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 5imply{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 allto positivesee why). real Therefore, numbers isthe an set abelian of integers group withunder 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.)giveis 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 addition a 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).