'st :mbHpttee containing S, in the scnHe of being ineluded in every subspace which contaim; S. 1'he coming theorem gives an alternative and more constructive d('J~criptioIl of rSJ(/i'); before obtaining this, one more definition is required. Definition 4-13. Huppose V(F) is AllY !inite MUIlI of thc form

It

vector spaee nnd X"

X2, •••

,X" E V.

where eneh

Ci E F is suid to be a linear combination (over F) of the vectors x". A linear combination is called trivial if all its coefficients 0 and nontrivial if at least one (~oefficient i8 differellt from zero.

X., X2, ••• , Ci

=

Using this new terminology, Theorem 4-8 may be rephrnscd in another way: W(/<') is a HuhHplwe of the vec:tor space V(F) if and only if W'is a nonempty 8uhsct. of V whieh if( dosed under t.Il1l formation of linear combinations. For our preSellt purpOHes, the significance of the notion of linear eombinations MtemH from the next result. Theorem 4-11. If S is a Ilonempty subset of the vector space V(F), then [8](F), the subspnce generated by S, cOllsiHts of all (finite) linear combinations of element,8 of S: •

[SI={ECkXkICkEF,XkEs,nEz+}.

256

4-2

VECTOR SPACES

Proof. For the moment, let us denote the set on the right by lin S. Evidently a linear combination of lin~llOmbinations of elements from S can again be written as a linear combination of members of S, so (lin S)(F) is itself a subspace of V(F). Moreover, for each % e S, we have % = be lin S, hence S !; lin S. As (8)(F) is the smallest subspace of V(F) to contain S, it follows that (8) s;;; lin S. On the other hand, since [SJ(F) is a subspace containing S, it must contain all linear combinations of elements of S. Accordingly, the inclusion lin S ~ (8) also holds and the proof is complete. Exampl.4-14. Let S be the subset of V,,(F) whose elements are the n vectors (l,O, •.. , 0), e2 (O, 1, ... , 0), ... , en = CO, 0, •.. , 1); in general, e. is

~l

=

=

the vector with 1 in the kth component position and 0 elsewhere:

k = 1,2, ... , n. Here [S](F), the subspace spanned by these vectors, is all of V,,(F). Indeed, for any n-tuple (alo a2, ..• , a,.) over F, we have (alo a2, ••. ,a,,)

+ a2(O, 1, ... ,0) + ... + a,.(0, 0, ... , 1) = aiel + a~2 + ... + a,.e" e (8). =

al(l,O, ..• ,0)

Let us lee what happens when we generalize this situation to the eequenee space V.(F). In this context, e. now denotes the infinite aequence whose kth term is I, while all other terms are 0: k

=

1,2, ...

As before, let S be the set of all elc. Recall that in forming the linear span [8], only finite linear combinations of elements of S are utilized. It follows therefore that [8](F) is not V .. (F), but rather the (proper) subspace consisting of those vectors having only a finite number of nonzero entries: [8]

=

{(ai, a2, .•• , a", 0, 0, ... ) I a. E F, n

e Z+}.

The first part of this example serves to suggest, as well as to illustrate, the next definition. Definition 4-14. A vector space V(F) is finitely generated when it contains a finite subset which spans V; if V is spanned by the vectors %It %2, ••• ,:t", this will be indicated by writing V = (%1,2:2, ••• , z,.].

The union of subspaces, unlike their intersection, is not necessarily a subspace. For instance, in the vector space M 2(F), take U to be the subSet consisting of all scalar matrices aeF,

4-2

ELEMENTARY PROPERTIES OF VECTOR SPACES

267

and W to be the set of all matrices of the form

beF. That U(F) and W(F) are both subspaccs of M 2(F) is easily verified. Note, however, that while the matrices and belong to U u W, their sum

( 1 0) + ( 0 1) _ ( 1 1) o

1

-1

0

-1

1

fails to be a member of the union. From this, we conclude (U U W)(F) is not a subspace of M 2(F). While the union of subspaces need not be a subspace, the subspace generated by the union always exists and equals the sum of the given subspace... We now establish this fact. Theorem 4-12. If U(F) and W(F) are 8ubspaces of the vector space V(F),

+

then (U W)(F) is the smallest subspace containing both U and W; in symbols, (U W)(F) = [U u W](F).

+ Since 1.£ + W = 11.£ + lw, any vector in U + W

Proof. can be expressed as a linear combination of elements from U U W. But Theorem 4-11 asserts that [U U W] consists of all such linear combinations, hence U W S;;; [U u W].• For the opposite inclusion, observe that both U s;;; U W and W s;;; U W, 80 their union U U W s;;; U W. In other words, (U W)(F) is a subspace of V(F) which contains U U W. As [U U W](F) is, by definition, the smallest subspace with this property, we conclude that [lJ U W] s;;; U + W.

+ +

+

+

+

This characterization of the linear span of U U W is usually much easier to apply in specific cases than the definition itself. In the space M 2(F), for example, where U and W consist of matrices of the form and respectively, we have

258

4-2

VECTOR SPACES

To see thill, one 11('1'<\ only observe that each of the matrices on the right can be written 111'1

+ ( 0 b) EU+ W. ( -ba ab) = (a0 0) a -b 0 Although caeh vcdor x belonging to the sum U + W admits a representation W with U E l! and wE W, this expression is generally not unique. The next theorem provides a necessary and sufficient condition for U and W to be uniquely determined by the vector x.

x

= U

+

Theorem 4-13. I",f, P(F) and W(F) h(' two HubHpaces of the vector spacn V(P). 'flu, II t.he fllllowillj.\I·OIulit.i()nH am (,quival(lIIt:

1) Un W = {O}. 2) Every vector x of the sum U + W is uniquely representable in the form x = U + w, where U E ll, wE W. Proof. We begin by assuming that Un W = {O}. vector x E U W has two representations

+

Ui

E U,

Suppose further that a Wi E

W.

Then UI - U2 = 11)2 - W\. But the left side is in II and the righ,t; side is in W, so both sides belong to l! n W. It follows therefore that or In other words, x is uniquely expressible in the form U + w. Conversely, aR.'!umc statement (2) holds and the vector Z E l! n W. We may then express z in two different ways as the sum of a vector in U and a vector in W, namely z = Z + 0 (here z E U and 0 EW) and z = 0 + z (here 0 E U and z E W). The uniqueness assumption of condition (2) then implies z = () or, rather, U n W = {O}. • Two comments on Theorem 4-13 are in order. First, even though the intersection of U Itnd W is not empty, we sometimes express condition (1) by referring to the subspaces IT(F) and W(F) as disjoint; needless to say, U and W can never be disjoint in the set-throretic sense, for every subspace must contain the zero vector. Secondly, st.atement (2) assumes, unnecessarily, that each vector of IT + W must have a unique representation. This condition could be replaced by the weaker requirement that only the zero vector is uniquely representable. For suppose a vector x E U + W has two decompositions UjE

Subtracting, we obtain

U,

wiEW.

4-2

ELEMENTARY PROPERTIES

m' VECTOIt SPACES

259

wlwm the difTprcnee~ UI -- U2, 101 - W2 lie in (! nnd W, respectively. The assumption that the zpro vector can be exprellscd ill only oue way as a member of (! + W would fOf<'e Ul - U2 = 0, tvl - 102 = 0 or Ut = Ua, tvl = tva. Whl'll t.h(~ equivalent eonditions of Theorem 4-1:J are sutillfied, the sum [T f- W is ('alled riil'('('.t ILnd symholized hy writing {! E9 W. If (T E9 W = V, we say oneh of t.he subspaeCti UW) and WW) i~ eOlllplementary [in V(F)] to the other. This coneept ill of sufficient importanee to rate a formal definition. Definition 4-15. Two subspaees U(F) and W(F) of the vector space V(F) are romplemenlary if U n W = {OJ and l T + W = V. '

Ollr /wxt lIH'(Jr(~m iH i/wludcd for eomplct.c/leSH; it. is 11Il ()f TIH,ormll -1 la ILlld IlCpdH lUI furUuH' justilic·atioll.

illlllledilLt(~ ('()u~quence

Theorem 4-14. Let U(F) ILnd W(F) be two subspaces of the vector space V(F). Every vector x E V is uniquely expressible in the form x = U 10, with U E U, tv E W, if and only if U(F) and W(F) are complementary sub:- • spaces.

+

We should call attention to the fact that a given subspace may possess several complementary subspaces. In the case of V 2(F), for example, take (J=

{(a,O)laEF),

W = {CO, a) I a E F},

W'

=

{(a, a) I a E F}.

It is not difficult to establish that the subspaces W(F) and W'(F) are both tOmplementary to U(F). What is quite signifieant is that every subspace ha.~ eomplementary subspaces, which, to be sure, are not necessarily unique. For the proof of this assertion, we resort to the powerful tool of Zom's Lemma (sec p. 185).

Theorem 4-15. Every subspace W(F) of the vector space V(F) has a complementary subspace. Proof. Define a family of suhsets of V by taking 5' =

{S I S

nW

= {O}; S(F) is a subspace of V(F)}.

This family is nonempty for, trivially, the set {OJ satisfies the defining properties. Now, consider any chain {Si} iu 5'. Our object, of course, is to show that US; is again a member of 5'. To achieve this, let x, y E USi and e, d E F. Then there exist indices i and j for which x E Si, Y E Sj. Because the collection {S;} forms a chain, either S; ~ Sj or else Sj ~ Si; say, Si ~ Sj, so that both x, y E Sj. Since Sj(F) is a subspa('c of V(F), we must then have ex dy E Sj ~ uS;. This proves that (uS;)(F') forms a subspaee of Yep). Moreover,

+

whence the union US; lies in 5'.

200

4-2

VECTOR SPACES

Thus, on the basis of Zom's Lemma, the family fF contains a maximal element U. Our contention is that the subspace U(F) is complementary to W(F). In order to establish this, it suffices merely to show that U W = V. To the contrary, suppose there is some vector x e V with x ft U + W. But then [U, z)(F), the subspace generated by U and x, will be disjoint from W(F), in the sense that [U, xl n W = {O}. (Verify this!) Since U is a proper subset of [U, x), we have arrived at the contradiction to the maximality of U in If.

+

Accordingly, V

=

U + W, and the proof is complete.

The foregoing ideas are easily generalized to finitely many subspa.ces: If WI (F), W 2(F), ... , W,,(F) are each subspaces of V(F), their sum is defined exactly as for two llubllpMClIl,

WI +W2 +···+W,,= {WI+W2+···+ w"l wieWi}. This may be denoted more compactly by :E Wi. As in Theorem 4-9, we can prove that (:E Wi)(F) is a subspace of V(F). The vector space V(F) is said to be the direct BUm of WI(F), ... , W,,(F), symbolized by v = WI ® W 2 ® ... ® WIt if and only if V = WI + W 2 + ... + WIt and W, n (:Ei.... Wi) this case too, every vector x e V has a unique representation

x with

Wi

e

W (i

=

=

{O}. In

= WI + W2 + ... + WIt

1,2, ... , n).

The concept of a quotient structure carries over to vector spaces as expected. In the present context, we encounter one slight, but highly important difference. When fonning quotient groups and quotient rings, it was necessary to introduce speoial lubaYltcm" (namely, nonnal aub/iCroupa, and ideals) in order to enlure that the operations of the quotient structure were well-defined. For vector spaces, no such distinguished subsystem need be defined. To be more concrete, let W(F) be an arbitrary subspace of the 'vector space V(F). Since (W, +) is a subgroup of the commutative group (V, +), we may fonn the quotient group (V /W, +). The elements of this group are just the cosets x + W, with coset addition given by (x

+ W) + (y + W) = x + y + W.

To equip V /W with the structure of a vector space, a notion of multiplication by scalars is introduced by taking e(x

+ W) = ex + W,

ceF.

As usual, we must first satisfy ourselves that scalar multiplication is unambiguously defined, depending only on the coset x + Wand scalar e e F.

4-2

ELEMENTARY PROPERTIES OF VEeroR SPACIIlS

This amounts to showing that whenever x + W = x' + W, th~n ex ex' + W. Our aim would obviously be achieved if we knew that ex - ex'

=

261

+W

r=

e(x - x') E W.

But this follows directly from the fact that x - x' E Wand that W(F) is assumed to be a subspace, hence closed under multiplication by scalars. Thus scalar multiplication in V /W is independent of coset representatives. It can easily be checked that (V /W)(F), with the operations 80 defined, forms a vector space over the field F. Since the pattern of proof is by now familiar, we omit the formal details. ThoM(! iciolUl may be (~onvoniolltly Immmnrized in the following thoorcm. Theorem 4-16. Let W(F) be a subl:lpacc of the vector space V(F) and the cosets x W, y W belong to V /W. If vector addition and scalar multi-

+

+

plication are given by (x

+ W) + (y + W) = x + y + W, e(x + W) = ex + W,

(e E F),

then (V /W)(F) is itself a vector spaoe, known u.s the quotient trpace of V by W. Example 4-15. For (V 3/W )(R'), where

a concrete example,

W

=

{(w, 0, 0)

IW

consider the

quotient space

E R'}.

In the situation at hand, the cosets of W in Va(R') take the form (x, 1/, z)

+W =

{(x

+ w, y, z) I W E

R'}.

The subspace W(R') may be viewed, geometrically, as the x-axis and the coset (x, 1/, .) + W as a lina through the point (x, 11,') parallel to this axi... Tho elements of Va/W would therefore consist of all lines parallel to the x-axis. Since such lines may be identified in one-to-one fashion with the points in the yz-plane, and these points in tum correspond to ordered pairs of real numberS, we 8.118ociatc (Va/W)(R') with the vector space V t(B'); that is, we can define a mapf: Va/W -+ V:a by f«x, y, z) + W) = (y, z). The notions involved will be formalized shortly.

PROBLEMS

In the exercises below, the symbol F denotes an arbitrary field. 1. If V(F) is a vI'r.tor space, establish that n(a) .. (nc)z ... e(nx)

Cor all :r: E V, c E F, and n E Z.

2112

4-2

Vy'I!'I'O" IWA($H

2. Prove that in any vector space V(P), if :1:, 'Y E V and c E P, then a) c(x - y) = cx - cy, b) ex = 0 implies eithl'r c ... 0 or:ll "" O. 3. In any vector spac,' 11(/<'), Hhow that the following cancellation laws hold: a) If x E l" wit.h .r ~ 0, t.hen CIX "" C2X implies CI '" C2. b) If x, y art' nonzl'm I'It'ments of V, tht'n cx = cy with c

~

0 implies z == y.

4. Prove that W(/<') is a subspace of the vector space V(P) if and only if ~ " JV!;;; V anll ax /1 E \I" for 1\11 x, 11 E W, a E !-'. ri. I,'or I'lI.l'h or till' rollowiug ,mlll \1', deu!rminc wlwl.llI'r I\' Ut') ill a 8ubllpacc of the vector KpacO V nUt'). a) W = {(aI, a2, ... ,an) I al a2 a" ~ O} b) Jr = {(ai, a2, ... ,an) I al = a2 = ... = an} c) Jr = {(aI, a2, ..• , an) I ala2 = O} d) II' = {(ai, a2, .•. ,an) I ak E Z for all k} 6. Let U(F) and II'(/") he slIhl'lPI1Ct'H of the veel,or "pace V(F). Prove that (U U Jr)(Ji') forms a Hubspace of V(F) ir and only if U!;;; \I' or II' ~ U.

+

+ + ... -/-

7. Determine whether II' (1<') ill a subspace of the indicated vector space: a) V 3 (F); for fixt'd scalars aI, a2, a3 E F, II' = {(XI, X2, X3) I alXI

+ a2X2 + a3Ia

= O}.

b) l'",(F); II' consists of sequt'nces where all but a finite number of the terms are equal to zero, that is, I\' = {(al, ...• a ..,O,O•...)lakEF,nEZ+}. I') Pix); I\" (~ .. nHiKtfl of all polynomials of degrllll greater than 4. d) V nW); for a fixed n X n matrix (aij).

II' = {x E V .. (P) I (aij)x = O).

e) !If a(F); JV consists of all matrices of the form

[: a:. :J'

a, bEF.

f) M n(F); II' conflists of all uPl>I'r t.riangular matrices of order

n.

8. Find all of the suh!lpacl's of V 2(Za) and V 3(Z2). 9. SUppoIIl' FW), II'(F), reF) art' subMpares of the vt'Ctor space l'(F). Prove that a) if U ~ }', II' ~ r, I,hcn U II' ~ r,

+

+ +

b) (U n r) (II' n 1') ~ (U 1\') n r, c) if II ~ }', tht'n U (II' n Y) = (U l\')

+

+

n Y. 10. I..ct VCR') he the vector space of real-vahll!d functions on the interval [a, b1 under pointwise addition and mUltiplication by a real number.

4-:!

UAKJo:H ANII IIIM~NSION

a) If Jr is the set of all functions! in V such that !(a) = 0, is IV(R') a subspace of V(R')? h) If 11' is the set of all functions/in V for which!(a) = !(b), is lI"(R') a subspace of V(R')? 11. In l'(Z3), dntermillc all of th(! vectors in the subspace "panned by two vectors Xl, X2 E V. 12. If V(P) is a vector space over F, and S, T nonempty subijets of V, establish that a) S ~ T implies IS] ~ IT1, '" 18 U 7'J - 181 I 17'), I') \1811 = 18), d) S ~ 17'1 illll'lil'>l lSI c I TI. 13. In the vector space V 3lF), define the subsets U and I\' by

+ b) I a, b E P},

U = {(a, b, a

\I' = {(e, e, c) IcE F}.

V(>rify that VaW) is Ow dircet slim of the lIuhspacl'f! U(/") and lI"(F), that is, Va=UEf>W. 14. Let V(R') be the vector space of all functions from R' into itself and U

=

{!E V I!(-x)

= !(x)},

II" = {f E V

I!( -x)

= -!(x)}.

Prove that V = U Ef> II".

4-3 BASES AND DIMENSION

Perhaps the most far reaching notion in the study of vector spa(!es and the prindpal t1wme of this 1I(,dion ill thnt of a hllllis. As will 1:10011 be evident, this concept is preeisely the tool needed to formalize the more or less intuitive idea of what is meant by the dimension of a space. A convenient starting point is the following definition. Deftnition 4-16. Let V(F) be II. vector space over the field F. A finite set {X., X2, ••• , Xn} of vectors from V is said to be linearly dependent (over F) X2, ••• , X,,; otherif the zero vector is a nontrivial linear combination of wise, the set {XI, X2, ••• , xn} is termed linearly independent.

x.,

Aecording to our dnfinition, {xt. and only if there exist scalars c., C2, that

,xn } ill a linearly uepenuent set if r.n E /<', lIot all of whi('h are zero, such

X2, ••• ••• ,

In the contrary ease, t.he zero vedor is expressible as a linmu eombination of independent vectors in only the trivial way where all t.he coefficients are zero; that ill to !lily, CIXI+ C2X2 + ... 1- ("".r" = 0 implies CI

=

C2

= ... =

Cn

=

O.

264 .

VIICl'OR SPACES

While linear dependence and independence are properties of sets of vectors, these tenns are frequently applied to the vectors themselves. Thus, we shall speak of XIt X2, ••• , z.. being linearly dependent or independent according as the set {XIt Xli, ... ,xn} is dependent or independent. Two examples should help to clarify matters.

Ex.m.... 4-16. In R'[x), the vector space of polynomials in z over R', consider the three polynomials 1

+ x+ 2x 2 ,

-4 + 5z + xli

2 - X+XlI,

of degree two. Since

the given polynomials are linearly dependent.

Ex.mple 4-17. The so-called unit vector8 ell ell, ... , en, where k

=

1,2, ...

,n,

fonn a linearly independent subset of V,,(F). In this case, tiel

+ C2tll + ... + cne" = (CII 0, ... , 0) + =

(0, C2,

••• ,

0) + ... +iO, 0, .•• , c,.)

(CII C2,"" cn);

hence the condition I:Z-I cj:ej: = (0,0, ... ,0) implies Cl

= ClI =

...

= c,. = O.

The essence of this concept lies in the fact that if {XII XI, ... ,z..} is a linearly independent set, then any vector X e [XII XI, ... , z,,) is uniquely expressible as a linear combination of the xj:'s. Indecd, suppose there are two such repreeentationa, X = bixi bllZlI + ... + bnz.. = CIXI ClIZlI + ... + c,.z..i

+

+

this leads to the relation (b l

-

CI)XI + (b ll -

CII)XlI

+ ... + (b" - c,.)z.. = O.

The hypothesized linear independence of the vectors XI, Xli, •.. ,z.. then (orces or

(or all k.

Definition 4-16, as stated, applies only to finite sets o( vectors. It is no great step to extend these concepts: Given a vector space V(F), a nonempty subset S ~ V is linearly dependent (over F) if it contains some finite subset which is linearly dependent. In the contrary case, S is a linearly independent set; that is to say, S is linearly independent if every finite sUbset is linearly

4-3

BASES AND DIMENSION

265

independent. While S may be infinite, linear dependence is a property of finite character. The following lemma is inherent in our definitions. Let V(F) be a vector spa.ce and 0 P! T ~ S ~ V. If S is a linearly independent set, 80 0.180 is the set T; conversely, if Tis a linearly dependent set, 80 also is the set S.

Lemma.

There is a minor point to be made here, namely, any set which contains the . zero vector is always linearly dependent. This follows from the fact 1(0)

+ OXI + ... + OXn = 0

ill Ii nontrivilll linear combination for arbitrary vectors Xl, X2, ••• , Xn • Accordingly, nil v('('tors in IL linl'arly independent l!Ct must be different from lero The set consisting of a single vector X is, in particular, linearly independent if and only if X ¢ O. • The next theorem in some sense justifies the use of the word "dependent," for it asserts that in a linearly dependent set one of the vectors belongs to the subspace spanned by the remaining ones; putting the matter informally, one of the vectors "depends" on the others. Theorem 4-17. A set of nonzero vectors {Xl, X2, ••• ,xn } is linearly dependent if and only if some vector (k > 1) is a linear combination of the preceding ones, x" X2, ••• ,X"_l'

x"

Proof. Suppose the vectors x" X2, nontrivial linear relation

••• , Xn

are linearly dependent,

80

there is a

where not all the c's are zero. Let k be the largest integer for which CAl ¢ O. If k = 1, we would have C1Xl = 0, hence Xl = 0, contrary to assumption. Thus k> 1 and

Since C;l exists in F, it follows that x"

= =

Ct l (-I)(C1Xl (-Ct l . CI)XI

+ CaXa + ... + C"-lX"_l) + (-Ct l . Ca).r2 + ... + (-Ct l . C"_l).l'k_l.

The vector x" i8 therefore expressible 8.8 a linear combination of its predC(~el!8Ors, as claimed. The converse is almost obvioml: If the vector Xli: depends linearly on XIt X2, ••• , Xk-It 80 that

26()

4-3

VECTOR SPACES

for suitable scalars b/c E P, then btxt

+ ... + bk-1Xk-1 + (-I)xk + OXk-t + ... + Ox" =

BecaullC tim (~oeffieicrtt of Xk if! nOllzero, {Xt, IICt of vcetors.

X2, ••• ,

O.

x .. } cOrtHtitutell a dependent

Theorem 4-18. If VW) iH a finitely generated vcctor space, say

then V is spanned by a linearly independent subset of these vectors. X2, •.. , In} is already indepencient, nothing n(J(~ds to be pmv('d. OtlwrwillC, Th('orem 4-17 implies that ROmf! ve"f,or Ik is 1\ linear (:omhina(.ion of .rt, .1'2 •••• , Xk_I' By hypothesis, any vector x in V can be written as a linear ('ombination of the n vectors Xt. X2, ••• , Xn; in this combination, Xk may hf! rt'pln('f!d hy n litmllr ('ombillation of Xt. X2, ••• , Xk-t. thereby showing x E [Xt, ... , Xk-t, Ik+t. . . . , In). The net relmlt is thn,t the n - 1 vectol'l! Xt, ... , Xk_l, Xk+t. . . . , In generate the space V(P). Next, examine t.he set {xt. ... , X/o-t, Xk+t. ... , xn} and repeat the process of removing a vector if it can be written as a linear combination of its predecessors. Continuing in this way, we eventually reach a subset

Proof. If th(' S4't [x I.

wiwrI' 1 ~ it < i2 < ... < im ~ n, of the original set of n vectors, still having lilwnr spall V aJl(1 su(,h thnt no Xi. is a linear combination of the preceding vect.ors. That the set· {Xi" Ii 2 , , •• ,Xi.. } is linearly independent follows immediately from Theorem 4-17, Example 4-18. As a simple illustration of some of t1wsc ideas, let us observe

that

l' 3 = [(1, 1,0), (1,0, 1), (0, 1,0), (1, 1, 1)].

The rml.soninJl; h('re is jmltifif!d by t.hc fa('t t.hat. e:wh of t1w unit, vectors et. e2, ea iM It lillcllr ('(lIllhillul.ioll of I.JII·MI~ Vl'dOrM: el

=

(1, 1,0) -

(0,1,0),

e2

=

(0, 1,0),

Ca =

(1, 1, 1) -

(1, 1,0).

Since V 3(P) is spanned by the ullit vedors, we infer that every clement of V 3 I\lUSt. 1)(':\ lilwar ('omhinatioll of (I, 1,0), (1,0,1), (0,1,0), (1,1,1), The lilJ('ar dl'l)('JI(\I'III'(~ or iIlC\I'JlC'tHII'II('c of t.his SC't. of ve(~t.ors is equivalent III Ilw exist.ellc'(! or 1I00I1'xisll'lI('(' of S('lIl:trl; (~I, 1'2, ra, (:4 (1I0t. nil zero) sudt thnt I't(1, 1,0) 11'2(1,0,1) 11:3(0,1,0) 11'.(1,1,1) Chl'ekillg ('OIllPOlll'lItl'l, we note that. the CI

+

C2

-I 1'4 = 0,

Ct

Ck'S

-I- C3 -I-

=

O.

must. sat.isfy the three equations

1'4

=

0,

4-3

267

BASES ANI> DIMENSION

In terms of an arbitrary choice of for instance, C4 = 1 leads to 0(1,1,0) -I- (-1)(1,0, I)

C4,

+

a solution is Cl = 0,

(-1)(0, 1,0)

C2

=

+- 1(1,1,1) =

-C4, C3

=

-C4;

(0,0,0),

whi('h, of eounw., impli!'s that t.he given set of gem'rut,or!! is linearly dependent. To obtain a linearly independent subset of the aforementioned vectors, we need only removc the element (1,1,1). In fact, (1, 1,1) is the only vector that can be written as a linear combination of those preceding it: (1, 1, 1)

Any

memh(~r

= (1,0,1)

+ (0,1,0).

(ai, a2, aa) of V 3 ean obviously be expressed as

wh('nec Va

=

[(1,1,0), (1,0,1), (0, 1,0)1.

The next, somewhat technical, result is the key to all that follows. theorem 4-19. (Steinitz Replacement Theorem). Let W(F) be 8. finitely gellerau~d subsplwe of t.Il(, vc(:tor I'lpnce V(F), W = Ix., :1:2, ••• ,xnl, and let {Y1o Y2, •.. , Ym} be any linearly independent subset of W. Then m of the Xt'S, say X10 X2, ••• , X m, may be replaced by Y .. Y2, .•• ,Ym, so that W = [y., . .. ,Ym, Xm+1o'" ,xnl, in particular, m ~ n.

Proof. Since {X., X2, ••• , x,.} spans W(F), the vector Yl E W can be expressed as 8. linear combination of the Xk'S:

Not all the cocffieients ak = 0, for otherwise Yl = 0, contradicting the linear independenee of t.he I!f1t [lit, 112, .•. ,11m}. R('illdexing, if llN·CR.'!firy, we may ILSSllIII(l thllt al ;t6 O. Nuw solve fUl' the V('I~t(IJ' of I in \.(~rllltl of 11 .. X2, ••• , .1:,,:

This relation permits us to replace a linear combination of X10 X2, ••• , Xn by a linear combination of the vedol'H YI, X2, .•• ,X"' and leads to the conclusion that W = rUI, 3'2, •..• x,,1. Rep(ml.l.he repilwenwn\, PI'OCNl:i with the vllC'l.or!1'l nlHl the Het [U .. .1:2 • ••• , x,,}. B('('IUI!'!e 112 belong!! to the Kubspac(' spanlled by y., X2, ••• , X n , we must have

for suitable scalars b., b2 • ••• , bn E F. The coeffieiellts b2 • ••• , b.. cannot all

268

4-3

VECTOR SPACES

be zero, for this would imply that b l r& 0 and, in tum, that bl 1l1

+ (-1)111 + Oys + ... + 011", = 0,

contrary to the independence hypothesis. Hence, one of the coefficients bl, ... , b" is nonzero; let us, for simplicity, take this to be bl. As before, we can solve for X2 in terms of the vectors YIo Y2, Xa, •.• , x" to obtain

W

=

[1Il> YI, Xa, ... , x,.1.

Continue in this manner: At each stage a y-vector can be introduced and an x-vector deleted so that the new set still spans W(F). If m were larger than n, after n steps all the Xk'S could be removed and the set {1Il> 112, •.• , 11..} would span W(F). Accordingly, the vector 11"+1 E W could be written as a linear combination 11"+1 = CIYI + C2112 + ... + c,.1I" with not all the c" being zero, since 11,,+1 po! O. Once again a contradiction to the independence of 111, Y2, ••. ,11.. would arise. It follows that the 1I'S must be exhausted before the x's (that is, n ~ m) and W = [YIo ••• , Y"., x..+1, ••• , x ..1. Of course, the possibility that n = m is not excluded; in this situation, the set {x..+1, ••• , x,.} is empty and the vectors 111, YI, .•. ,1/" themselves span W(F).

Corollary. If a vector space V(F) is spanned by n vectors, thea any set of n 1 vectors from V is linearly dependent; in particular, any n 1 vectors of the n-tuple space V,,(F) are dependent.

+

+

Proof. A set of n + 1 linearly independent vectors from V would be impossible, since the theorem would imply n ~ n + 1, an obvious contradiction.

The Replacement Theorem has several notable consequences, but these will have to await further developments. DefInition 4-17. A bam for a vector space V(F) is a linearly independent· subset of V that spans the entire space V(F). Example 4-19. The familiar unit vectors e" e2, ... , ell of V,,(F), the space of n-tuples of scalars, form a basis (see Examples 4-14 and 4-17). Hereafter, we shall refer to this particular ba.sis as the natural or standard ba8i8 for V,,(F). Example 4-20. For a more general example, consider the vector space V.(F) of infinite sequences of elements from F. If

k

=

1,2, ... ,

then the set 8 = {e" e2, ...} is linearly independent. These vectors do not constitute a basis for V.. (F), however, since it was shown earlier that the linear span [81 is a proper subset of V•. Example 4-21. Let M,,(F) be the vector space of n X n matrices over a field

F. This space has a basis consisting of the n 2 matrices EiJr where EiJ is the

4-3

BASES AND DIMENSION

269

square matrix of order n having 1 as its ijth entry and zeros elsewhere. Any matrix (aij) E M .. (F) can obviously be written as (aij)

=

(ll1EU

+ auEu + ... + a,...E .....

Moreover, (ai;) = 0 if and only if all = au Ell, Eu, ... , E .... are linearly independent over F.

= ... = a,... =

0, hence,

Example 4-22. One final example: Consider F[x], the vector space of polynomia.ls in x with coefficients from F. A basis for F[x] is formed by the set

S

=

{I, x, x 2 ,

••• ,

x .. , . ..}.

+

+ ... +

By definition, each polynomial p(x) = aol alX a,.x" of F[x] is a linear combination of elements from S. The independence of S follows from the fact that, for any finite subset the relation holds if and only if Cl

=

C2

= ... =

CIc

= o.

These are but a few examples of the more frequently encountered bases and should amply illustrate the concept; as we continue our discussion, additional examples will appear. We ought to point out several things. First, Theorem 4-18 may be rephrased 80 as to assert that any vectOr space which is spanned by a finite subset, linearly independent or not, possesses a finite basis. Since a basis S for a vector space V(F) is by necessity a linearly independent subset of V, it is possible to express each vectorj Vasa linear combination of elements from S in exactly one way. The u ique scalar coefficients which occur in this repreeentation are called the c dinateB of x with respect to the given basis. Thus the notion of a basis enables us to coordinaliie the space. • Finally, let us observe that a. given vector space may have more than one basis. Example 4-18, for instance, shows that the vectors Xl

=

(1, 1,0),

X2

=

(1, 0, 1),

xa

=

(0,1,0)

constitute a second basis for Va(F). In this case, an arbitrary vector (alJ all, 4a) of Va can be written as (aJ, all, aa)

=

(al - aa)Xl

+ 4aX2 + (all -

al

+ aa)xa.

While the coefficients in the above linear combination are uniquely determined, they obviously differ from those which represent the same vector relative to the standard basis ell e2, ea; roughly speaking, a vector has different coordinate. with respect to different bases.

270

4-3

We next dispose of a natural question that arises here: is it possible to obtain a basis for a given vector space? A closely related question is this: presuming one can seleet two differ('nt bases for a space, must each contain the same number of c1!'n1('nt.s? When our vector space is the zero space {O} (F), no subset is linearly independent and certainly no basis exists. On the other hand, t.he coming th!'or!'m guarantees that a nonzero space will always have a basis. The proof ill II. stl'llightforward application of Zorn's Lemma. Theorem 4-20. a basis.

(Ba.~i.~

Theorem). Every nonzero veetor space V(F) possesses

I'roof. Let a he the family of all linearly independent Hubsets of V. If x "" 0, then {:r} E «, so t.hat. (t is plainly nonempty. Our immediate aim iH to show that for any chain of Sl't.s f11 j} in a. their union UA j also helongs to a. To do so, we aSSlIllH' th .. vl'etors .rl, X2, ••• , .rll E UA j and that. UH' lilwar combination ('I.rl 1 r2'(2 1 ... 1 r":e,, = O. Now, each vllct.or Xk lieH in some member Aik of {/1,}. As : A,: forms a chain, one of the sets Ai" A ;2' .•.• A i. contains all the others. call it A j'. This means that the given vectors Xlr X2, ••• , Xn arc nil in A j'. Bill. the linear ilHlependence of A j' then impli(~s that the Hcalar coefficients rk = 0 for all k = 1, 2, ... ,n. Thus the union uA i is itself a linearly independent subset. of V, whence a ml'mher of thc family a. The hypoth!'s('fol of Zorn's Lemma being Ratifolfied, there exists a maximal clement S in a. As a m('mber of a, S is a linearly independent ""subset of V, folO to complet!' the proof it remains simply to establish [S) = V. To see this, let x be any v!'dor of V not in S. Because th!' set S' = S U {x} properly contains S, it mUfolt be linearly dependent (the maximality of S enters here). Therefore, for folome finite suhs!'t :!lI, 112,' .. ,lIm1 of S. It dependence relation

exists in which not all the cocffieicnts are zero. Were a = 0, we would contradict· th~ lill<'ar indl'II('II
= (-a-I. al)lIl -t (-- a-I.

a2)112 -1-"'1 (--a-I. am)Ym

or x E [.'I). WI' eOlll'hlll(' from this that t.he lin('ar spnll of 8 is the whole folpac(' V(F), finiHhing the proof. The ItrgllJnf'nt lIS<'d in Th('orem 4-:W slIgg('st,s t.he following ehll.rnderizntion :l hasis ror 1,111, Vf'C'lor Spill'" reF) if llllli only if S i~ IL nlllxilllal linearly in
of a hasis: S is

4-3 for some finite set E F. But then

DASES AND DIMENSION

{XI, X2, ••• ,Xn }

271

of vectors from S and suitable scalars

CIc

is a nontrivial dependence relation among veetors of S u {x} and leads to the eondusion t.hat S U {xl is linearly dependent. A set. S of generat.ors for It vedor space V(F) is termed minimal if every suhset. fortllt'd frolll S hy removing one or more vectors fails to 81)l\n V. This idea gives rise to another characterization of a basis:

Theorem 4-21. A set S is It bw;is for a vector splu:e V(F) if and only if S is It minimal g('nemt.ing set. of V. The proof of til!' t.heon'nl is rOlltin<', and we 1<'lLve til(' det.ails to the reader. WI' arl' 1I0W pn·pan·1I III ('iiI ahlilih tllP Illain t.hl'lIl'('nl of I his iiI,('tion: all hlU!
Theorem 4-22. If V(P) is u. finitely g<'lH'rtlted nOllzero vee tor spaee, then any two bases for V(F) have precisely the same number of clements.

Proof. Let {XI, :r2, ... , xn) he one basis for thl' spaee V(F) and {YI, Y2,' .. ,Ym} be unother. Beeause the vedors YI, !/2, ... , 11m are linearly independent and V(F) is spannell hy :rl, X2, ..• , Xn , it follows from t.h(~ Rephwcmcnt Theorem t.hat t.he integer m is no larger than n, tlmt. is, m ~ n. Simply revertling the roles of {XI, X2, , .. ,xn } and {Yb Y2, .•. , Ym) in the argument, we also condude n ~ In, whClII'e n = m. Theorem 4-22 ean be used to define dimenllion. For if II vedor space V(F) has a finite basis, then every ballill for the IlJlace wiII have the sallle finite number of clemen til. This unique integer is called t.he dimension of V(F) anti dellignatcd by dim V. Although UII' notion of hmlis iH IlOt ltpplimhlc to the z(~\'O space, it is (~IIKt.omary t.o t.n'llt. :0) (P) lUI finit.e-dinwllliiollal, wil h Zf'ro dimellsion. A (nollzcro) v('ctor sJllwe ill ~tid to be injinite-dill/.fIt.~il}Ttal if it. ill not spanned by any fillite sllhiil~t., that. is, if it iH not finite-dimellsiollal. Not every spnce is of finil.t, dilll('IISioll, II..'! ('videlll·(·d hy F[.rj, t.I\(' vpdo\' KpIU'(' of' polynomials ill x. III lIIost of what. follows, we lihall for 1.1", sakI' of I·(llll'ppl.llal HilllJlli"ily eOllfille ollr alll'lItioll t.o fillitl!-dilll('IIHiolllll VI'I'I.OI· liJlIU~I'S. '1'111' IIl'xl I hl'fl 1'('11 I illdi":lII'K t.Iml al II':llit r'JI' lillill'-,liIlII'IIHioll:l1 HJlIU'I'H, Oil£' "lUI f01'1II haiil'" without. 100 I11I1I'h dilli,·ulty.

Theorem 4-23. Let \'(F) 1)(' It fillite-dillll'lI"ional V('dOl'''JlIU'I', liay dim V = n. Thl'll I) I'vl'ry iiI't. of n Vf'f'lors whil'h KJlall V iH a haHiH, /I lilll'ady illlll'Jllmcll'lIt V\'dorH frolll \' is

:!) ('V\'ry Ii!'t of

Il

hllSiK.

272

4-3

VECTOR SPACES

Proof. Suppose V = [Zl, Z2, ..• ,Zt&J. According to Theorem 4-18, some subeet of {Zh Z2, ... ,Z,.} is a basis for V(F). But the previous result implies that this subeet must contain n elements and accordingly is the entire eet {Zh Z2, ... , z,.}. For a proof of the second assertion, let Zh Z2, ... ,Z.. be n linearly independent vectors of V. If Z E V is arbitrary, the set {z, Zh ... , z,.} is dependent, since the mwcimalllumoor of linearly independent vectors in V is n. Thus some nontrivial dependence relation exists among theee vectors: ex

+ C.Zl + ... + C,.z,. = o.

Were the coefficient c = 0, a contradiction to the linear independence of Zh Z2, ... ,Z.. would arise. Hence c ~ 0, and we may solve for the vector Z in terms of Zit Z2, ... , z .. to obtain

This argument shows that

making the eet {Zit Z~h ... ,z,.} a ; ; . for V(F).

...

Theorem 4-18 told us that it is ·ble to chooee a basis for a vector space V(F) from any set of generators of V. In the opposite direction, the coming theorem asserts that any linearly independent subset of V is either a basis or else can be extended to a basis for V(F), in the sense that vectors may be added to it to fonn a basis. Theorem 4-24. If V(F) is a finite-dimensional vector space and {ZltZ" ... ,Z,.}

is a linearly independent subset of V, then there exist vectors 1I,.+h ... ,II... such that {Zl, ... ,.z,., 1I.. +h ... ,II..} forms a basis for V(F). Proof. The proof is short: Since V(F) is finite-dimensional, it has a finite basis IIh IIllt •.. ,II..· As V = [lIh 112, ... , II..], we may apply the Steinits Theorem to replace n of the II's by z's and obtain a eet {ZI, •.• , z,., 1I"+l, ••. , II..} whose linear span is atill V. But any generatin, set with m elements in an m-dimensional space is a basis.

Corollary. Every basis for a subspace of a finite-dimensional vector space can be extended to a basis for the entire space. . Example 4-23. As a particular case of this last point, consider the vector space M,(F) and the three matrices

4-3

BABES AND DDlJ!lNBION

273

To check the linear independence of these matrices, let

On equating corresponding entries, we see that Cs - 0,

which implies CI = C2 = Ca = O. Theorem 4-24 now tells us that the given matrices will form at least part of a basis for M 2(F). We shall leave the details to the reader and content ourselves with this one comment: the addition of any fourth matrix having a nonzero element in the (2, 2)-position will yield a linearly independent set. For instance, among the many possibilities, the matrices

are linearly independent and, being four in number, must therefore be a basis for the 4-dimensional space M 2(F). This illustrates the wide latitude of choice for bases of M 2(F). We terminate the present section by giving some useful results concerning the dimensions of subspaces of a given vector space. The reader should first prove the following theorem. Theorem 4-25. If W(n is a subspace of a finite-dimensional vector space V(F), then 1) W(F) is also finite-dimensional with dim W S dim V. 2) dim W = dim V if and only if W = V. The next theorem relate~ the dimensions of the sum and intersection of two subspaces. Theorem 4-26. If U(F) and W(n are 8ubspaces of a finite-dimensional vector I!plwe V(F), then dim (U

+ W) = dim l! + dim W -

dim (U

n W).

In particular, dim (U E9 W)

= dim U + dim W.

Proof. Before entering into the details of the proof, we observe that by Theorem 4-25 the four subspaces involved in the statement of our theorem all have finite dimension. Now, let {xa. X2, ••• ,x,,} be a basis for (If n W)(F). Accord-

274

4-3

VECTOU SPACES

ing to Theorem 4-24, there will exist vectors u., U2, ••• , u'" such that {X., •.. , Xn, u., ... , u"'} is a basis for the subspace U(F) and vectors 1/'., 11'2, ••• , //', ~\wh that {Xb' •• , X"' WI, ••• , w,} i~ a bu.sis for the subspace W(F). Combine these two bases into a single set

Becausc the finlt m + n vectors of the foregoing set are a basis for U(F) alld th(' last TI -I r vect.ors are a basis for W(F), any c1f1ment of the sum (! + W may I)(~ cxprN;sc!\ ItS It lincar eomhirmtion of the vectors of thi" sct, that is

F / W = [UI, ... , U"',

XI, ••• ,

xn ,

w., ... ,w,l.

We wish to show that the vectors on the right are also linearly independent and conRequent.iy a 1>I\sis for the subspace (U + W)(F); once this is established, it would follow that dim (fT

+ W)

=

m+n

= dim

II

+r =

(m

+ dim W

+ n) + (n + r) - dim (11

- n

n W).

Thereforc, let us suppose

Settinp; 2 =

C111'I

+ ... + C,II'"

we would t.hen have

I fIlII / ... t- bnxn) or 2 E [X., . .. , In, ttl, ••. , uml = [T. Since the vector 2 ulao belongs to W, it lIIust hI' 1\ lilll'ar (·llIl1hiIlHj.ion of Hw hnsiIl1'1!'lIl1'lIt.S X" X2, ••• , Xn of I,he Imbllpal'1' z~·

- (alul -/ .... / a,nu",

(i! n W)(/"), lIay,

so that

But the set {XI, ••• , X"' till, ... , til,} is linearly independent, being a basis for the subspace lV(F), hell(~e d l = ... = dn = CI = ... = Cr = O. In conjunction wit-h the IinC'ar indl'pcIHI!'n('e of fx., ... , Xn, U" ••• , u m }, this fon~es al = ... = am = II. = ... = b" = O. We havl' thus sueeccded in provinj!; that t.I\I~ vcclol"ll I l l , " " Itn" .r" ... , .r", !t., arc linearly independent, !l.S required.

w" ... ,

Theorem 4-27. If W(F) is a subspace of It finite-dimensional vcetor spa(:c V(F), then the quotient space (V /W)(F) is allID finito-dirnelll~ional and

dim V = dim W

+ dim V/W.

4-3

BASES AND DIMENSION

275

Proof. Let {Xl. X2, ••• ,XII} be a basis for W(F). By adding vectors Yt. Y2, .•. , y"" we may extend this set to a baliis {x!, ... , X,,, Yh .•• ,y",} for the whole !!pacc V(/t'). Any vector x E V can then be written in the form x = atXl a"x" b1Yl b",y", for appropriate choice of coefficients. Since x - (b'Yl b",y",) E W, the coset x + W is expressible

+ ... +

+

+ ... + + ... +

as

In othn!" words, i1w demcnl.li Yl + W, 712 + W, ... ,!1m + W span the quotient Rpuee (V /W)(P). The remainder of t.he proof amounts to showing th('HC coset!! to be linearly independent and heIl(~e a basi!! for (V /W)(F). To see thill, we suppose

where, of course,

t')

+W =

W is the zero clement of V /W. Thus

elY I

and must he

:~

+ C2Y2 + .. , + CnY", E W

linpur ('ombilllLtioll of the basis v('ctors

Xl, X2, •• , ,

x" of W, say

But the lincar indepcndence of the set {x!", . , x"' YI, .•. , y",} then implies 1:, ~ •.• = em = (I, = ... = till = O. The foregoing urgulllcnt. indicates that the quotient space (V /W)(F) has a basis eonsisting of the m cosets Yl + W, Y2 + W, ... , y", + W. The conclusion of Uw U\('orem followH imml'diutcly from this, sirwil dim V /W

=

m

=

(m

+ n)

-

n

=

dim V - dim W.

Example 4-24. We iIIustmte the above by looking again at Va(F) and the one-dimensional subspace W(F), where W

=

{(a, 0, 0) I a E F}.

In this case, the quotient space (Va/W)(F) has dimension 2, for the equation of Theorem 4-27 reads dim V 3/W

=

dim V 3

-

dim W

=

3 - 1 = 2.

To actually obtain a basis for the quotient space, one need only employ the procedure of the theorem. First, extend the basis of W(F)-that is, the vector (1,0, O)--to a basis for the entire space V(P), say by adjoining vectors (0, 1,0) and (0,0,1). The corresponding cosets (0,1,0)

+W

and

(0,0,1)

+W

276

4-3

VECTOR SPACES

then serve as a basis for (Va/W)(F). Indeed, it is easy to show that any element (CllI Cl2, CIa) W of Va/W m&y be written in the form

+

(CllI CI" CIa)

+ W = CI,[(O, 1,0) + W) + Cla[(O, 0, 1) + W).

PROIUMS

In the problems below, F will denote an arbitrary field.

1. For each of the following vector spaces, determine whether the lets listed are linearly dependent or independent. a) V.(F): {(I, 0, 0, 0), (1,1,0,0), (1, 1, 1,0), (1,1,1, I)} b) F[%]: {%' % - 1, %2 - % - 2, %2 + %+ I} c) Va(Za): {(4, 1,3), (2,3, 1), (4, 1, On d) Va(C): {(I, 2 i, 3), (2 - i, i, 1), (i, 2 3i,2)}

+

+

+

2. Prove that if each vector form % - 41%1 42%, are linearly independent.

+

% e [%1, %" ••• ,%..] + 4,.2: .. (4t F),

+ ...

e

is uniquely representable in the then the vectors %1, %2, ••• , %..

3. Given vectors %1, %2, ••• ,%.. e V, establish the &8IIertioDB below: a) If %i ... %j for some i ~ j, the set {%1, %2, ••• ,%..} is linearly dependent. b) If {%1' %2, ••• ,%..} forms a linearly independent set and aI, a2,'" •• , a ..-1 then

e F,

is also an independent set of vectors. c) If {%l' %2, ••• ,z ..} is linearly independent while {%1, ••• , %., %.. +1} is linearly dependent, then the vector %.+1 e [%1, %" ••• , %J. 4. Show that the vectors (3 - i, 2 2i,4), (2,2+ 4i, 3), and (1 - i, -2i, 1) form a buis for the apace Va(C) and determine the coordinatee of each of the .tandard basis vectors (1,0,0), (0,1,0), and (0,0,1) with respect to this basis. .

+

o.

a) Find a basis for the vector space C(R') and all bases for the space Va(Z,), b) For what values of 4 do the vectors (1 a, 1, 1), (1, 1 a, 1), and (1,1,1 a) form a basis of V,eR'>?

+

+

+

is a basis for the vector space Va(R'>. Verify that the sets and {%1, %1 + %2, %1 %, %a} &lao serve as bases of Va(R'). Is this situation true in the space Va(Za)?

6. Assume

{%1, %2, %a}

{%1 + %2, %, + %a, %a + %I}

+ +

7. Prove that the subspace of F[%] consisting of all polynomials of degree at most n is finite-dimenaional. 8. If diag Jf.. denotes the set of all diagonal matrices of order n (over the field F), show that (diag M ..)(F) is a subspace of the vector space M ..(F) and determine its dimension. I

9. Prove that if W(F) is a proper subspace of the finite-dimensional vector space V(F), then dim W < dim V.

LINEAR MAPPINGS

10. Assume the space V(n is finite-dimensional with basis {~l, ~2, ... ,z,,}. If W.(n is the subspace generated by the vector Zk (Ie - I, 2, ... , n), verify that V - WIE9 W2E9 .•• E9 W". 11. Let U(n and W(n be subspaces of V,,(F) such that dim U > n/2, dim lV > ,,/2. Show Un w pi {OJ. 12. Suppose {~l, ~2, ... ,~,,} is a basis for the subspace U(n of V(F) and {Ill, 112, ••• ,II.. } is a basis of the subspace W(F). Given that the set {~l, ... , X", Ill, .•. ,II..} forms' a basis for the entire apace V(F), prove that V - U E9 W.

+, .)

13. Let (F[x), be the ring of polynomials in X over F and p(~) E F[~) be a polynomial of degree n. If «p(~», +, .) is the principal ideal generated by p(x), establish that (F[~1I(p(~»)(F) is a veotor space of dimension n. [Hint: Consider the cosets 1 + (p(~», Z + (p(x», ••• ,X,,-1 + (p(x) ).] 14. Determine the dimension of the quotient space (Va/W)(F), where the set W is defined by W - {(a, h, a+ h) I a, h E F}. 15. In the vector spaCe M,,(n, let T ..(F) be the subspace of upper triangular matrices [matrices (a.,) such that a., - 0 for i > j) and T~(/") be the subspace of lower triangular matrices [matrices (ai,) such that a.i == 0 for i < j), Find dim T", dim T~, dim (T" n n), dim (T.. + T~), and verify the truth of Theorem 4-26 in this particular case. 4-4 LINEAR MAPPINGS

In this section, which is our last, we examine the vector space analog of the familiar homomorphism concept. Since a vector space V(F) is comprised of two algebraic systems, a group (V, +) and a field (F, +, .), there may be BOrne initial confusion as to what operations are to be preserved by such functions; the answer is only those operations which explicitly involve vectors: vector addition and scalar multiplication. Traditionally, vector space homomorphisms are called linear mappings or linear transformations, and we adhere to this terminology. DefInition 4-18. Let V(F) and W(F) be vector spaces over a field F. A function f: V - W is said to be a linear mapping from V(F) into W(F) jf f(x

+ y) = f(x) + f(y)

and

f(cx)

= cf(x)

for all vectors x, y E V and all scalars C E F. The set of linear mappings from V(F) into W(F) wiIl subsequently be designated by L(V, W). Simply put, a linear mapping from the space V(F) into W(F) is a homomorphism from the additive group (V, +) into the additive group (W, +) which, at the same time, preserves scalar multiplication. This is plainly equivalent to the single requirement that f(az

+ by) = af(x) + bf(y)

278

VECTOR SPACES

for all x, Y E V and a, b E F. Suffiee it to say, the above definition makes sense only when both vector spaces are taken over the same field. Before proceeding to the theory of linear mappings, let us illustrate to some extent the great variety of possible examples. Example 4-25. Let M m,,(F) be the vector space of all m X n matrices over a field F, and let (ai;) he!\ fixed m X m matrix (again, over F). Define a function I: 1If rn" -+ 111 mn by

Then I is seen to be a linear mapping, because fHII,j) I

II«(:,}»

= (aij)' [r(lli,)

-

r{a,;)· (/Iii)

I II(C,,»)

I-

lI(aj/) . (r.ii)

.,.,- r!«aij» /- 1I/«r.iJ»' Example 4-26. In the Rpll('e V",(F) of infinite Heqlwnees of elements from a field F, we define thc shifl function f as follows: for each sequence x = (at. a2, aa, ... ), take I(x)

=

(a2' a3, a4, ... ).

The reader may easily verify that I is a linear mapping from VlI (F) to V..,(F). In fact, any power of I is again linear, Hinee

Example 4-27. Next consider F[x], the space of polynomials in the indeterminant x with coefficient,s from F. A lillear mapping 011 F[x] is given by means of Uw Ho-{:nlInd dijJerentiat1:on functilln. That. iH, for lUI arhitrary polynomial p(x) = ao --j- alx + a2x2 + ... -f- a"x" ill ~'[x], let I(p) = a1

+ 2a2x + ... + nanx,,-l.

Example 4-28. One more example: Suppose W(F) is a subspace of the vector space V(F). Then the familiar natural mapping natw: V -+ V IW defined by taking natw (x) = x + W

is a linear tranHformation. By virtue of the definition of the operations in (V IW)(F), we easily dlCck that natw (a.x

+ by) =

+ by + W = a(x + W) -I- bey + W) = ax

a

natw (x)

+ b natw (y).

One fact which follows almost immediately from Definition 4-18 is that if dim V is finite, then any linear transformation f from V(F) into W(F) is com-

4-4

LINEAR MAPPINGS

279

pletely described hy specifying its values on a basis for V(F). For suppose {Xli X2 • ••• ,;r n } is It basis of the space V(F) j then Mch vector X in V has the form for suitable t;("alars ak E F. By the linearity condition, extended to n vectors,

Thus,! is completdy determined onee its cITed on the bu.'lis vectors Xl, X2, ••• , Xn is known. The next. theorem indicates that this effeeL may be prcserihed nrhil.rllrily.

Theorem 4-28. 1,,'1. V("'IIII'HIII\I'"

:.r I, .r~, ...• ;r,,:

1"(/<') 111111 :111, II~, • ••

w(/,'). '1'111'11 I.\wn. iH l·xlIl!t.ly

(lI1()

be!

,/I,,:

IL 1111

bMiH

fOl·

nl"l,il.nLl'Y

t,lIn

fillit~,-clillloIlHi()tll\l

Hill. Ill' "

VC,d.III·H frllIll

Iillellr II IllppiIIg f E LO', W) sllch thnt (k=l,:l, ... ,n).

Proof. To prove that there d(){.s exist such a lllapping f, we proceed as follows: Since {Xb X2, ••• , xn) is u basis for V(F), each vector x E V is uniquely expressible in the form

Let us simply define a function f: V -

W at the vector x by taking

J(x) = alYI -j- a2Y2

+ ... + anY.. ·

To see that this function is actually linear, let x, Y E V, where

Then

and 'so, hy definition, f(x

+ y) = =

+

+

+

(al b1)YI + ... (an bn)Yn (alYI + ... anYn) + (b1Yl + ...

+

+ bnYn) =

f(x)

+ f(y)·

One establishes similarly that f(ex) = rf(x) for each scalar e ill F. Now, suppose 0 is any other lillcnr t.mnsformutioll from V(F) to W(F) with the propert.y that O(x,,) = Yk. Thcll, for any vcet.or x = alxl + a2x2 + ... + anx,,, till' lillL'urit.y of (f implies o(x)

= =

+ a20(x2) + ... + anU(xn) alYI + a2Y2 + ... + anYn = f(x).

alo(xl)

280

VECTOR SPACES

We therefore conclude that the linear mapping I for which I(z,,) - 11" is the only one possible. To see how closely linear mappings arc associated with matrices, let us now suppose the space W(F) is also finite-dimensional, say, with basis {Yh Y2, ..• , y",}. Then each vector I(Xi), being in W, is a unique linear combination of these basis vectors: I(Xi)

'" ai;Yi, = :E

(j

=

1, 2, ... , n).

i-I

(Observe that, in contrast with many texts, the summation is on the first index of 4;j.) In this fashion, we produce an m X n matrix (aii) called the matrix representatiun of I relative to the bases {Xl, X2, ••• ,XII} and {Yt. Y2, .•. , y",}. Notice particularly that the matrix (aii) depends on the pair of bases used for V(F) and W(F) as well as oni; any change in the basis elements, even in their order, would lead to a different matrix. Accordingly, the same linear mapping (whose definition, after all, does not depend on any basis) might be represented by several matrices. Now, any vector X E V can be expressed as a linear combination of the basis {zt, X2, ••• , XII} : . , Hence,

Once the bases are picked, the effect of I on each vector of V is therefore completely determined by the scalars Ci and the representing matrix (4;j). On the other hand, two bases {Xl, X2, ••• , X..} and {Yt. Y2, ... , y",} for V(F) and W(F), respectively, along with an m X n matrix (4;/) of elements from F, give rise to a unique linear transformation IE L(V, W). All we need do is define the proposed mapping on the basis of V(F) by specifying • I(z;)

'" aiiYi, = :E i_I

then extend I to all of V by the condition that it be linear. The reader may well feel, in view of these remarks, that it is unnecessary to distinguish any longer between a linear mapping and its representing matrix, and that all subsequent results could be phrased in the language of matrices. The utility of this approach is diminished by the fact there is no unique correspondence between linear mappings and matrices; before we can introduce the matrix representation, we must first pick bases for the underlying vector spaces, and the matrix depends strongly on the bases chosen. Since most aspects of the theory are best treated as independent of any basis, we shall study linear mappings in the abstract, without reference to matrices.

4-4

LINEAR MAPPINGS

281

As a linear mapping I e L(V, W) is, in particular, a homomorphism of the additive group (V, +) into the additive group (W, +), the results of Chapter 2 may be utilized. I·'or instance, by Theorem 2-40, we already know that I will be a one-to-one function if and only if ker (f) = {OJ; as usual, ker (f) consists of all vectors mapped onto t.he zero element of W: kcr (J) = {x e V If(x)

=

O}.

Recall also that 1(0) = 0, whence ker (f) oF- 0, and f( -x) = -/(z}. By this stage, the first two parts of the following theorem should come as no surprise; their demonstration is not difficult a.nd we ask the reader to fill in the details. Theorem 4-29. Let V(F) and W(F) be vector spaces over the field F and the mapping I e L(V, W). Then I} (ker (J)(F) i~ a subspace of V(F); 2} (f(V»(F) is a subspace of W(F); 3) dim V = dim ker (f) + dimf(V) [Sylvester's Law]. Proof. ,We establish assertion (3) only in the case where dim V is finite. Suppose first that ker (f) oF- {O}, so that a basis {Xlt X2, ••• , z,.} may be selected for the subspace {ker (f»(F). Using Theorem 4-24, this set can now be extended to a basis {Xh ••• ,z,., Yl, ... , Y... } for the entire space by adding new vectors Yl, Y2,"" Y.... Given any element Y e/(V), there will exist some x in V such that Y = f(x). In terms of the basis for V(F), the vector x is representable as

x- =::- 41Xl

+ ; .. + a,.z,. + blYI + ... + b".y....

But,/(x,,) = 0 for k = 1,2, ... , n, which implies Y

= lex) = =

+ ... + a,.f(x,.) + bt/(Yl) + ... + b... f(y ...} b./(YJ) + ... + b... f(y ... )·

at/(xI)

From this, we deduce that

We maintain further that the vectors f(Yl), I(Y2), ..• ,/(y... ) are linearly independent and consequently a basis for (f(V» (F). Assume some linear combination of them is t~e zero vector, say '.,;

Then,/(cIYI

+ C2Y2 + ... + c...y",) = 0, or expressed otherwise, CIYI + C2Y2 + ... + c...y ... e ker (f).

.

282

v

~;C'r()1l

4-4

foll'ACI';H

Since XI, XZ, ••• ,X" are a basis for (ker (f)) (F), there must exist scalars dJ, ... , d" such that

Were any of th!,H(~ N)('f1i(~i!'ntH nonzero, Il. eontradiction to the linear independenee of t.he lief, {x I, ••. , X n, Y I, ••• , Ym} would result. Hencc CI = ... = C m = 0 and we have Slle('('eded in proving that the m vectors f(Yl), f(Y2), ... , f(Ym) form a basis for the image space (J(V»)(F). This means dim V

= n+m=

dim ker (f)

+ dimf(V).

If ker (f) = {O), f hNl eSH(mtildly the same argument shows that f maps any hlLHil'! for r(F) ont.o IL hasil'! for (J(V») (F); thui\uim V = dimf(V), establishing Hylv('sf('r's LuI\' for t.hit' ('HKI' nlKo.

!

Corollary. Let V(P) ILnd W(F) he tinite-dimensional vector spaces with

dim V = dim W, and let f E L(V, W). and only if f maps onto V.

Then f is a one-to-one function if

Proof. To begin with, I'IUppose f is one-to-one, so that ker (f) = {O}. Then dim ker (f) = 0, and Sylvester's Law reduces to dim W = dimf(V). It now follows from Th('orem 4-2!i that W = f(v), whence f is an onto mapping. Conversely, if the range of f is all of W, the same equation yields dim. ker (f) = 0, or equivalently ker (f) = {O}, so f is one-to-one.

Remark. Neither one-to-one nor onto implies the other for linear mappings between spaces of infinite dimension. Example 4-29. We illustrate Sylvester's Law with the linear mapping f: V 4 -+ V 3 given by

The kllrncl of J iH el~Hily 1'41l1'l1 to lin thn flCt of VClltOrH ()f tho t,ypc (a, a, a, a), while itH range is all of V 3' Hcncc, dim 1'4 = 4 = 1 -I- 3 = dim ker (f) -I- dim V 3' The terms isomorphil' alHl isomorphism have the obvious meanings for V(·I·tor KIIIU·C'S. Two V<'dor spaees V(F) and W(F) ILre isomorphic, written V(F) ~ lV(F), if and only if there exists It one-to-onc mappingf: V -+ Wof V onto lV whidl preS('rv('s the basic vector space operations; that is to say, we rcqllin' f E 1.( r. lV). AI'I wit h our earlier Kysiems, Sill'll It funetion f is eltlled un isomo/'ph iSIll. The ('ollling t.heorem il'l of ('oJll'li
4-4

1,INt:AIt MAI'I'INUH

Theorem 4-30. If V(F) is a finite-dimensional vector space of dimension

n,

then V(F) is isomorphic to V,,(F). Proof. For Hakn of argument, let {XI, X2, ••• ,xn } be n basis for the space V(F). Then cnch veetor x E V has n uniqun repro8(Jntat.ion of the form

Since the n-tuple of scalars (a" a2, .•. , an) is uniquely determined by X (relative to the given basis), we may define a function f: V - V n by taking

=

f(x)

(a" a2, ... , an).

A routine (lnl(mlntioll eHtahlhdw8 that this mapping dTe(lts the required isomorphism. Corollary. Two fillil.n-dimnnHionnl vC(!tor HplweH V(/<') unci W(F) arc

morphic if and only if dim V

180-

= dim W.

Proof. If dim V = dim W = n, then each spaee is isomorphic to Vn(F), hence isomorphic t.o one another. Conversely, if V(F) ~ W(F) under the linear mapping f, Sylvester's Law yields

dim V

=

dim knr (f)

+ dimf(V) = dim {OJ + dim W =

dim W.

A remark in passing: The isomorphism between V(F) and V.. (F) was obtained by arbitrarily choosing one particular basis for V(F) in preference to all others. If we were to confine further study of finite-dimellsional vector spaces exclusively to the n-tuple spaces V.. (F), as the above theorem suggests, we would always be restricted to a prescribed basis. Since this would be contrary to our policy of giving, so far as possible, a basis-free treatment of vector spaces, we shnll for thn most part ignore Thl.'Orem 4-30. The corollnry 1.0 Thnorem 4-:JO ennblm~ 11M to establish qllotilllij. HPIU!()H nllIl (!omplmnelltnry HuhHpaC()H.

I~

relationship between

Theorem 4-31. Let U(P) and WW) be c:omplemnntnry Hubspaces relative

to th() finite-dimensional ve(,tor spac:e V(F); that ill, V (V /W)(F)

~

=

U(F).

Proof. We nlretuiy know from Theorem 4-26 that dim V Appealing to Theorem 4-27,

dim V/W

=

II 9 W. Then

dim V - dim W

Hence, the last corollary implies that (V /W)(F)

= ~

dim

=

dim U

+ dim W.

(T.

lI(F), as alleged.

Corallary. All subspaces eomplementary to a given subspace are isomorphic.

VEcroa SPACES

80 rar we have gathered certain infonnation concerning individual linear mappings, but have imposed -1)0 structure on the set L(V, W) itsele. This will now be remedied by showing that L(V, W) inherits a natural vector space structure from the underlying spaces V(F) and W(F). The sum 1+ g or two linear mappings I, g e L{V, W) is defined, as one might expect, by the rule (I g)(z) = I(z) g(z) for all z in Vj similarly, when c e F, the scalar product cJ is given by means or (cJ)(z) == cj(z). (We shall not bother to distinguish between the various uses of the sign, since the context will ordinarily suffice to make clear which spaces are involved.) With these operations, one can verify that L(V, W)(F) does indeed satisfy all the axioms for a vector space. What little difficulty there is arises in showing that L(V, W) is closed under the operations. But this poBe8 no real problem, for if z, y E V and a, b E F, the fact that I and g are themselves linear implies that

+

+

+

(f

+ g)(az + by) =

+ + + + + + + + + + + +

I(ax by) g(az by) = af(x) bl(y) ag(x) bg(y) = a(j(x) g(x») b(j(y) g(y») = a(f g)(z) b(1 g)(y),

whence 1+ 9 E L(V, W). An equally easy computation, which we omit, establishes that cl lies in L(V, W) for each scalar c. The point of all this is summarized by the following theorem. Theorem 4-32. Let V(F) and W(F) be vector spaces over the field F. With

addition and scalar mUltiplication defined in the usual way for mappings, L(V, W)(F) is itself a vector spo.cc. We can prove considerably more by taking the underlying spaces to be finite-
dim L(V, W) = dim V· dim W. Prool. Assume {Zh Z2, ..• , z,.} and {Yh Y2, ••• , y",} are bases for V(F) and W(F), respectively, 80 that dim V = n, dim W = m. Our strategy is to con-

struct a specific basis of nm elements for L(V, W)(F). According to Theorem 4-28, for each pair of integers (i, j), where 1 SiS n, 1 S j S m, there is a unique linear mapping hi: V -+ W satisfying the conditions !ii(X,,)

=

ai"Yi

if i '" k, = {o . . Yi

If

,== k.

The contention is that these nm functions serve as a basis for the space L(V, W).

4-4

LINEAR MA.PPINGS

First, let f be an arbitrary linear transfonnation from V(F) into W(F). For each index i (1 ~ i ~ n), fez,) lies in W, hence is a linear combination of the basis vectors 11" 112, ••• , 11"" say G'I

eF.

In this fashion, we produce a set of nm scalars Gil; the problem is now to IIhow that f can be represented as

Evaluating the right-hand side of this equation at the basis elements Zll leads to

'" GlliYi = =~

I(XII).

i-I

Since the linear mappings I and Ei.l (E:.. l t!.tj/ii) both have the same effect on the basis of V(F), they must be identical. From this, we conclude that the functions hi span the space T.I(V, W)(F). . To establish the linear independence of the hi> suppose some linear oombination of them is the zero mapping: . (Cij

E F).

If this expression is evaluated at the vector ZII, it follows, as above, that

However, {y" 112, ••• , 11",} is a basis of W(F), hence a linearly independent set, which fonles Clll = ... = CII". = O. By varying Ie, we conclude that all the coefficients Cij must be zero, and the theorem is proved in its entirety. Let us take a closer look at two special classes of linear mappings which occur frequent.Iy, namely L(V, V) and L(V, F). In view of our latest results, we aln!udy know that f.l(V, V)(F) is Il. Ve(ltor "pace over F, and if V(F) is of finite dimension n, then dim L(V, V) = n 2 • There is, however, more algebraic structure 011 L(V, V) than the vector space structure just described. By re-

286

VECTOlt

4-4

SPAC.~

HI riding the imllJl;l' "'PIWI', we arc ahle to introduce stilll~nother hinnry operation: composition of mappings. Indeed, if j, g E L(V, V), it is ensy to see that (f. g)(ax

+ by) =

j(g(ax

+ by»

+ bg(y» = aj(g(x» + bj(g(y» = a(f. g)(x) + b(f. g)(y) = j(ag(x)

for all x, y in V and all a, b in F. This establishes the linearity condition, thereby showing j. (I to be a member of L(V, V). TIll' haHi(· alg('braic properties of the sum and composition operations arc contained in the following theorem; we omit the proof, which consists of little more than eopying th(' r('sult.s of Example 3-6. Theorem 3-34. For each vector space V(F), the triple (L(V, V), ring with identity.

+,.)

IS

a

There is another important fact which is suggestcd by this discussion. Corollary. If GL(V) denotes the set of invertible mappings in L(V, V), thcn (GL(V), .) forms a group, called the general linear group.

Theorem 4-:J4 can he sharpencd considerably by limiting 01U' attention to finite-dimensional spaces.

+,.) is isomorphic to

Theorem 4-35. If dim V = n, then (L(V, V), (Mn(F), +,.), thc ring of n X n matrices over F.

for V(F). Wo daim thnt thc function to ench linear mapping its matrix reprcsentation relative to this particular hasis is 8. (ring) isomorphism of (L(V, V), +,.) onto (Mn(F), +, .). From our previous discussions, it is already known that the correspondence is one-to-one and onto M n(F); what remains to be checked here is whether preserves the ring operations. To begin with, suppose the mapsj, g E L(V, V) po&'lCSS representing matrices (aij) and (bj j ), respectively, so that

[)rooj. Fix n hasis (XI, X2, •••., J"n} <1>: L( V, V) -+ M n (F) whieh afI.'1igml

n

I(xj) =

n

2: ajjXi,

g(Xj) =

i_I

Then, for the sum j

2: iljjXi,

(j = 1,2, ... ,n).

i_I

+ (/, we have =

n

n

n

i ...... l

i-I

i==1

2: ajp:j + 2: bjjxj = 2: (ajj + bjj)xj.

It follows from what is meant by the matrix of a linear transformation relative

4-4

LINEAR MAPPINGS

to lL given hnllis that. (aii I- lI ij ) mUllt be the matrix ('orr('/iponding in cOIll;equencc, (f

+ g) =

(aij

+ bij) =

(a,j)

+ (b ii ) =

(f)

287

to! + g;

+ (g).

The problem of representing the composition fog is a little more complica.ted. First, we compute (f g) (Xj) : 0

(f g)(Xi) 0

=

=

f(g(xj»)

f ( t bkiXk)

=

k-I

t

bki!(Xk)

=

k-1

t

bkj(t aikXi).

k_l

i_I

By reversing the order of summation, this can bc expr(f. g)

=

(t

a,kbk j )

=

(a'i) . (b i ;) = (f) . (g),

k=1

implying that is a ring homomorphism from (L(V, V), +, 0) (111 n(f), +, .); this substantiates our contention and proves the theorem.

Ollto

A Uiseful cOJlsequcnce of this last th('orcm is that, if dim V = n, the genera.l linear group-now !\C'not.ed by (OL .. ( V), 0) ~- ill isomorphic to the group of all nOlIl-lingullU' n X n IIlntri(~(~s ovm' F. The proof of Theorem 4-3.5 brings out a point worth mentioning: The operations for mat.riees were purposely definr.d to agrC'r. with our operations for linear mappings. To be precise, Definition 4-8 was formulated in such a manner that the matrix of a product of two linear transformations would be the product of their representing matrices. When V(F) is of finite dimension n. there is another natural way of relating the algebraic structure of L(V, V) to that of AI n(F). Namely, since dim fAV, V)

=

112

= dim AI .. (F),

fAV, V)(F) mul AI .. (F) IIIUSt. also I}(' isolllorphi(' liS vedOl' splwell; in fact, a stmightJorw/1J"(1 ('alcuilltion shows the funet.ioll outaill(·d in Theorem ..-a!) to be a Vl'ctor space isomorphism, not only a ring iHolllorphism. The structure of L(V, V) can be approached f!"Om olle more direction. For this, let liS define what. is meallt hy all (assoeiativ(~) aJI(·Jwa over a field.

Deftnition 4-19. A vee tor SPIU'!' V(F) i:,llittid t.o he lUI alyeill"a ovel' tltefielrl F if its elements can be multiplied in Iml'h It way that I'(F) becomes a ring in

VEC'rOa SPACES

which scalar multiplication is related to ring mUltiplication (denoted by .) by the following mixed a88ociatif1e law: e(% . y)

=

(C%)' Y =

%.

(cy)

(%,yeV;eeF).

Our immediate goal is to establish that, for any vector space V{F), the ring (L(V, V), 0) has the structure of an algebra over F. In light of earlier results, all that really need be demonstrated here is the mixed assoeiative law. To obtain this, we let e e F,f e L(V, V), and % E V; a straightforward calculation then yields

+,

(j. (eo»(x)

= f«eo)(x» = f(cg(%» =

cj(g(%})

=

e(f g)(%). 0

Since this equality holds for every x E V, we conclude that

f

0

(cg)

=

e(J. g).

Verification of the property (ef) • g = e(J • g) follows in a similar vein and is left as an exercise. I··urther examples of rings which are also algebras are the ring (M ,,(F), -t, .) of matrices of order n, the ring of real-valued functions on as well as the real and complex number fields. As the reader is 80 well aware, since an algebra is basically a rinI and vector space combined, any homomorphism between aIiebras must preserve both the ring and vector space operations; more specifically, if f is a homomorphism between two algebras over the same field, then

n',

f(x

+ y) = f(x) + f(y) ,

f(% • y)

= f(x) • fey),

f(C%)

=

ef(x).

The following example affords a good illustration of this idea. Let T 2( n') denote the set of all 2 X 2 upper triangular matrices • with elements from

Example 4-30.

n':

It iM already known that T 2 (R') is the HOt of clements of a noncommutative ring with identity [Problem 13(b), Section 4-1] as well as of a vector space over the real numbers [Problem 7(f), Section 4-2]; in other words, (T 2 (R'), +,.) may be regarded as an algebra over R'. Our purpose here is to obtain a complete description of the homomorphisms from this algebra into the real numbers. AH 1\ Mimplifying devi(~e, note that upon setting

_-(0o 0) ,

y-

1

4-4

LINEAR MAPPINGS

289

each element of T,(R') may be expressed in the form

(: :) =

al + bX + (c - a)Y.

Now, if I is any nontrivial homomorphism from the algebra (T 2 (R'), +,.) into the algebra (R', +.. ), then

1(: :) =

ai(l)

+ bl(X) + (c -

a)/(y).

Since I(T) = I(T) . l{l) for all T E T,(R'), and I is not identically zero, we must have 1(1) = 1. What can be said about the functional value I(Y)? Observe first that the matrix Y is idempotent with respect to multiplication; this means I(Y) = I(Y') = I(Y)', which in tum implies I( Y) can only assume the values 0 or 1 (Problem 2, Section 3-3). Finally, the fact that the matrix X is nilpotent (specmc&lly, X 2 = 0) leads us to conclude I(X) = O. The effect of these remarks is that

1(: :)

= a or c.

This suggests consideration of mappings /;: T 2 (R') --. R' (i by

=

1,2) defined

A routine check establishes that I. and It actually are algebra homomorphisms from (T 2 (R'), +,.) into (R', +, .); barring the trivial homomorphism, our argument shows them to be the only such functions. Now is perhaps an appropriate point at which to say a brief word about

r-

idempotent linear mappinglJ, that is, functions IE L(V, V) such that I (hore /2 IltallUIi (or / • f). Our object ill to show the intimate connection be-

tween transformations of this type and direct sum decompositions of the vector space. The precise statement follows. 4-36. If V = U $ W, then the function/w: V - W which assigns to each vector x E V its uniquely determined component X2 in the representation x = Xl + X2(Xl E U, X2 E W) is an idempotent linear mapping with

Theorem

ker (fw) = U,

Iw(V)

=

W.

Conversely, every idempotent linear mapping IE L(V, V) defines a direct

290

VECTOU SPACES

sum decomposition

v=

ker (f) C!> f(V).

Proof. We start hy showing that the function fw, as defined in the statement of the theorem, is actually in [,(V, V). Suppose x, Y E V, so that x = UI + Wit Y = u21· '/1'2, with 14" E P, Wt E W (k = 1,2). Theil for scalars a, b,

wh('rf' au, -\ bU2 E U, alOI

+ bW2 E W,

This mealls that

fw(a;r I hy) = aWl I bW2

~

ajw(x) I IJfw(Y),

hell(,l~ fw E L(V, V). In fad, morc is true: Since U = 14 + 0 and W = 0 + W serve as the uniquc dncompo!litiolls for elements 14 E 1rand wE W, we must htwe 1) fw(u) = 0 if alld only if 14 E P, 2) fw(w) = W if and only if til E W.

This leads to the mo!:!t significant property of the function fw, its idem potency ; given x = It -+ w, with U E U and W E W, . fa,{x)

= fw(Jw{x» = fw{w) =

W

= !w{x),

resulting in If., = fw. For the eOnVI!J'IIe, suppose that f E /,( V, V) with P = f, Taking U = x - f(x) and 10 = f(x), we may write each vector x E V as a sum x = U w. By hypothesis I is an idempotent map, so

+

feu)

=

f(x) - f2(X) = 0,

while few) = f2(x)

This shows that V = ker (f)

W

=

f{x)

=

10.

+ W, where the set W is given by =

{x E V I/(x)

=

x}.

Our cont.ent.ion is that the foregoing l-1um is direet; in other words, ker (I) n W

=

{o}.

Thi!:! is easy: If a vector x lies in ker (f),f(x) = 0, whercas if x is in W,f(x) = x. Thus, whenever x belongs to both ker (f) nnd W, we must have x = O. To ('Oll('hlll" 1.1t" proof, it rl'mnini'l to h" 1-11'1'11 that, lV = f(v), The in('lll!lioll JV k f( il'lllli imm(~(lintc COnSl''1I1CIH'(' of the definition of the set W; Uw r(!VI~rs(! ineiullion il'l ('(lluLlly ohvioul-1, !lilw(' til(' (',!uutioll f2 = j Jl\elU11'I f(x) E W for every x E V,

n

4-4

LINEAR MAPPINGS

291

It is natural to think of the function fw as projecting V(F) onto the subspace For thiN roUI!()Il, the term 111'ojectiCln iH (mstomurily used to refer to any idempotellt lilll'nr nlllppillg 011 V(F). Sirwe every Huhspaee of a vector space f>OHHesscs a (~()llIplemcllt./lry Hubspnec, Th(lor('m 4-36 asscrts the exitltence of a proj(·etion of t.h(l foIJlIL(~(! ont.o any of it.1I IIl1hHpaccs; IwmlleHH to MY, (:omplementtl Iir(l not Illliq\l(~, 110 t.henl mlLy (lxiHt. s('veml proj('(~t.ionR onto the same 8ubspru:c. For a simple CMe of the above idea, considcr the projection f: V. -+ V 4 defined by W(F).

It ill fnirly (!vid('Ilt. tlmt

=

ker (f)

{(O, 0, aa, a.) I a3, a. E F}

and

The reader may verify that t.he corresponding sllbspaces are actually complementary. Throughout the remainder of this section, we restrict our attention to linear mappings on V (F) which assume values in the associated scalar field F [if scalar multiplication is defined to be the multiplication in F, then F may be viewed as /I. ono-dim(lnsional Ve(ltor space over itself]. It is customary to use a special terminology for such scalar-valued functions and to refer to them 8.B linear functionals, or merely functionals, on V(F). Definition 4-20. The vector spaee L( V, F)(F) eonsisting of all linear funetionals on V(F) is called the dual space (often called the conjugate space) of V(F) and is denoted by V*(F). Example 4-31. Let (a;j) be an n X n matrix. The sum of all elements on the main diagonal is known as the trace of the matrix and is represented by tr (aij): n

tr (aij)

=

1: au· k_1

It is not difficult to verify that the function tr: M n -+ F, defined in this way, is a linenr functional 011 the spnce 111n(F); for, if (aij) , (b ij) are two matrices of order n alld r, 8 E F, then

t.r [r(aij)

+- s(b ij) J =

tr (raij

+ sb ij) =

n

1: (rau + sbu ) k_1

=

r

t

k-l

au

+s

t

k_1

bu

= r tr (ai;) + 8 tr (b i ;).

292

VECTOR SPACES

From Theorem 4-33 we already know something about the space V*(F) j namely, that if V(F) is finite-:dimensional, then dim V* = dim V. Moreover, the proof of this theorem provides an explicit way of constructing a basis of V*(F): Given a particular basis {Xl. X2,' •• ,x..} for V(F), a unique linear functional h is defined on V by prescribing (j

=

1,2, ... ,n).

Theorem 4-33 then tells us that the n functions It, f 2, ••• , f" form a basis for V*(F), the so-called dual bC&8ie to XIo X2, ••• , x.. or dual basis of V*(F). The definition of h is somewhat opaque, but it may be interpreted in a way that is more illuminating. Since any vector x E V is of the form x = alXl + aJX2 + ... + a"x", it is clear that

hex) =

.

.

L a;!;(x;) = ;-1 L a;6i; = ai· ;-1

From this, we see that the functional h has the effect of assigning to each vector x the coefficient of Xi in the representation of x as a linear combination of the basis vectors. A suggestive name for the mappings f, might be to call them coordinat.e functional8 (with resPect to a fixed basis, of course). An interesting observation which we shall use almost immec!jately is the following. Lemma. If V(F) is a finite-dimensional vector space and x is any nonzero vector of V, then there is some linear functional f e V* for which f(x) po!! O.

Proof. Since x P&!! 0, there exists a basis {Xl, X2, ••• , x..} of V(F) with Xl = X (Theorem 4-24). tf {fir f2, ..• ,f.. } is the corresponding dual basis for V*(F), we then have

Since the dual space V*(F) is a vector space, V*(F) has a dual of its own; we shall denote this latter space by V**(F) and refer to it as the second dual of V(F). If V(F) is of finite dimension n, we know that V*(F) has the same dimension, hence in tum dim V** = n. By the corollary to Theorem 4-30, this equality of dimensions implies that V(F) and V**(F) are isomorphic. There is one perfectly natural and useful isomorphism between these spaces, the so-called canonical imbedding j the full story is told below. Given a vector x e V and functional f e V*, f(x) is a scalar. Although we have grown accustomed to thinking of this as a function of x for fixed f, let us now invert this usual practice and allow f to range over V*, while holding x fixed [many authors would emphasize this by writing x(f) in place of f(x)]. Specifically, define the function T",: V* --+ F by T",(f)

=

f(x),

fe

V*.

LINEAR MAPPINGS

293

It is easy to see that T., is actually a linear functional on V·, for T.,(af

+ bg) =

(af

+ bg)(x)

= af(x) + bg(x) =

aT.,(f)

+ bT.,(g).

T '" defined in this way, is called the evaluation functional induced by the vector x.

The choice of imbedding should now be clear, for what is more natural than to &88OCiate x with F ". Precisely: Theorem 4-37. If V(F) is a finite-dimensional vector space, then V(F) V··(F) via the mapping 4J(x)

=

~

T.,.

ProoJ. At the outset, note that for any J E V·, we have T",,+bll(f)

= J(ax + by)

= aJ(x) + bf(y) = aT,,(J) + bTII(J) =

(aT"

+ bTII)(f),

whence the relation T ""+"11 = aT., + bTII' This leads directly to the conclusion that the function 4J is linear: . 4J(ax +by)

=

Tu+" = aT.,

+ bTII = a4J(z) + b4J(y).

The one-to-one nature of 4J is established by showing ker (f) = {O} or, in the present context, if J(z) = 0 for every f in V·, then z = OJ but this is precisely the content of the previous lemma. Finally, the equality of the dimensions of V(F) and V**(F), together with the fact that 4J is a one-to-one function, necessitates that 4J map onto V*· (see the corollary to Theorem 4-29). The theorem is now fully proved. The import of Theorem 4-37 lies in the following rather remarkable corollary which asserts that all functionals on V· can be obtained by evaluation at elements of V. Corollary. If V(F) is finite-dimensional, then each linear functional in V·· is of the form T., for some unique vector x E V.

In practice, one usually identifies the vector z with the fun~tional T., defined by it. When dim V is fillite, we thereby abolish V*·(F) and regard V(F) as the space of linear functions on V·. From this point of view, V(F) and V*(F) have a natural symmetry and we may justifiably speak of them as being dual spaces (to each other). In the infinite-dimensional case, however, it is not true that the set V" is exhausted by the functionals T.,j our mapping 4J is into, not onto, so that V(F) can only be regarded as a vector subspace of V**(F). As a parting shot, we give further evidence of the symmetric relationship between V(F) and V*(F) by showing that every basis for V*(F) is the dual of a basis for V(F).

294

VECTOR SPACES

Theorem 4-38. If V(F) is a finite-dimensional vector space, then any basis for V*(F) is the dual of some basis for V(F).

Prool. Let {!J,/2,"" In} be a basis for V*(F). We can then find a basis {T., T 2 , ••• , Tn} in V**(F) which is dual to this givcn basis; in other words, T i (!;) = 6ij. But, according to the preceding corollary, there exist vectors x., X2, • •• , Xn in V with Ti(f) = T.if) = I(Xi) for all I E V*; in particular,

taking I

= Ii. we obtain

(i, j

whencc

{XII X2, ••• ,

=

1, 2, ... , n),

xn} is a basis for V(F) having {fl, 12,

••• , In}

as its dual.

PROBLEMS

Unless indicatl'd otherwise, VW) and Jl"W) denote vector spaees over an arbitrary field P. 1. Using nothing other than Definition 4-18 and Theorem 4-7, show that if the mapping/E ,,(V, l\"), then/CO) .. 0 andf(-x) .. -/(x) for each x E V.

2. Determine whieh of the following functions are linear mappings of V s(R') into

itself: a) I(al, a2, a3) = (a2, -aI, a3), b) I(al, a2, a3) = (ai, 0, as), c) I(al, 42, as) = (ai, al + a2, al a2 as), d) I{al, a2, as) = (al 2a2 - a3, -al a2, 2al a2). 3. Assume that IE l.. (lr, Jr) and U(F) is a vector subspace of lr(F). Prove that 1-1(U)(F) ill a subspace of V(I"), where, as usual,

+ + +

+

+

l-l(U) = {xEVl/tX)E U}. 4. Let I be a linear transformation from V(P) into JrW). If (j(Xl),!(X2), .•• .f(x,,)}· is a linearly independf'nt loubset of W(Jl), prove that the set {Xl, X2, ••• ,x.) is also independl'nt. From this, cl(',hwe that dim/(li) ~ dim V.

+

+ ... +

+ +

+ ... + + ... +

a2x 2 a"x" E R'[x), the vector space of polynomials in x over R'. Determine which of the funetions below are linear mappings of R'[x) into itself: a) I(p) = aoz alx 2 a2xs a"x"+! I b) I{p) = ao 41X2 a2x· a"x2 .. I

5. Let p(x) .., ao -I- alx

+

+

+ ... + n ~ xn+! 2 3 +l' 6. Let the mapping I E 1.. ( V, V) and S denote the set of vectors of V which are left fixed by I: S = {x E V I/{x) "" x}. c) 1(P) '" aoz+ al x 2 + a2 x s

Verify that S(F) forms a subspace of the vector space V(F).

4-4

1,IIIIEAIt MAPPINGS

295

7. Prove that if f E L( V. Jr). then I is a one-to-one function if and only if the vectors l(xl). I(X2) • •.. • !(x ure linearly independent whenever XI. X2 • •••• x.. are linearly indl'pendent. R. Show hy exn.mple that the concluHion of the corollary to Theorem 4-29 is false if V (P) iH infinite-dimensional. A )

9. Suppose the mapping f E /,( V.1\"). with dim V a nonzero vector Xo E V for which I(xo) = o.

>

dim It". Show that there exists

10. Obtain the Fundaml"ntal Homomorphism Theorem for Vector Spacl's: If I is a linear mapping from the vector space V(F) onto the vector space W(F). then (V /ker (f»(F) ~ W(F). ) 1. Let V(F) be finite-dimensional with basis {XI. X2 • •.•• x ..}. and let VI. Y2 • .••• y .. be any n pll'IIl1'nt,H of V. If thtl function!: V -+ V i~ defined by taking !(nIXI

1 ... -I a.x ..) -

alYI

+ ...

I· a.y ...

prove that f is ~ linear mapping; determine when f will be an isomorphism. 12. Prove that if the mapping f E 1.( V. V) is such that ker (j) = ker (p). then V = ker (j) E9 f(V). 13. For a fixed element a E F. define the scalar tran8formation f.: V --+ V by

f ..(x) = ax, Given that R = {fa I a E F} and R' = R -

xE V.

{O}. show that

+,

+, .)

a) the tripl!! (R, 0) form~ a !!ubring of (J,(V, V), illomorphic to W. +.. ); b) the pair (R', .) is a normal subgroup of the linear group (GL(V), .); in fact, R' = cent GL(V). 14. A linear mapJlingfE !.(V, V) is said to be nilpotent if f" = 0 for some nEZ+. If fill nilputent anti if r-J(Xn) ;&! 0, prove that {xu, f(xo), P(xo), •.• , r-l(xo)}

is a linearly independent set of vectors. 15. Prove that t.he reRIlIt. of Theorem 4-31 holdH in gent'ral; in other word!!, if V = 11 E9 1\', I'Kt,ILhliKh t.hat, (V /1\') (f') ~ U W) regardll'K!I of the dinlllDsion (finite or infinitl') of V(P). [/lint: Consider the restriction nat", I V.l 16. Con Rider the trace functional tr as defined in Example 4-20 of the text. In the vect.or space Mn(R'), prove that ' a) tr «a;j) . (b;j» = tr «b;j) . (a;;», b) tr «a;j) . (b;j) . (a;j) -I) = tr (b;;), c) tr «a;j) . (ai;)I) = 0 if and only if 17. SUPpoRe {Xl, X2, ••• , x.l iM a basig {fl, /2, . , . , I,,} is the corresponding vector x E V,

x

whenever (a;j) is nonsingular, (aij) = O.

of the finite-dimensional space V(F) and dual basis oC V*(F). Show that for each

.

=

~ 1.(X)Xi, k_l

296

VECTOR SPACES

while for each functional I E V*,

. I

=

. :E l(x.)/•. • -1

18. Let V(F) be a finite-dimensional vector space over F. Prove that a) if Xl, X2 E V with Xl '" X2, then there exists a linear funotional IE V* for which l(xI) '" l(x2); b) if W(F) is a proper nonzero subspace of V(F) and the vector Xo e: w, then there is 80me/E V* such that/(xo) - 1,/(x) - 0 for all x E W. (Hi"': Given any baais {XI, ••. ,x..} of W(F), the set xo, Xl, ••• ,x.. ia linearly independent, hence contained in a baais for V(P); now, utilize the corresponding dual baais for V*(F).) 19. If W(F) is a subspace of the vector space V(F), the annihilator of lV is the set W.1 defined by lV.1 z: {f E V* I/(x) = 0 for all X E lV}. Assuming V(F) is finite-
+

0) (V /W)*(F) ~ W.l(F), (V /lr.l) (F) ~ W*(F); d) WH ... W. (Hint: Use part (b).)

20. Let U(F) and W(F) be two subspaces of the finite-dimensional space V(F). Establish the following facts concerning the annihilators of U and W:

a) U.1 - W.l if and only if U - W. b) (U + W).l ... U.l n W.l; (U n W).l '" U.1 + Jr.l. c) If V - U lV, then V* .. UJ. lVJ..

e

21. For each linear mapping IE L(V, function F: W* -+ V* defined by

e lh,

the tranaposll (adjoint, dual) . of I is the gE W*.

Given that V(F) and 11'(F} are finite-dimensional, show that a) is a linear mapping from 11'*lF) into V*(F), b) ker (fT) - 1(v)J.; ker (f)J. - IT(W*), c) L(V, lV) ~ L(W*, V*) under the mapping that sends each functional IE L(V, 11') to its transpose IT. 22. Let {Xl, X2, ••• , X ..} be a basis for the finite-dimensional vector space V(F), and let {fl; 12, ... ,I..} be the corresponding dual basis for V*(F). Suppose that (ail) is the representing matrix, relative to {Xl, X2, ••• , x ..}. of the linear mapping IE L(V. V). Prove that the transpose IT of f is represented by the matrix (ail)' relative to {fl, h • ... , I,,}. 23. If the functionals f,g E V* are such that ker U) !;; ker (g). prove that there exists a scalar a for which f = ago

r

Selected R.ferenc••

Our purpose here is to present a list of suggestioDil for collateral reading and further study. Those works classified under General References roughly parallel the content of this book; the specialized sources more fully develop topics mentioned in the text and will carry the reader considerably beyond his present knowledge.

Gen_IR......nc •• . 1. A. A. ALBERT, Fundamental Concepti of Higher Algebra. Chicago: The University of Chicago Preas, 1956. 2. W. BARNES, Introduction to Abstract Algebra. Boston: Heath, 1963. 3. R. BEAUMONT and R. BALL, Introduction to Modern Algebra and Matrix ThMry. New York: Holt, Rinehart and WiDilton, 1961. 4. G. BIRltH()FF and S. MACLANE, A Survey of Modern Algebra, 3rd ed. New York: Macmillan, 1965. 5. R. DEAN, Elements of Abstract Algebra. New York: Wiley, 1965. 6. R. DUBlscH, Introduction. to AbBtract Algebra. New York: Wiley, 1965. 7. I. N. HERSTEIN, Topic' in Algebra. New York: Blaisdell, 1964. 8. N. JACOBSON, Lectu.ret in Abstract Algebra, Vol. I, Basic Concepts. Princeton: Van Nostrand, 1951. 9. S. LANG, Algebra. Reading, Mass.: Addison-Wesley, 1965. 10. D. J. LEWIS, Introduction to Abstract Algebra. New York: Harper and Row, 1965. 11. N. McCoy, Introduction to lIlodern Algebra. Boston: Allyn and Bacon, 1962. 12. G. MosTOw, J. SAMPSON, and J. P. MEYER, Fundamental StrUcturet of AlgebrlJ. New York: McGraw-Hill, 1963. 13. H. PALEY and P. WEICHSEL, A First Cou.ru in Abstract Algebra. New York: Holt, Rinehart and Winston, 1966. 14. R. E. JOHNSON, University Algebra. Englewood Cliffs, N. J.: Prentice-Hall, 1966. 15. S. WARNER, Modern .·llgebra, 2 vols. Englewood Cliffs, N. J.: Prentice-Hall, 1965. 16. J. E. WHITESITT, Principles of Modern AlgebrlJ. Reading, Mass.: Addison-Wesley, 1964. 17. O. ZARISKI and P. SAMUEL, Commutative Algebra, Vol. I. Princeton: Van Nostrand, 1958. 297

298

SELEcrED REFERENCES

Group Th.ory

18. C. CURTIS and I. REINJo:R, Repre8entation Theory oJ Finite GrOUp8 and Associative Algebras. New York: Interscience, 1962. 19. M. HALL, The Theory

0/ Groups.

New York: Macmillan, 1959.

20. A. KUROSH, The Theory oj Groups, 2nd ed. New York: Chelsea, 1960. 21. W. LED.:RMANN, Introduction to the Theory 0/ Finite Groups, 5th ed. New York: lnterscience, 1964. 22. J. ROTMAN, The Theory o/Groups: An Introduction. Boston: Allyn and Bacon, 1965. 23. K SCHENKMAN, Group Theory. Princeton: Van Nostrand, 1966. 24. W. !;COTT, Group Theory. l<:nglewood Cliffs, N. J.: Prentice-Hall, 1964. 25. H. ZAS8ENHAUfI, The Theory oJ Groups, 2nd ed. New York: Chelsea, 1958. Ring. ond FI.ld.

26. I. ADAMSON, Introduction to Field Theory. New York: Interscience, 1964. 27. E. ARTIN, Galois Theory, 2nd ed. Notre Dame, Ind.: University of Notre Dame Press, 1955. 28. N. McCoy, Rings and Ideals. Buffalo: Mathematical Association of America, 1948. 29. N. McCoy, The Theory

0/ Ring•.

New York: Macmillan, 1964.

30. D. NORTHCOTT, Ideal Theory. Cambridge, England: Cambridge University Press, 1953. 31. A. ROBINSON, Numbers and Ideal•. San Francisco: Holden-Day, 19.§5. UnoarAlg.bra

32. C. CURTlli, Linear Algebra: An Introductory Approach. Boston: Allyn and Bacon, 1963. 33. P. HALMOS, Finite Dimensional Vector Spaces, 2nd ed. Princeton: Van Nostrand, lUSH. 34. K. HOFFMAN and R. KUNZE, Linear Algebra. Englewood Clifftl, N. J.: PrenticeHall, 1961. 35. S. LANG, Linear .-1lgebra. Reading, Mass.: Addison-Wesley, 1966. 36. D. RAIKOV, Vector Spaces. Groningen: P. Noordhoff, Ltd., 1965. 37. F. STF.WART, Introduction to Unear Algebra. Princeton: Van Nostrand, 1963.

s.. Th.ory and Function. 38. P. HALMOS, Naive S/'t Theory. Prim'cton: Van Nostrand, 1960. 39. N. HAMILTON and J. LANDIN, Set Theory: The Structure oJ Arithmetic. Allyn and Bacon, 196!. 40. R. STOLL, Set Theory and Logic. San Francisco: Freeman, 1963. 41. P. SUPPES, ,·txiomatic Set Theory. Princeton: Van Nostrand, 1960.

Boston:

Index of Special Symbol. and Notation.

In the list below, numbers refer to the page on which the symbol is first defined or used. A ~ B, :t C B Sl't ..t is im'luoeo (properly ineluoerl) in 8t't R, 2 .1 U /I IInilln .. f 111'1>1 .1 RIIII n, :J ..t () B intersection of sets A and B, 3 A - B relative complement of set B in set A, 3 .-1 B sum of !lets .1 and B, 161 .-1 A B symmetric difference of sets 1t and B, 36 A X B Cartesian product of sets A and B, 8 -A complement of the set A, 3 {a} singleton a (set consisting of the element a), 2 (a] equivalence class determined by the element a, 9 a E A, a ~ .1 element a belongs (does not belong) to the set A, 1 a- b element a is equivalent to element b, 9 alb, atb a divides (does not divide) b, where a, b are integers, 20 a == b (moo n) integer a ill congruent to integer b modulo n, 52 gcd (a, b) greateKt common divisor of the inttlgers a and b, 21 lcm (a, b) least common multiple of the integers a and b, 26 (a, b) ordered pair of a and b, 6 [)" n, domain ami range of the fun(ltioll I, 13 I( A) direct image of the set A under the function I, 17 I-I (A) invf'rl!f1 image of the set A under the function I, 17 l inverse of the function I, 16 II A restriction of the function I to the set A, 17 /. g composition of the functions I and g, 15 iA ident,ity funetion on the ACt .-\, 17 P( A) power set of the fI(.t .-1, 5 " I'mpty Ill.t, 2 Q set of rational number:!, 1 R'-+ set of real (positive real) numbers, 1 U universal set, 2 Z ACt of in teg('rs, 1 Z., Z., Z+ sets of even, odd, and positive integers, 10 Z.. . set of integers modulo n, 55 addition (multiplication) modulo n, 56, 146

+

r

..

n',

+., ',.

[a,

bJ

a.1/

Notation. from Group Theory a. b • a-I. b- I , 85 I'IISI'\.

flf til(' >III hp;roll II (/[, .), 7Ii 2tJ9

300

INDEX OF SPECIAL SYMBOLS AND NOTATIONS

«a), .) A(G)

(A", .) C(a)

cent.

0

8

f. FQ (O/H, 0) (0 X 0 ' ,') ([0,0], .) [G:H)

H.K homG

1(0) ker (f) nat.H N,,(H) N(H) 0(0) (f.

(sym 0, .) (S", .)

(Z".+,,) ~

cyclic subgroup generated by the element a, 69 set of automorphill.lD8 of the group (G, .), 91 alternating group on "symbols, 126 centralizer of the element 4 in 0,74 center of the group (0, .), 66 identity element of the group, 32 multiplication or translation function induced by 4, 50 set of multiplication functions on 0, 50 quotient group of (0, .) by the subgroup (H, .), 84 direct. product group, 52 commutator subgroup of the group (0 • • ), 86 index of a subgroup {H, .} in (G, .), 129 product of scull and K, 71 set of homomorphislD8 of (G, .) into itself, 90 set of inner automorphislD8 of the group (G, .), 108 kernel of a homomorphism f, 94 natural mapping of 0 onto G/ H. 96 normalizer of H in K, 129 normalizer of H in the group (G, .), 88 order of t.he group (0, .), 129 inner automorphism induced by a, 108 symmetric group of the set 0, 62 8ymmctric group on " Iymbols, 69 group of integers modulo '" 56 is isomorphic to, 97 N...... on. '!om II... Theory

«a),+,') ann I

(C.+, .) degf(z)

a

liEf) 12

11·12 B[z)

(B·, .)

(B/I.+ • .) radB ZI

(Z".+ ..,·,,)

principal ideal generated by thc element a, 158 annibilator of the set I, 169 complex number field, 173 degree of the polynomialf(z), 199 derivative function, 218 direct sum of ideals, 171 product of ideals, 171 set of polynomials over B in the indeterminant. z, 198 group of invertible elements of the ring (B, 143 quotient ring of B by the ideal I, 160 radical of the ring (B. 191 set of integral multiples of the identity element, 153 ring of integers modulo n. 146

+, .),

+..),

Notation. from Vector Spaco Th. .ry (a.) (/Iii) (aij) ,

diag M"W) dim V 8il

n-dimensional vector, 235 m X n matrix. 238

transpose of the matrix (aij), 248 set of ·diagonal matrices of order n over F, 248 dimension of the vector III)8Ce VW), 271 the Kronecker delta, 241

INDJilX OF SPECIAL SYKBOLS AND NOTATIONS

IT I"

(GL(V),o)

I"

L(V, W) M",,,(F) ltl,,(F)

M:(F)

o tr (au) T.(F)

T. UEa W V*(F) (V/W)(F) V..(F) V.(F)

Wl. [%1, ••• , %,,1

transpose (adjoint) of the linear mapping I, 296 . projection onto the subspace W(F), 289 general linear group, 286 identity matrix of order ", 241 set of linear mappings from V(F) into W(F), '1:17 set of (or vector space of) m X,. matrices over F, 239 set of (or vector space of) square matrices of order,. over F, 241 set of nonsingular matrices of order,. over F, 244lero vector or matrix, 236 trace of the matrix (0.'/),291 set of upper triangular matrices of order,. over F, 248 evaluation functional induced by the element %, 292 direct sum of the subspaces U(F) and W(F), 259 dual space of the vector space V(F), 291 quotient (vector) space of V by W, 261 set of (or vector space of) rHlimensional vectors over F, 236 vec\Or space of infinite sequences over F, 251 annihilator of the subspace W(F), 296 linear span of the vectors %1, %2, ••• , %",256

301

302

INDEX OF SPECIAL SYMBOLS AND NOTATIONS

The following chart should help the reader to visualize the interrelations among the various algebraic IlYfltcmll considered in the text.

Commutative _i· groups

Vector spaces

Division rings

Algebras

INDEX

INDEX

abelian (commutative) group, 35 abelianization of a group, 87 absorption law, 11 addition modulo n, 56 additive group of a ring, 142 adjoint transformation, 296 algebra, associative, 287 Boolean, 224 fundamental theorem of, 204 of linear mappings, 288 alternating group, 126 annihilator ideal, 169 annihilator of a subspace, 296 arithmetic, fundamental theorem of, 23 associative law, 30 automorphism, 91 inner, lOS axioms, for a 800lean algebra, 224 for a groUI), 35 for a ring, 142 for a vector space, 249 basis,268 theorem on existence, 270 binary operation, 27 Boolean alpbra, 224 Boolean ideal, 234 Boolean homomorphism, 234 Boolean ring, 220 cancellation law, 45 Cartesian I)roduct, 6 Cauchy's theorem, 133 Cayley's theorem, 101 center, of a group, 66 of a ring, 168 centralizer, 74, 131

chain, for a group, 117 of sets, 6 characteristic of a ring, 152 class,congruence,54 conjugacy, 132 equivalence, 9 class equation, 132 closure condition, 28 . commutative group, 35 commutative law, 32 commutative ring, 143 commutative semigroup, 32 commutator, of two elements, 85 subgroup, 86 commuting elements, 32 complement, of an element, 224 relative, 3 of a set, 3 complementary subspaces, 259 . complex numbers, 173, 211 compoeite number, 21 composition chain, 118 compoeition of functions, 15 congruence, modulo a subgroup, 73 modulo n, 52 conjugate elements, 115, 131 conjugate subgroup, 129 conjugacy class, 132 constant polynomial, 198 coordinate functionals, 292 coset, 75 cycle, 60 cyclic group, 69 degree of a polynomial, 199 DeMorgan's rules, 5 dependent set of vectors, 263 305

306

INDEX

derivative (differentiation) function, 218,278 derived Kroll", lUI

prime, 180 of quotients, 179 ""Jitting, 2] 2

dial(onal mal.rix, 241'1 difference, of ring clement!!, 14~ of sets, 3 dimension of a vector space, 271 direct product, 52, 155 direct sum, 171,259 disjoint sets, 3 di8joint sub8paces, 258 distributive law, 141 division algorithm, for integers, 20 for polynomials, 200 division ring, 182 divisor of zero, 148 domain of a function, 13 dual basis, 292 dual space, 291 duality, principle of, 225

linit.I,ly KI'III'ral.I·t\ vIm tor Hpal!ll, 25f\ fixed points of a homomorphism, 102, 170, 294 function, 13 domain, range, 13 equality of, 14 extension,restrietion, 17 functional value (image), 13 identity, 17 into, onto, 13, ]4 one-to-one, 16 operations for, 14, 15 functional, linear, 291 fundamental homomorphism theorellUl, 106, 166, 295 fundamental theorem of algebra, 204 fundamental theorem of arithmetic, 23

Eisenstein criterion, 219 element, conjugate, 131 idempotent, 45 identity, 33 inverse, 34 nilpotent, 155 order of, 70 prime, 190 empty (null) set, 2 endomorphism, 90 equality, of functions, 14 of sets, 2 equivalence class, 9 equivalence relation, 9 equivalent composition chains, 120 Euclid's lemma, 22 evaluation functional, 293 even permutation, 126 extension field, 207 extension of a function, 17

general linear group, 286 generating set, 68, 158, 255 generator, of a eyclic group,-69 ol a principal ideal, 158 greatest common divisor, 21, 217 group, 35 alternating, 126 center of, 66 commutative, 35 cyclic, 69 finite, 70 general linear, 286 of inner automorphisms, 108 of integers modulo n, 56 of invertible elements, 143 Klein four-, 98 order of, 70 'P-, 130 quotient, 84 simple, 81 solvable, 123 symmetric, 59, 62 of symmetries of the square, 49

factor theorem, 104 family of sets, 5 Fermat's little theorem, 219 field, 172 extension, 207 finite, 183

homomorphism, Boolean algebra, 234 group, 89 kernel of, 94, 163

INDEX

ring, 161 trivial,90, 161 Vl'ctor HI"~"'" 277 hUIIUIIUClrphiHIII l,hlllll'UIII, fllr groull", 1011 for ringR, 166 for vector spaces, 295 ideal,156 annihilator, 169 generated by a set, 158 imbedding in a maximal ideal, 185 maximal, 184 minimal, 194 nil, 195 primary, 195 prime, 188 principal, 158 proper, 155 trivial, 155 idempotent element, 45 identity, ring with, 143 semigroup with, 83 identity element, 33 identity function, 17 identity matrix, 241 image, direct, 18 inverse, 18 imbedding, 164 inclusion (injection) map. 17 independence, linear, 263 indeterminant, 198 index set, 6 index of a subgroup, 79 inner automorphism, 108 inner product, 236 integers, group of, 56 modulo n, 55 modulo a prime, 174 ring of, 146 integral domain, 149 integral powers of an elemcnt, 48 intersection, of ideals, 157 of sets, 3 of subgroups, 66 of subs paces, 255 inverse element, 34 inverse mallping, 16 irreducible polynomial, 205

307

iaomorllhiRm, of groups, 97 of rings, 164 of veetor RPM-CK, 282 Jordan-Holder theorem, 121 kernel of a homomorphism, 94, 163 Klein four-group, 98 Kronecker delta, 241 Kronecker's theorem, 208 Lagrange's theor~m, 79 leading coefficient, 199 least common multiple, 26, 159 left coset, 75 left distributive law, 141 left ideal, 157 left multiplication function, 50, 170 length, of a chain, 117 of a cycle, 60 linear combination, 255 linear dependence (independence), 263 linear functional, 291 linear mapping, 277 linear polynomial, 205 linear space, 249 linear sum, 254 local ring, 196 mapping; Bee function mathematical system, 29 matrix,237 diagonal,241 dimensions, 237 entry (element) of, 237 identity, 241 nonsingular, 243 operations for, 238, 239 representation of a linear mapping, 280 scalar, 248 skew-symmetric, 249 square (of order n), 238 symmetric, 249 trace of, 291 transpose of, 248 triangular, 248, 277 zero, 238 maximal element principle, 185

308

INDEX

maximal ideal, 184 maximal normal subgroup, 119 member (element) of a set, 1 minimal ideal, 194 monic polynomial, 199 multiple root, 218 multiplication table, 28 multiplicative semigroup of a ring, 142 natural (canonical) mapping, 96, 165 nil ideal, 195 nilpotent element, 155 nilpotent mapping, 295 normal chain, 117 normal subgroup, 80 normaliser, 88, 129 null set, 2 one-to-one function, 16 onto function, 14 operation, associative, 30 binary, 27 commutative, 32 distributive, 141 well-defined, 83 order, of a group, 7l> of a IfOUP element, 70 ordered n-tuple, 6 ordered pair, 6 ordered triple, 6 partition, 10 permutation, 58, 62 cycle, 60 even, odd, 126 group of, 62 transposition, 61 two-line form, 58 p-group, 130 polynomial, 196 constant, 198 degree of. 199 division algorithm, 200 irreducible, 205 leading coefficient. 199 linear, 205 monic, 199 quadratic, 207

ring of, 197 root of, 202 polynomial function, 218 power eet, 6 primary ideal, 196 prime element, 190 prime ideal, 188 prime factorisation, 23 prime field, 180 prime number, 21 principal ideal, 158 domain, 189 ring, 159 projection mapping, 291 proper subset, 2 quadratic polynomial, 207 quotient, 202 quotient group, 84 quotient ring, 160 quotient space. 261 quotients. field of, 179 radical, 191 range of a function, 13 refinement of a chain, 117 relation, binary, 8 equivalence, 9 reflexive, 9 symmetric, 9 transitive, 9 relative complement, 3 relatively prime. 22 remainder, 202 representative. of a congruence class, 54 of a coset. 75 restriction of function, 17 ring, 142 Boolean, 220 commutative. 143 diviHion, 182 with identity, 143 of integers modulo n, 146 of linear mal>llings, 286 10cal,l96 of n X n matrices, 242 of polynomialA, 197 principal ideal, 159

a

quotient, 160 semisimple, 191 simple, 155 root, 202 multiple, 218 scalar, 249 scalar matrix, 248 scalar multiplication, 250 scalar tranHformation, 295 second dual, 292 semigroup, 31 commutative, 32 with identity, 33 semisimple ring, 191 set, Cartesian product, 6 complement, 3 difference of, 3 disjoint, 3 empty, 2 equality of, 2 intersection of, 3 member of, 1 power, 5 singleton, 2 symmetric difference, 36 union of, 3 universal, 2 shift function, 278 simple group, 81 !!imple ring, 155 singleton, 2 skew-symmetric matrix, 249 solvable chain, 123 solvable group, 129 span, linear, 255 splitting field, 212 exifltence of, 212 uniqueness of, 216 standard (natural) basis, 268 Hteinit.z replacement theorem, 267 Stone representation theorem, 223 tlubfield, 176 subgrollp, 64 commutator, 86 conjugate, 129 (~yclic, 69 normal,80

Sylow, 134 torsion, 88 trivial, 64 subring, 150 Hubs pace, 253 annihilator of, 296 Sylow p-subgroup, 134 Sylow theorems, 135, 137 Sylvester's law, 281 symmetric difference, 35 symmetric group, 59, 62 symmetric matrix, 249 symmetries of the square, 49 theorem; Bee also fundamental theorems, basis, 270 Cauchy's, 133 Cayley's, 101 correspondence, llO Euclid's, 24 factor, 104 Fermat's little, 219 Jordan-Holder, 121 Kronecker's, 208 Krull-Zorn, 185 Lagrange's, 79 remainder, 203 Steinitz replacement, 267 Stone representation, 223 Sylow's, 135, 137 torsion-free, 88 torsion subgroup, 88 trace of a matrix, 291 transformation, linear, 277 transpose, of a linear mapping, 296 of a matrix, 248 transposition, 61 triangular matrix, 248, 277 trivial homomorphism, 90, 161 trivial subgroup, 64 trivial subring, 150 trivial subspace, 259 union of sets, 3 unique factorization, for integers, 23 for polynomials, 206 uniqucncHH, of identity elements, 33 of inverse elements, 41

310

INDEX

unit vcctorll, 264 univerllall!Ct, 2 vectors, 235, 249 components of, 235 rlimenMion of, 235 inner product, 236 linearly independent, 263 operations for, 235 vector space, 249 basis for, 268 dimension of, 271 generators for, 256

linear span, 255 subspace, 253 well-defined operation, 83 well-ordering principle, 19 zero divisor, 148 zero element of a ring, 142 zero matrix, 238 zero polynomial, 197 zero ring, 147 zero vector, 236 Zorn's lemma, 185