CHAPTER THREE

Groups

Introduction

Some of the standard topics in elementary group theory are treated in this chapter: sub- groups, cyclic groups, isomorphisms, and homomorphisms. In the development here, the topic of isomorphism appears before homomorphism. Some instructors prefer a different order and teach Section 3.6 (Homomorphisms) before Section 3.5 (Isomorphisms). Logic can be used to support either approach. Isomorphism is a special case of homomorphism, while homomorphism is a generalization of isomor- phism. Isomorphisms were placed first in this book with the thought that “same structure” is the simpler idea.

Both the additive and the multiplicative structures in Zn serve as a basis for some of the examples in this chapter.

3.1 Definition of a Group

The fundamental notions of set, mapping, binary operation, and binary relation were pre- sented in Chapter 1. These notions are essential for the study of an algebraic system. An algebraic structure, or algebraic system, is a nonempty set in which at least one equiva- lence relation (equality) and one or more binary operations are defined. The simplest struc- tures occur when there is only one binary operation, as is the case with the algebraic system known as a group. An introduction to the theory of groups is presented in this chapter, and it is appropriate to point out that this is only an introduction. Entire books have been devoted to the theory of 138 Chapter 3groups; Groups the group concept is extremely useful in both pure and applied mathematics. A group may be defined as follows. 3. G has an identity element e. There is an e in G such that x * e 5 e * x 5 x for all Groupsx [ G . !Week 4 Definition 3.1 I Group Definition4. G contains 3.1 Groupinverses. For each a [ G, there exists b [ G such that a * b 5 b * a 5 e. Suppose the binary operation p is defined for elements of the set G. Then G is a group with respectThe to phrasep provided “with therespect following to p” shouldfour conditions be noted. hold: For example, the set Z of all integers is a group with respect to addition but not with respect to multiplication (it has no inverses 138 Chapter 3for1. Groups elementsG is closed otherunder thanp. 6That1). Similarly,is,x [ G and the yset[ GG 5 imply{1, that21}x is* ya isgroup in G. with respect to multiplication2. p is associative but not. For with all respect x, y, z toin addition.G, x * (y *Inz )most5 ( xinstances,* y) * z. however, only one binary operation is under consideration, and we say simply that “G is a group.” If the binary operation3. G has isan unspecified, identity element we adopte. Therethe multiplicative is an e in G notationsuch that and x use* e the5 ejuxtaposition* x 5 x for137 allxy to indicatex [ G. the result of combining x and y. Keep in mind, though, that the binary operation is4. notG necessarilycontains inverses multiplication.. For each a [ G, there exists b [ G such that a * b 5 b * a 5 e.

Definition 3.2 I TheAbelian phrase Group “with respect to p” should be noted. For example, the set Z of all integers isDefinition a group with 3.2 Abelianrespect to Group addition but not with respect to multiplication (it has no inverses forLet elementsG be a group other with than respect61). Similarly,to p. Then theG is set called G 5 a {1,commutative21} is a groupgroup ,with or an respect abelian to† multiplicationgroup, if p is butcommutative. not with respect That is, to xaddition.* y 5 y *Inx most for all instances, x, y in G. however, only one binary operation is under consideration, and we say simply that “G is a group.” If the binary operationRemark is unspecified, we adopt the multiplicative notation and use the juxtaposition xy toExample indicate the 1 resultWe canof combining obtain some x andsimpley. Keep examples in mind, of groups though, by that considering the binary appropriate operation issubsets not necessarily of the familiar multiplication. number systems. 1) The set of integers is a group under the OPERATION of addition: a. The set C of all complex numbers is an abelian group with respect to addition. I Definition 3.2 Web. AbelianhaveThe setalready Q GroupϪ seen{0} of that all thenonzero integers rational under numbers the OPERATIONis an abelian of additiongroup with are respectCLOSED, to ASSOCIATIVE, have IDENTITY 0, and that any integer x has the INVERSE −x. Because multiplication. † Letthe Gsetbe of a integersgroup with under respect addition to p .satisfies Then G allis calledfour group a commutative PROPERTIES, group it is, ora group!an abelian groupc. The, if psetis R commutative.+ of all positive That real is, numbersx * y 5 yis* anx forabelian all x ,groupy in G with. respect to multiplica- 2) Thetion, set but {0,1,2} it is not under a group addition with isrespect not a togroup addition, because (it has it no does additive not satisfy identity all and of theno groupadditive PROPERTIES: inverses). it does not have the CLOSURE PROPERTY (see the previousI Examplelectures to see1 why).We can Therefore, obtain some the simpleset {0,1,2} examples under ofaddition groups is by not considering a group! appropriate subsetsThe of followingthe familiar examples number givesystems. some indication of the great variety there is in groups. (Notice also that this set is ASSOCIATIVE, and has an IDENTITY which is 0, but does not haveExamplea. The the set INVERSE C 2 ofRecall all PROPERTYcomplex from Chapter numbers because 1 thatis an − a 1abelian permutation and −2 group are onnot with a in set therespectA isset!) a one-to-one to addition. mapping b.fromTheA onto set QA andϪ {0} that ofS (allA) denotesnonzero therational set of allnumbers permutationsis an abelian on A. We group have with seen respect that S( Ato) is3) closedThemultiplication. set with of integers respect tounder the binarysubtraction operation is not+ of a group mapping, because composition it does and not that satisfy the all of the group PROPERTIES: it does not have the ASSOCIATIVE PROPERTY (see the operation + is+ associative. The identity mapping IA is an identity element: previousc. The set lectures R of all to positive see why). real Therefore, numbers is the an set abelian of integers group with under respect subtraction to multiplica- is not a group!tion, but it is not a group with respectf + IA 5 tof 5additionIA + f (it has no additive identity and no for additiveall f [ inverses).(A), and each f [ (A) has an inverse in (A). Thus we may conclude fromI (Notice alsoS that this set is CLOSED,S but does not haveS an IDENTITY and therefore also doesresultsThe not in following haveChapter the 1 INVERSEexamples that S(A) PROPERTY.)give is a group some withindication respect of to the composition great variety of mappings. there is in However groups. S(A) is not abelian since mapping composition is not a commutative operation. I Example 4) The set 2 of Recallnatural from numbers Chapter under 1 that additiona permutation is not on a a setgroup,A is abecause one-to-one it does mapping not fromsatisfyExampleA onto all of A3 andtheWe thatgroup shallS( APROPERTIES: )take denotes A 5 the{1, set 2, it of 3}does all and permutationsnot obtain have anthe explicit onIDENTITYA. We example have PROPERTY seen of thatS(AS) (see .( AIn) istheorder closed previous to define with lectures an respect element to to see f theof why). S binary(A )Therefore, , we operation need to the specify+ setof mappingof f (1),naturalf(2), compositionnumbers and f(3). under There and addition are that three the is not a group! operationpossible choices+ is associative. for f(1). The Since identityf is to mapping be bijective,IA is an there identity are element: two choices for f(2) after

(Notice also that this set is CLOSED,f + I AASSOCIATIVE,5 f 5 IA + f but does not have the INVERSE PROPERTY†The term abelian becauseis used in none honor of of theNiels negative Henrik Abel numbers (1802–1829). are inA biographicalthe set.) sketch of Abel appears on forthe alllast pagef [ ofS (thisA), chapter. and each f [ S(A) has an inverse in S(A). Thus we may conclude from results in Chapter 1 that S(A) is a group with respect to composition of mappings. However S(A) is not abelian since mapping composition is not a commutative operation. I1

Example 3 We shall take A 5 {1, 2, 3} and obtain an explicit example of S(A) . In order to define an element f of S(A) , we need to specify f(1), f(2), and f(3). There are three possible choices for f(1). Since f is to be bijective, there are two choices for f(2) after

†The term abelian is used in honor of Niels Henrik Abel (1802–1829). A biographical sketch of Abel appears on the last page of this chapter. Groups !Week 4 5) The set of rational numbers with the element 0 removed is a group under the OPERATION of multiplication:

We have already seen that the set rational numbers with the element 0 removed under the OPERATION of multiplication is CLOSED, ASSOCIATIVE, have IDENTITY 1, and that any 1 integer x has the INVERSE . Because the set of rational numbers with the element 0 x removed under multiplication satisfies all four group PROPERTIES, it is a group!

6) The set of rational numbers (which contains 0) under multiplication is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the INVERSE PROPERTY (see the previous lectures to see why). Therefore, the set rational numbers under multiplication is not a group!

(Notice also that this set is CLOSED, ASSOCIATIVE, and has an IDENTITY which is 1.)

7) The set of rational numbers under division is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the ASSOCIATIVE PROPERTY (see the previous lectures to see why). Therefore, the set of rational numbers under division is not a group!

(Notice also that this set is not CLOSED because anything divided by 0 is not in the set, does not have an IDENTITY and therefore also does not have the INVERSE PROPERTY.)

8) The set of natural numbers under division is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the IDENTITY PROPERTY (see the previous lectures to see why). Therefore, the set of natural numbers under division is not a group!

(Notice that this set does not have the CLOSURE, ASSOCIATIVE or INVERSE PROPERTIES.)

9) The set of integers under multiplication is not a group, because it does not satisfy all of the group PROPERTIES: it does not have the INVERSE PROPERTY (see the previous lectures to see why). Therefore, the set of integers under multiplication is not a group!

(Notice also that this set is CLOSED, ASSOCIATIVE, and has the IDENTITY ELEMENT 1.)

Definition A permutation of a set is a bijection (one to one and onto) of this set onto itself, i.e.,

S(A) = { f f : A → A and f isbijective }.

Example*

2 Groups7 Permutations !Week 4 7 Permutations The aim of this section is to discuss a large class of groups, called permutation groups. The aim ofThe this central section result is to of discuss this section a large is class Cayley’s of groups, theorem, called which permutation states that groups. actually any group The centralis result contained of this in section this class is Cayley’s of group. theorem, Thus, which permutation states that groups actually giveany a genericgroup example of a is containedgroup. in this class of group. Thus, permutation groups give a generic example of a group. 7.1 Permutations 7.1 PermutationsHomework*. Take A = {1,2,3} and obtain an explicit example of S(A). Definition 7.1. A permutation of a set is a bijection of this set onto itself. Definition 7.1. A permutation of a set is a bijection of this set onto itself. Note that 1 Note that (i) if α : X X is a permutation, than the inverse function α− : X X is also a 1 (i) if α : X X is a permutation,→ than the inverse function α− : X X is also a→ permutation;→ → permutation; (ii) if α : X X and β : X X are permutations, one can form a composition (ii) if αα : Xβ : X X Xand, which→β : X is alsoX aare permutation. permutations,→ one can form a composition α β : X ◦X, which→ → is also a permutation.→ ◦ →Theorem 7.2. Let A be a nonempty set, and let S be the collection of all permutations of TheoremTheoremA 7.2.. ThenLet ASLetbeis A a a be groupnonempty a nonempty under set, the and set, composition letandS letbe theG be collection. collection of allof all permutations permutations of of A. Then G A. Then Sis isa agroup group under under the the composition composition . ◦ Proof. Clearly, defines a binary◦ operation on S. We need to check the three axioms G1, Proof. Clearly, defines a binary operation on S. We need to check the three axioms G1, G2, G3 of the group.◦ G2, G3 of the group.◦ (G1) We have seen in section 1 that composition is associative. (G1) We have seen in section 1 that composition is associative. (G2) Consider the mapping ι : A A such that for all x A, ι(x)=x. Then for any (G2) Consider the mapping ι : A A such that→ for all x A, ι(x)=x∈. Then for any α S one has α ι = ι α→= α. Thus, ι is the identity∈ element on S. α S one has∈ α ι = ι α =◦ α. Thus,◦ ι is the identity element1 on S. 1 1 ∈ (G3)◦ For◦ any permutation α S, its1 inverse α− 1 satisfies α− 1 α = α α− = ι. Thus, (G3) For any1 permutation α S, its inverse∈ α− satisfies α− α = α α− ◦= ι. Thus,◦ 1 α− is the inverse of α∈ with respect to the group operation.◦ ◦ α− is the inverse of α with respect to the group operation. Example**. Construct the Cayley (multiplication) table for S(A) in Example*. Notation NotationBelow we willBelow denote we will composition denote composition of permutations of permutationsα and β by αβα andratherβ by αβ rather than α βthan. α β. ◦ ◦ DefinitionDefinition 7.3. 1. Let7.3.A be1. any Let set.A be The any group set. of Theall grouppermutations of all permutations of A is called of a A is called a symmetricsymmetric group on A groupand denoted on A and by denotedSym(A). by Sym(A). 2. Any subgroup2. Any of subgroup Sym(A) of(including Sym(A) Sym( (includingA) itself) Sym( isA called) itself) a permutation is called a grouppermutation on group on the set A. the set A. 3. If A is the3. If finiteA is set the1 finite, 2,...,n set 1of, 2n,...,nelements,of n thenelements, Sym(A) then is commonly Sym(A) is denoted commonly denoted { }{ } Sn. We haveSn. already We have dealt already with the dealt groups withS then. groups Sn.

PropositionProposition 7.4. The order 7.4. ofThe the order group ofS theis groupn!=1Sn2 is n!=1(n 1)2 n. (n 1) n. n · ····· −· ······ − · Proof. TheProof. problemThe of problem computing of thecomputing number the of elements number in ofS elementsn is the same in Sn asis the the problem same as the problem of computingof computing the number the of numberdifferent of ways diff theerent integers ways the 1, 2,...,n integerscan 1, 2 be,...,n placedcan in thebe placedn in the n blanks indicatedHomework**.blanks (using indicated each Construct (using integer eachthe just Cayley integer once): (multiplication) just once): table for S(A) in Homework*.

Example 4 (Homework) 12... n12... n . . ! ... ! " ... "

25 25

3 140 Chapter 3 Groups

140 Chapter 3and Groups inverses of elements. An element e is a left identity if and only if the row headed by e at the left end reads exactly the same as the column headings in the table. Similarly, e is a rightand inverses identity of if andelements. only if An the element columne headedis a left by identitye at the if top and reads only exactly if the row the headed same as by thee rowat the headings left end in reads the exactly table. If the it exists, same as the the inverse column of headings a certain in element the table.a can Similarly, be founde is by a searching for the identity e in the row headed by a and again in the column headed by a. right identity if and only if the column headed by e at the top reads exactly the same as the If the elements in the row headings are listed in the same order from top to bottom as the row headings in the table. If it exists, the inverse of a certain element a can be found by elements in the column headings are listed from left to right, it is also possible to use the table searching for the identity e in the row headed by a and again in the column headed by a. 140 toChapter check for 3 commutativity. Groups The operation is commutative if and only if equal elements appear If the elements in the row headings are listed in the same order from top to bottom as the in all positions that are symmetrically placed relative to the diagonal from upper left to lower elements in the column headings are listed from left to right, it is also possible to use the table right. In Exampleand inverses 3, the of group elements. is not An abelian element sincee is athe left table identity in Figure if and 3.2 only is if not thesymmetric. row headed For by e to check for commutativity. The operation is commutative if and only if equal elements appear Groupsexample, g at + ther 2 5d left end is in reads row exactly 5, column the ! same 3, and asr the2 + columng5s headingsis in row in 3, the column table. Similarly, 5. Weeke is a4 in all positionsright that identity are symmetrically if and only if the placed column relative headed to by thee at diagonal the top reads from exactly upper the left same to lower as the Exampleright. In Examplerow 4 headingsLet 3, theG be group in the the set table. is not of If abeliancomplex it exists, since numbers the the inverse table given of in a Figure certainby G 5 3.2 element{1, is not21,asymmetric.cani, 2 bei}, found where For by example, g + r2 5dis in row 5, column 3, and r2 + g5sis in row 3, column 5. i 5 21,searchingand consider for the the identity operatione in the of row multiplication headed by a andof againcomplex in the numbers column headedin G. The by a. If the elements in the row headings are listed in the same order from top to bottom as the table in Figure 3.3 shows that G is closed with respect to multiplication. Exampleelements 4 Let inG thebe column the set headings of complex are listed numbers from left given to right, by itG is5 also{1, possible21, i, to2 usei}, the where table !Multiplication in G is associative and commutative, since multiplication has these prop- i 5 21,toand check consider for commutativity. the operation The operation of multiplication is commutative of ifcomplex and only ifnumbers equal elements in G. appear The erties in the set of all complex numbers. We can observe from Figure 3.3 that 1 is the iden- table in Figurein all 3.3 positions shows that that are G symmetricallyis closed with placed respect relative to tomultiplication. the diagonal from upper left to lower tity elementright. and Inthat Example all elements 3, the group have is notinverses. abelian Each since theof 1 table and in 2Figure1 is its 3.2 own is not inverse,symmetric. and For i !Multiplication in G is associative and commutative, since multiplication has these prop- and 2i are example,inverses gof+ eachr2 5d other.is in Thus row 5, G columnis an abelian 3, and r2 group+ g5s withis in respect row 3, columnto multiplication. 5. erties in the set of all complex numbers. We can observe from Figure 3.3 that 1 is the iden- tity3 element1Example and2 that1 all4 elementsLeti G be2 thehavei set inverses. of complex Each numbers of 1 andgiven 2 by1 isG 5its {1,own2 1,inverse,i, 2i}, andwhere i and 2i are iinverses5 21 of, and each consider other. Thusthe operation G is an ofabelian multiplication group with of complex respect tonumbers multiplication. in G. The 11table 2in 1Figure 3.3i shows2 thati G is closed with respect to multiplication. 3 1 !Multiplication21 i in G is2 associativei and commutative, since multiplication has these prop- 21 211erties in the set2 ofii all complex numbers. We can observe from Figure 3.3 that 1 is the iden- 1121 i 2i iitity element2i and2 11that all elements have inverses. Each of 1 and 21 is its own inverse, and i and 2i are inverses of each other. Thus G is an abelian group with respect to multiplication. 221i 2211ii21ii21 I Figure 3.3 I ii3 2i 1 22111 i 2i 1121 i 2i 2i 2iiIt is an immediate1 21 corollary of Theorem 2.28 that the set I ExampleExample 5 (Homework)5 I Figure 3.3 21 2112ii Zn 5 0 , 1 , 2 , c, n 2 1 ii2i 211 ofExample congruence 5 classesIt is an modulo immediate n forms corollary an abelian of Theorem group with2.28 respectthat the to set addition. I 2i 2ii1 53 42314 3 4 3 46 I I Figure 3.3 Zn 5 0 , 1 , 2 , c, n 2 1 Example 6 Let G 5 {e, a, b, c}withmultiplicationasdefinedbythetableinFigure 3.4. of congruence classes modulo n forms an abelian group with respect to addition. I Example 5 It is an immediate53 4 3 corollary4 3 4 of Theorem3 46 2.28 that the set ? eabc Zn 5 0 , 1 , 2 , c, n 2 1 ExampleExample 6 (Homework) 6 Let G 5 {e, a, b, c}withmultiplicationasdefinedbythetableinFigure 3.4. of congruence classes modulo n forms an abelian group with respect to addition. I eeabc 53 4 3 4 3 4 3 46 ? eabc aabceExample 6 Let G 5 {e, a, b, c}withmultiplicationasdefinedbythetableinFigure 3.4. bbceaeeabc ? eabc cceabaabce I Figure 3.4 eeabc bbcea From the table,aabce we observe the following: I Figure 3.4 cceabbbcea 1. G is closed under this multiplication. cceab I FigureFrom2. e 3.4theis the table, identity we observe element. the following: 1. G is closedFrom theunder table, this we multiplication. observe the following: 2. e is the 1.identityG is closed element. under this multiplication. 2. e is the identity element.

4 3.1 Definition of a Group 141

3.1 Definition of a Group 141 Groups3. Each of e and b is its own inverse, ! and c and a are inverses of each other. Week 4 3.1 Definition of a Group 141 3.4. EachThis ofmultiplication e and b is its is own commutative. inverse, and c and a are inverses of each other. This multiplication is also associative, but we shall not verify it here because it is a labori- 4. This multiplication3. Each of e isand commutative.b is its own inverse, and c and a are inverses of each other. ous task. It follows that G is an abelian group. I This multiplication4. This ismultiplication also associative, is commutative. but3.2 we Properties shall not of verify Group Elementsit here because145 it is a labori- I ous task. It followsThis multiplicationThe that table G is inanis also Figureabelian associative, 3.5group. defines but we shall a binary not verify operation it here becausep on the it is set a labori- S 5 33.ExampleLet G be the 7 set of all matrices in M3(R) that have the form {A, B, C, Dous}. task. It follows that G is an abelian group. I Example 7 The table in Figure1 ab 3.5 defines a binary operation p on the set S 5 {A, B, C, D}.Example 7 The table01 in Figurec 3.5 defines a binary operation p on the set S 5 * ABCD{A, B, C, D}. 001 *ABCABfor arbitraryABCD real numbers a, b, and c. Prove or disprove that G is a group with respect to multiplication.* ABCD BCDBA 34.ABCABProve or disproveABCAB that the set G in CExercise 32S is a group with respect to addition. 35.BCDBACABCDProve or disproveBCDBA that the set G in Exercise 33 is a group with respect to addition. 36.CABCDDABDDFor an arbitraryCABCD set A, the power set p(A) was defined in Section 1.1 by p(A) 5 I Figure 3.5 {X X A}, and addition in p(A) was defined by 8 DABDD I FigureDABDD 3.5 I Figure 3.5 0 X 1 Y 5 (X c Y) 2 (X d Y) From the table, we make the following5 (X 2 Yobservations:) c (Y 2 X). From the table, we make the following observations: From1. a.S theProveis closedtable, that wepunder(A make) is pa. group the following with respect observations: to this operation of addition. 1. S is closed under p. 2. b.CIfisA anhas identityn distinct element. elements, state the order of p(A). 1. S is closed2. underC is an pidentity. element. Sec. 1.1, #7c @ 37.3. WriteD does out notthe haveelements an inverseof p(A) sincefor theDX set 5A 5C{hasa, b ,noc}, solution. and construct an addition 2. Ctableis anfor identityp3.(AD) usingdoes element. additionnot have as an defined inverse in since ExerciseDX 536.C has no solution. Thus S is not a group with respect to p. I Sec. 1.1, #7c @ 38.3. DLetdoesA 5 Thusnot{a, b have,Sc}.is Provenot an ainverse groupor disprove with since respectthatDX p( A5to) ispC. a hasgroup no with solution. respect to the operation I of union. Thus S is not a group with respect to p. I DefinitionSec. 1.1, #7c 3.3Definition @ Definition39.I LetFinite 3.3A 5 3.3 I{Group,a, FinitebFinite, c}. Prove Infinite Group,Group, or disprove InfiniteInfiniteGroup, that Group,Group, Orderp(A) is Order Orderaof group a Groupof ofwith a a Group Grouprespect to the operation of intersection. IIf a group GIfhas a group a finiteG has number a finite ofnumber elements, of elements,G is calledG is called a finite a finite group group,ora,oragroupgroup of of finite finite Definition 3.3 Finite Group,order.Thenumberofelementsin Infinite Group, OrderG isof called a Group the order of G and is denoted by either order.Thenumberofelementsino(G)or G .IfG does not haveG is a finite called number the order of elements,of GGandis called is denoted an infinite by group either. 3.2 IfPropertieso( aG group)or GG.Ifhas Gof adoes finiteGroup not number haveElements of a finite elements, numberG is of called elements, a finiteG is group called,ora an infinitegroup of group finite. order.Thenumberofelementsin0 0 G is called the order of G and is denoted by either oSeveral(G)or consequencesG0 .If0ExampleG does of the8 not definitionIn have Example a of finite a3, group the number group are recorded of elements, in TheoremG is 3.4. called an infinite group. Example 8 In Example*, the group G = {e,σ } has order o(G) = 2 . In Example 5, Example 8 In Example 3, the group G 5 e, r, r2, s, g, d o Z = n0 . 0The set Z of all integers is a group under addition, and this is an exaple of an ( n ) G 5 e, r, r2, s, g, d Example has8 orderIn Example o(G) 5 6. 3, In the Example group 5, o(Zn5) 5 n. The set Z6 of all integers is a group under ad- Strategy I infiniteParts a group.and bdition,of the and next this theorem is an example are statements of an infinite about group. uniqueness, If A is an and infinite they can set, be then (A) furnishes has order o(G) 5 6. In Example 5, o(Z ) 5 n. The set Z of all integers is a groupS under ad- proved by thean standard example type of ofan uniquenessinfinite group.G 5proof:n5e, Assumer, r2, s that, g, twod6 such quantities exist, I Propertiesanddition, then and prove thisof the Group is two an toexample Elementsbe equal. of an infinite group. If A is an infinite set, then S(A) furnishes hasan exampleorder o(G of) 5an 6.infinite In Example group. 5, o(Zn5) 5 n. The set Z6 of all integers is a group under ad-I dition,Theorem and 3.4 this Properties is an example of Group of an infinite Elements group. If A is an infinite set, then (A) furnishes I Exercises 3.1 S Theorem 3.4 an exampleProperties of an of infiniteGroup Elements group. I True or False LetExercisesG be a group with3.1 respect to a binary operation that is written as multiplication. Label each of the following statements as either true or false. Truea. The or identity False1. elementThe identity e in G elementis unique. in a group G is its own inverse. Exercises 3.1 21 Labelb. For each each ofx2.[ theIfG,G following theis inversean abelian xstatements group,in G is thenunique. as x eitherϪ1 ϭ x truefor all or x false.in G. Truec. For or eachFalse x [ G,(x21)21 5 x. 1. The identity element in a group G is its own inverse. Label each of the following statements as either true or false. 2. If G is an abelian group, then xϪ1 ϭ x for all x in G. 1. The identity element in a group G is its own inverse. 5 2. If G is an abelian group, then xϪ1 ϭ x for all x in G. 146 Chapter 3 Groups

d. Reverse order law. For any x and y in G,(xy)21 5 y21x21. e. Cancellation laws. If a, x, and y are in G, then either of the equations ax 5 ay or xa 5 ya implies that x 5 y.

Uniqueness Proof We prove parts b and d and leave the others as exercises. To prove part b, let x [ G, and suppose that each of y and z is an inverse of x. That is, xy 5 e 5 yx and xz 5 e 5 zx. Then 146 Chapter 3 Groups y 5 ey since e is an identity Groups !Week 4 5 (zx)y since zx 5 e 21 21 21 d. Reverse order law. For any5 xzand(xy) y inbyG associativity,(xy) 5 y x . e. Cancellation laws. If a, x5, andz(e)y aresince in G xy, then5 e either of the equations ax 5 ay or xa 5 ya implies that x 5 y. 5 z since e is an identity. 21 Uniqueness ProofThus y 5Wez ,prove and this parts justifiesb and thed notationand leave x theas othersthe unique as exercises. inverse of To x inproveG. part b, let (p q) ⇒ r x [ GWe, and shall suppose use part thatb ineach the of proof y and ofz partis and .Specifically,weshallusethefactthatthein-inverse of x. That is, verse (xy)21 is unique. This means that in order to show that y21x21 5 (xy)21,weneedonly ¿ to verify that (xy)(y21x21) 5xy 5e 5e 5(y2yx1x 21and)(xy ).xz These5 e 5 calculationszx. are straightforward: Then (y21x21)(xy) 5 y21(x21x)y 5 y21ey 5 y21y 5 e and y 5 ey since e is an identity 5 (zx)y since zx 5 e (xy)(y21x21) 5 x(yy21)x21 5 xex21 5 xx21 5 e. 5 z(xy) by associativity The order of the factors y251 andz(ex) 21 insince the reversexy 5 e order law (xy)21 5 y21x21 is crucial in a nonabelian group. An example5 z wheresince (xy)2 e1 is2 anx 2identity.1y21 is requested in Exercise 5 at the Thusend ofy this5 z ,section. and this justifies the notation x21 as the unique inverse of x in G. Part e of Theorem 3.4 implies that in the table for a finite group G,no element of G (p q) ⇒ r We shall use part b in the proof of part d.Specifically,weshallusethefactthatthein- verseappears (xy )twice21 is unique.in the same This meansrow, and that no in element order to showof G thatappearsy21x 2twice1 5 (inxy the)21 ,weneedonlysame column. ¿ toThese verify results that (canxy)( bey2 extended1x21) 5 eto5 the(y statement21x21)(xy in). the These following calculations strategy are box. straightforward: The proof of this fact is requested in Exercise 10. (y21x21)(xy) 5 y21(x21x)y 5 y21ey 5 y21y 5 e

Strategy I andIn the multiplication table for a group G, each element of G appears exactly once in each row and also appears (exactlyxy)(y21 xonce21) 5 inx each(yy21 )column.x21 5 xex 21 5 xx21 5 e.

Remark. TheAlthough order ofour the definition factors y 2of1 anda groupx21 in is the a standard reverse orderone, alternativelaw (xy)21 forms5 y21 canx21 isbe crucial made. inOne a nonabelian of these is group.given inAn the example next theorem. where (xy)21 2 x21y21 is requested in Exercise 5 at the end of this section. Theorem 3.5 TheoremI PartEquivalent e 3.5of TheoremEquivalent Conditions 3.4 Conditions implies for athat Group for in athe Group table for a finite group G,no element of G appears twice in the same row, and no element of G appears twice in the same column. TheseLet G resultsbe a nonempty can be extended set that isto closedthe statement under an in associative the following binary strategy operation box. Thecalled proof multi- of thisplication. fact is Thenrequested G is ain group Exercise if and 10. only if the equations ax 5 b and ya 5 b have solutions x and y in G for all choices of a and b in G.

Strategy I In the multiplication table for a group G, each element of G appears exactly once in each row and also appears exactly once in each column.

Although our definition of a group is a standard one, alternative forms can be made. One of these is given in the next theorem.

Theorem 3.5 I Equivalent Conditions for a Group

Let G be a nonempty set that is closed under an associative binary operation called multi- plication. Then G is a group if and only if the equations ax 5 b and ya 5 b have solutions x and y in G for all choices of a and b in G. 6 3.2 Properties of Group Elements 147 p ⇒ (q r) Proof Assume first that G is a group, and let a and b represent arbitrary elements of G. Now a21 is in G,and so are x 5 a21b and y 5 ba21. With these choices for x and y,we have ¿ ax 5 a(a21b) 5 (aa21)b 5 eb 5 b

and

ya 5 (ba21)a 5 b(a21a) 5 be 5 b.

Thus G contains solutions x and y to ax 5 b and ya 5 b. (q r) ⇒ p Suppose now that the equations always have solutions in G. We first show that G has an identity element. Let a represent an arbitrary but fixed element in G. The equa- ¿ tion ax 5 a has a solution x 5 u in G. We shall show that u is a right identity for every element in G. To do this, let b be arbitrary in G. With z a solution to ya 5 b,wehave za 5 b and Groups !Week 4 bu 5 (za)u 5 z(au) 5 za 5 b. Thus u is a right identity for every element in G. In a similar fashion, there exists an element v in G such that vb 5 b for all b in G.Thenvu 5 v,sinceu is a right identity, and vu 5 u,sincev is a left identity. That is, the element e 5 u 5 v is an identity ele- ment for G. Now for any a in G, let x be a solution to ax 5 e, and let y be a solution to ya 5 e. Combining these equations, we have

x 5 ex 5 yax 5 ye 5 y,

and x 5 y is an inverse for a. This proves that G is a group. Remark. In a group G, the associative property can be extended to products involving moreIn than a group three G, the factors. associative For example,property can if a 1be,a 2extended,a3, and a to4 products are elements involving of more G, than then applicationsthree factors. of For condition example, 2 in if Definition a1, a2, a3 ,3.1 and yielda4 are elements of G, then applications of con- dition 2 in Definition 3.1 yield

a1(a2 a3) a4 5 (a1 a2)a3 a4

and 3 4 3 4

(a1 a2)(a3 a4) 5 (a1 a2)a3 a4. These equalities suggest (but do not completely prove) that regardless of how symbols 148 Chapter 3 Groups 3 4 of grouping are introduced in a product a1a2a3a4, the resulting expression can be 148 Chapter 3reduced Groups to Definition 3.6 DefinitionI Product 3.6 ProductNotation Notation (a1 a2)a3 a4. Definition 3.6 I Product Notation Let n be a positive integer, n $ 2. For elements a1, a2, , an in a group G, the expression With these observations in mind, we make3 the following4 c definition. a1a2 can is defined recursively by Let n be a positive integer, n $ 2. For elements a1, a2, c, an in a group G, the expression

a1a2 can is defineda recursively1 a2 cak ak 1by1 5 (a1 a2 c ak)ak11 for k $ 1.

a1 a2 cak ak11 5 (a1 a2 c ak)ak11 for k $ 1. We can now prove the following generalization of the associative property. We can now prove the following generalization of the associative property. Theorem 3.7 I Generalized Associative Law Theorem 3.7 TheoremI Generalized 3.7 Generalized Associative Associative Law Law Let n $ 2 be a positive integer, and let a1, a2, c, an denote elements of a group G. For any positive integer m such that 1 # m , n, Let n $ 2 be a positive integer, and let a1, a2, c, an denote elements of a group G. For

any positive integer m such(a1 a that2 c 1 #am)(mam,11n,c an) 5 a1 a2 c an.

(a1 a2 c am)(am11 c an) 5 a1 a2 c an. Complete Proof For n $ 2, let Pn denote the statement of the theorem. With n 5 2, the only possi- Induction ble value for m is m 5 1, and P2 asserts the trivial equality Complete Proof For n $ 2, let Pn denote the statement of the theorem. With n 5 2, the only possi-

Induction ble value for m is m 5 1, and P2 asserts(a1 )(thea2 )trivial5 a1 aequality2. 7

Assume now that Pk is true: For any positive(a1)(a2) integer5 a1 a2 .m such that 1 # m , k,

Assume now that Pk is true:(a1 a2 Forc anyam positive)(am11 c integerak) 5 masuch1 a2 cthata 1k.# m , k,

Consider the statement P(ak1 a12,andc letam )(mabem1 1a cpositiveak) 5 integera1 a2 csucha kthat. 1 # m , k 1 1. We treat separately the cases where m 5 k and where 1 # m , k. If m 5 k, the desired equality Consider the statement P ,and let m be a positive integer such that 1 # m , k 1 1. We is true at once from Definitionk11 3.6, as follows: treat separately the cases where m 5 k and where 1 # m , k. If m 5 k, the desired equality

is true at once from Definition(a1 a2 c a3.6,m)(a asm1 follows:1 c ak11) 5 (a1 a2 c ak)ak11.

If 1 # m , k, then (a1 a2 c am)(am11 c ak11) 5 (a1 a2 c ak)ak11.

If 1 # m , k, then am11 c ak ak11 5 (am11 c ak)ak11

by Definition 3.6, and consequently,am11 c ak ak11 5 (am11 c ak)ak11

by Definition ( a1 a3.6,2 c anda mconsequently,)(am11 cak ak11)

5 (a a c a ) (a c a )a ( a1 a2 c1 a2m)(am11m cma1k 1ak11) k k11

5 (a a c a )(a c a ) a by the associative property

5 (a11a22 c amm) 3(am11 c ak)ak1k1114

5 a a c a a by P 5 (a1 a2 c ak )(k1a1 c a ) a by thek associative property 3 1 2 m 3 m11 k 4 k114 5 a a c a by Definition 3.6. 5 a1 a2 c ak1a1 by P 33 1 2 k 4 k11 4 k

Thus P is true5 awhenever a c a P is true, and the proof of bythe Definition theorem is 3.6. complete. k11 3 1 2 k1k41 Thus Pk11 is true whenever Pk is true, and the proof of the theorem is complete. The material in Section 1.6 on matrices leads to some interesting examples of groups, both finite and infinite. This is pursued now in Examples 1 and 2. The material in Section 1.6 on matrices leads to some interesting examples of groups, both finite and infinite. This is pursued now in Examples 1 and 2. Example 1 Theorem 1.30 translates directly into the statement that Mm3n(R) is an abelian group with respect to addition. This is an example of another infinite group. Example 1 Theorem 1.30 translates directly into the statement that Mm3n(R) is an When the proof of each part of Theorem 1.30 is examined, it becomes clear that each abelian group with respect to addition. This is an example of another infinite group. group property in M (R) derives in a natural way from the corresponding property in R. When the proofm of3n each part of Theorem 1.30 is examined, it becomes clear that each group property in Mm3n(R) derives in a natural way from the corresponding property in R. Groups !Week 4

Example 1 M m×n (R) is an abelian group with respect to addidtion. This is an example of another infinite group.

Example M  , M  , M  , M  is a group with respect to addition. m×n ( ) m×n ( ) m×n ( k ) m×n ( )

Example The nonzero elements of M n (R) do not form a group with respect to multiplication.

Example 2 The invertible elements of M  , M  , M  , M  , M R form a group n ( ) n ( ) n ( k ) n ( ) n ( ) G with respect to matrix multiplication.

8