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2 Someexamplesofgroups (iv) Someidentity elementehasthe followingproperty: given 1 any element a inathere isat least onesolution x to the = equation ax e, each solution being called an inverse to a and written as a-I. Some examples of groups Itisimportant tonote that wearenot assuminguniquenessof identities and inverses in a given group. Another fact that requires carefulconsideration isthat weare usingwhat mightbe calledright identities andright inverses. Supposewemakejust = one changein the above axioms,replacing the equation ax e SINOEexamples areofgreat importance inthe theory ofgroups in (iv)by xa = e, sothat (in an obvious sense)we are dealing we devote this chapter to a survey of the examples that are with right identities and left inverses; then many systems that co~monly encountered and used. We must of courseindicate are far from being groups satisfy the new axioms. Wemay for w~ch of the several possible definitions of 'group" wehave in instance take any set S,suchasthe set ofallteapots, anddefine mind, and the reason for our particular choice issimplicity in products by putting xy = xfor eachpair x,y ofelements ofS. proving that certain systems weshallpresent areinfactgroups. Axioms(i)and (ii)arereadily verified,and everyelement y ofS Ifthe reader is~amiliarwith most ofthe facts in this chapter serves asan identity asrequired in (iii). To solvethe equation he mayyet findIt ofvalue asasystematic revision ofhisknow xa =y forxwith agiveny, wesimplytake x = y. The axioms ledge. aretherefore satisfiedifSnon-empty. But Sdoesnot constitute a group if it has more than one element, for the equation DEFINITIO..N: A group is a set and a binary operation+ in that set satisfying certain conditiona.f that is the elements ax= y has no solution x ify =F a. a Wepassontoexamplesofsystemsthat aregroups. Themost a, b (in that order) in the group define an element usually written as ab and usually called the product of a and b the familiar and readily available aregroups ofnumbers. binary operation being (conventionally) called multiplic~tion. Example 1.01. The set Z of allintegers with addition asits The conditions are: operation formsagroup. Tohold fast to the conventions inour (i) aisclosedunder multiplication. definition would involve writing 3+7 as 3.7 and calling 10 (ii) The associativelawformultiplication holds, that is the 'product' of 3 and 7. We therefore have every excuse to a(bc) = (ab)c callthe binary operation addition and to usethe familiar nota tion in this special case. The group axioms are easily verified: for allelements a,b,oinG. (iii) There is at least one identity element in a, that is an (i)issimplythat the sumoftwointegers isaninteger, and (ii)is element esuch that ae =a for all ain a. wellknown, plausible, and a consequence of any axiom system forintegers. Wecantake the number 0asanidentity, forsurely t For more details ~bout binary operations seethe Appendix. a+O=afor allelements aofZ. As for (iv), aninverseofais . + The reader whoWIshesto assumemore, for instance that identities and = mverses havosomeofthe two-sided and uniquenees properties enumerated in -a sincewehave, just as simply, a+(-a) O. Thoo,,?ms2.01-2.05, may do so. He willnot then have to prove the corre Example 1.02. Theset Qofallrational numbersformsagroup sponding theorems, but he wiUhavethe additional task ofverifying that all the ~xampl~ presented inthisandsubsequent chapters aregroupsinthesense under addition, the proof being similar to that for Z. ofhis definition. 858187 B Some exa1111Jle.osf groups 3 4 Some examples of groups Example 1.03. The set R of allreal numbers forms agroup Example 1.07. The non-zero complex numbers form agroup under addition. Againthe reader should be ableto findaproof under the usualmultiplication. quite like that given for Z. Example 1.08. The complex numbers of unit modulus form Example 1.04.Theset Cofallcomplexnumbersformsagroup a group under the usual multiplication. under addition. Example 1.09. For each prime number p there is asubset of Itmay be wellto mention somesets ofnumbers that do not the complex numbers that forms a multiplicative group of form groups. The odd integers do not form a group under someinterest. Let Z; denote the followingset: addition (though the evenintegers do); a sufficient reason for {z:z ECand zP"= 1for someinteger n}. this is that (i)doesnot hold; another reason is that there is no identity element, which impliesthat (iii)and (iv)cannot hold. (Note that the positive integer n depends on the element e.) The non-negative integers donot form agroup under addition Weverifythe axiomsindividually forthis important group. Let for in their case (iv)doesnot hold (it is an empirical fact that Z1'Z2beelements ofZ;withzf" = 1,zf"= 1. Then (ZtZ2)P' = 1 Z; many naturally occurringsetswithmultiplication arenotgroups whereristhe largerofmandn,so isclosedunder multiplica because (iv) fails). The integers with the binary operation of tion. The associative lawholds simplybecauseitisalwaysvalid Z; subtraction donot form a group because the associative law is for multiplication of complex numbers. Clearly contains not valid; for instance 5-(4-1)= 2and (5-4)-1= o. the complexnumber 1,which willserve as an identity element. Since multiplication is the other familiar operation on (real IfzP' = 1then (l/z)p·= 1,so ifzis in Z; sois lIz-note that Z; and complex) numbers, it is reasonable to ask what sets of z =f=O-and l/z will act as an inverse of z. Therefore is a numbers form groups when the group multiplication is taken multiplicative group. to be the ordinary multiplication of numbers. The set of all A slight knowledge of complex numbers reveals that the Z; real numbers is not a candidate for group status in this sense elements of are precisely all those numbers ofthe form for~tco~taiDBthe number 0and (since the number 1is clearl; 2k7r .. 2k7T oos-+~sm-- an identrtyelement) werequire for axiom (iv) a solution x to p" p" the equation Ox=1;and becausethere is no such x wedo not have a group. forn =0,1,2, ...,where k isan integer with 0 :::;;k <p»; Further important types of groups have residue classes Example 1.05. The set ofpositive rational numbers forms a instead ofnumbers as elements. Aresidue classmodulo n is a group under (ordinary) multiplication. To see this, note that certain equivalence class of integers:'] thus the residue class (i) and (ii) are easily checked, take e= 1 in (iii), and take (aJmodulo n isthe set ofall solutions x to the congruenoe o. a-I = l/a in (iv); l/a exists since a =f= We remark that the x == amodulon, non-zero rationals form agroup for similar reasons. where n is a fixed positive integer. More explicitly but less Example 1.06. The set of positive (or of non-zero) real precisely (in the striot mathematioal sense)wehave numbers forms a group under multiplication of numbers. As (aJ ={..,.a-n, a,a+n, a+2n, ...}. the reader should have bynowacquired somefacility inproving such a statement there isno need to give further details. t As explained inthe Appendix. Some examples of groups 5 6 S01T/,e examples of groups The set ofresidue classes modulo nisdenoted by Zn;itcontains The subject ofmultiplication ofresidue classes and the forma nelements, each being an infinite set of integers. tion of multiplicative groups requires some care. The usual definition ofmultiplication isthe obvious one: Example 1.10. We define a binary operation called addition on residue classes by the following equation: [abJ== [aJ[bJ. It isessential toverify that this isaproper definition. This may [a+b] == [aJ+[bJ. be carried out in a way similar to that explained in the case of This.merely says that the sum of the residue classes containing addition, the key step being the remark that the integers a and brespectively is to be that containing a+b. (a+kn)(b+ln) == ab+(la+kb+kln)n yve ought first, however, to have verified that the result [a+b] and so [(a+kn)(b+ln)J == Cab]. ISthe same whateverrepresentatives we choose in [a]and [b],so The reader ought to fillin all the details for himself. that the definition depends onlyonthe classescontaining aand b, Unfortunately this multiplication of residue classes will not not on these numbers themselves. This weproceed to do. Take generally yield groups. Just as the number 0 cannot occur in any element of[a];it willbe a+kn for someinteger k. Similarly multiplicative groups ofnumbers sothe residue class [OJcannot an arbitrary element of [b]will be b+ln. Thus occur in multiplicative groups of residue classes when n> 1, [a]== [a+kn], [b]== [b+ln]. forhow can w*e' find an inverse element for [O]? Remember that [O][x] == [0] [1] for all x. There is another complication. We require [a+knJ+[b+lnJ == [a+bJ Consider residue classes modulo 4, for which if we are to have a proper definition ofsum. But [2][2] == [4] == [OJ; and sothe product oftwo non-zero classes may be [0]' We must [a+kn]+[b+ln] == [(a+kn)+(b+ln)], ~xclude [OJfrom any group, and having done that there are still and since (a+kn)+(b+ln) == a+b modulon cases when we do not have closure. We can, however, salvage something. we have [(a+kn)+(b+ln)] == [a+bJ, Example l.ll. The set of non-zero residue classes modulo p and so all iswell. forms a group under multiplication, pbeing any prime number. J The group axioms are easily verified now that the delicate The associative law and the existence ofan identity element [1 matter ofdefinition has been dealt with. Closure isobvious, the are obvious facts, but axioms (i) and (iv) require care in their associative law is well-known or alternatively follows from the consideration. Let [aJ be a non-zero residue class modulo p. corresponding axiom for integers, [0] is an identity element This is precisely the same as supposing that p does not divide [-a Jisan inverse for [aJ. Therefore wehave agroup ofresidu; the integer a. Anelementary and well-known result ofnumber classes, denoted by Z,,, with addition as its binary operation. theory+ states that forsuitable integers u,vwehave the equality Notice that it contains precisely nelements where nis an arbi au+pv == 1. .trary positive integer. We therefore have a solution to the Thus in terms ofresidue classes modulo pwehave problem of finding a group with a given finite number of ele [aJ[uJ== [auJ== [au+pvJ == [l]. ments. t Seethe Appendix. Some examples of groups 7 8 Some examples of groups Nowconsider(i),the closurerequirement, and supposethat [a] THEOREM1.12. The set of allmappings of afixed (non-empty) and [b] are non-zeroresidue classesfor which [a][b] = [OJ. It set into itself forms a semigroup -0 followsfromthe existence ofu and from the commutative and To be quite explicit about the multiplication of mappings associativelawsthat used here, let .pand .pbe two mappingsofour fixedset X into [b] =[a][u][b] =['u][a][b]=[u][O]=[0], itself; then 4,.p is defined by putting x(#) = (x.p).pfor each and wearrive at acontradiction. Therefore (i)holds. Sincewe element x in X. havefoundaninverse[u] to [a], wehave alsoverified(iv). This Wenowhaveameansofconstructing many semigroups. For showsthat wehave a multiplicative group whoseelements are groups ofmappings,however,werequire inversesofmappings, the p-I non-zeroresidue classesmodulop. andwerecallthat amapping fromasetX into X hasaninverse if and onlyif it isboth one-one and cnto.] This explains our Allour examples sofar have been groups with the property interest in the concept of 'permutation'. that everypair ofelements commutes, that isab isalwaysequal to 00, and this, of course, has been a consequenceofthe com DEFINITION. Apermutation ofa set isa one-one mapping of mutative laws for addition and multiplication of real and the set ontoitself. complexnumbers. We omit detailed verification. Example 1.13.ThesetSxofallpermutations ofa(non-empty) set X forms a group. Itis easy to seethat Sx is closed,and DEFINITION. The elements a, b of the group G commute if (sinceSx is a subset of the set of all mappings of X into X) ab =ba. Thegroup Giscommutative orabelian ifeverypair of associativity is a consequence of Theorem 1.12. An identity its elements commutes. element ofSx isthe permutation which leavesevery member L Apair ofelements ofthe form z, xx in any group commutes of X fixed, and which therefore has the property that pL =P because ofthe associative law. foreachelementpofSx. Finally, sincepis one-oneandontoX, There are many groups that are non-abelian, and they are each permutation p h~san inverse mapping p-l, with pp-l = L, oftenbestpresented asgroupsofmappingsofonesortoranother. and it iseasilyverifiedthat p-l is one-one and onto X, that is This seemsto be their natural context. To make this remark p-l is amember of Sx' precise, we recall that there is a natura.l multiplication of Thisgroup Sx isknown as the symmetric group onthe setX. mappings and that this multiplication is associative; details In a sensethat willlater be made precise,the structure ofthe are tobe foundinthe Appendix. The namesemigroup isgiven group Sx dependsonlyon the number ofelementsin X. Since to any non-empty set-with an associative multiplication under there are n! distinct permutations ofa finiteset ofn elements, whichit isclosed. Thus everygroup isasemigroupwithcertain weseethat Sx hasn! elements ifX hasn elements. additional properties; note that axiom(iii)impliesthat nogroup WhenX isfiniteand smallit ispracticable towritedownthe isempty. Onthe other hand many semigroups (forexample, elements ofSx explicitly and to carry out exploratory calcula the even integers with the usual binary operation of multi tions. IfX ={I,2,3}then Sx is plication) failto satisfy group axioms (iii)and (iv). {L, (23),(31),(12),(123),(132)} These remarks, together with the facts presented in the in the usual permutation notation. This particular Sx isnon- Appendix, should make the following theorem and its proof apparent. t Seethe Appendix. Some examples of groups 9 10 Someexamplesofgroups abelian, sincewehave, using the usual multiplication ofmap mapping; clearlyeach translation hasan inverse,whichisagain pings, (23)(31)= (132), a translation; clearly each reflection c/> has c/> as an inverse, because # = t. (31)(23)=(123), Hencewefind two more groups. sothat (23)(31)=1= (31)(23). Example 1.16.Thesetofalltranslations of.!l'(withtheusual Example 1.14. Let Rx be the set of permutations p of the multiplication) formsan abelian group. Itisnot hard tovisual setX suchthat eachpleavesallbut afinite number ofelements izethe group properties for this set. of X fixed. Itturns out that Rx is an interesting group. To prove closure the reader should show that if upis the set of Example 1.17. The set ofalloongruencemappings ofthe line elements of X that are displaced by the permutation p then .!l'formsagroup. Acongruencemapping may be presented as a combination of suitable translations and reflections, which u"'Pt S; up!Uup,. The other axioms are readily verified. This group Rx isknownasthe restrictedsymmetric grouponthesetX. should enable the reader to seehowthe effectof a congruence mappingmaybe'undone' andaninverseconstructed. Theother Example 1.15. Let Ax bethe setofpermutations pofX such laws are easierto grasp. Among points that the reader might that eachpleavesallbut afinite number ofelementsofX fixed, ponder on is the fact that the product of two reflections is a and hasevenparity (that is,p isthe product ofanevennumber translation. of transpositions). Onceagain the reader is invited to prove The congruence mappings of the plane & into itself are a thegroupproperties ofAx undertheappropriate multiplication. little more complicated to describe. There are: Thegroup Ax iscalledthe alternating group onthe setX. (i) Translations-a mapping c/>of& into & isa translation Wenext considercertain permutations ofrather specialsets, if the segment joining P to Pc/>isofconstant length and namelythesetsofpoints makingupEuclidean spaceofone,two, direction for allpoints P. or three dimensions. Initially our account of these groups of (ii) Rotations about a fixed point O-a mapping c/> of F/' mappings will be geometrical and tentative, and it should not intoF/'isarotation if P and Pc/>areat the samedistance be regarded as rigorous. After these intuitive considerations from 0 and ifthe angle subtended at 0by P and Pc/>is weshallexaminegroups ofmatrices, both thosethat arisewhen constant for allpoints P. coordinates are introduced in Euclidean geometry and those (iii) Reflectionsin a fixed line.!l'-a mapping c/> ofF/'into & that result fromgeneralization. isareflectionin.!l' if eachpoint P maps onto its mirror Let .!l'be a line-a one-dimensional Euclidean space. Itis imagein.!l'. obviousinthe intuitive sensethat twosorts ofcongruence(that We note several of the numerous groups formed by sets of is,distance-preserving) mappingsof.!l'onto .!l'exist: congruencemappings ofF/'. (i) Translations-a mapping c/>of.!l' into.!l'isatranslation if c/>movesevery point afixeddistance to the right or left. Example 1.18.Thesetofalltranslations ofF/'formsanabelian (ii) Reflectionsin afixed point O-a mapping c/>of.!l' into .!l' group. is a reflection if c/> maps each point P into the point P,p which ison the other side of0 but at the same distance from O. Example 1.19.Thesetofallrotations ofF/'aboutafixedpoint o Clearly there is a translation t which acts as an identity formsan abelian group. Some examples of groups 11 12 Some examples of qroup« Example 1.20. The setofallcongruencemappings off!Jforms mappings are rotations of f!J about 0 through the angle 2kTr/n agroup. Note that this groupincludesallrotations about every for k=0.1.2,...,n-l, and reflections in the linesjoining 0 to point 0off!J. eachvertex and to the mid-point ofeach side. Itwillbe found that there are 2n such mappings and that these form a group, Inthe case of congruence mappings of three-dimensional aso-calleddihedral group. space 9' into 9' weindicate somefamiliar types: Weshallnowtry tointroduce somemathematical rigour into (i) translations; our geometrical considerations. We take coordinates in the (ii) rotations about afixedline; usual fashion, though we shall not try to relate all the groups (iii) reflections in afixedplane. that now force themselves on our attention to Examples It isaremarkable factthat anycongruencemapping of9' which 1.16-1.24. isacombination oftranslations and rotations and.whichleaves .z Coordinates for arise from the real numbers. Groups onepoint fixed isequivalent to a rotation about a linethrough t corresponding to the congruencegroups 1.16and 1.17are: that fixedpoint. The following groups of congruence mappings of 9' are of Example 1.26. Thesetofallmappings4>d'withdreal,suchthat interest. x4>d =x+d forallrealnumbers z, forms an abeliangroup. The verification Example 1.21. Thesetofalltranslations of9'formsan abelian isomitted. group. Example 1.27.Thesetofallmappingsofthepreviousexample Example 1.22. The set ofallrotations of9' about afixedline together with all mappings <Pa of the type x<pa =;= 2a-x forms forms an abelian group. agroup. The verification isomitted. Example 1.23. The set ofall rotations of 9' leaving a given In higher dimensions points are regarded as ordered sets of point fixed forms a group. real numbers when coordinates are introduced, and we write Example 1.24. Theset ofallcongruencemappings of9' forms such ordered sets as row vectors.. Now an nXn matrix is a group. essentially a mapping of the set of row vectors into itself, the Many other important groupsariseassetsofmappings which row vector being multiplied by the matrix in the usual way. leavesomegeometrical figureinvariant. Weshall giveonecase Let therefore Yen denote the set of allrow vectors such as whichgives agroup that weshall mention several times, refer x =(x1,XZ ••••,x,,), ring the interested reader elsewhere'[for afuller account. where XllXZ, ...,X" are real numbers; and let .At", be the set of Example 1.25.The setofallcongruencemappings off!J'which allnxn matrices, sothat an element A of .AI" has the form leavefixedaregular n-sidedpolygon isagroup. Wemean that the polygon is unchanged as a whole, but not necessarily unchanged point by point; and of coursenmust be an integer A= greaterthan 2. Ifthecentreofthe polygonis0then the required t Aproofmay be foundinJ. L. Syngeand B. A. Griffith,PrillcVples oj mechaniC8,p.280 (3rded.,McGraw·HilI. 1959). t For instance to P. S. Alexandroff,Introduction to the theory oj groups, a"l a,,2 Chapter 6(Blackie. 1959). with the entries ailreal. Some examples of groups 13 14 Someexamplesofgroups Then A definesamapping of&In into gen' namely the mapping Example 1.31. Theset ofmatrices whichtakes each x into xA where {(~ ~):x real and non-zero} xA =(aUXl+aZlXZ+· ..+anlxn, ...,alnxl+aznxz+ ...+annxn)' formsan abeliangroupunder matrix multiplication. Theproof Example 1.28. Theset ..Itnofallnon-singular nXn matrices dependsonthefactthat withrealentriesformsagroupunder matrix multiplication. The closurelawisobvious. Theassociativelawfollows,forexample, X x)(y y) = (XY xy) (0000 00' from the facts that amatrix represents amapping and that the product oftwomatricesrepresents theproduct ofthe associated whioh proves closure, shows that an identity element is the mappings. The unit matrix serves as an identity element, and matrix inwhichx =1,and indicates howto findinversessince an inverse of the element A is the inversematrix A Sowe -1. have a group. Our final example is of lesser interest than multiplicative Example 1.29. The set ofall real nXn matrices with deter groups ofmatrices. minant 1formaagroupunder matrix multiplication. The least obviouspoint intheproofofthis statement isthe closurelaw. Example 1.32.Theset ofallmXn matrices formsanabelian groupunder matrix addition. Theentries mayberational num Many variations onthese examples arepossible. In addition bers, real numbers, complexnumbers, etc. to having infinitely many possibilities for n, we can choose the rational numbers or the complex numbers or other sets Problems of numbers for entries. We can even choose residue classes (Harderproblemsarestarred) modulo somepositive integer. 1. Discussthe followingsystemswith a viewto determiningwhichof Example 1.30. The following set of matrices, in which i themaregroups. denotes a complex number for whichiZ = -1,forms a group (i) Thepositiverealnumbers,withthebinaryoperationofdivision. (ii) Theintegersthat aredivisibleby 10,withaddition. under matrix multiplication: (iii) Thenon-zerorationalnumbers,with abtaken asthe 'product' i) -i) ofa and b. {(oI 0l') (-10 -10') (-10 01') (01 -10)' (i0 0' (0-i 0 ' (iv){a+b..J2:a andbrational,not bothzero}withthebinaryopera- tion ofnumbermultiplication. 0) (i O)} (v)Theintegers,with2a+3btakenasthe' product' ofaandb. -i (o i!' 0 -i . (vi)Therationals,witha+b-ab taken asthe 'product' ofaandb, (vii)The set ofall 2x2singularmatriceswith real entries, under Theverification isomitted. Thisgroup iscalledthe quaternion matrix addition. (viii)The set of all vectors in three-dimensionalEuclidean space, group. withthe binaryoperationofvectorproduct. The multiplicative semigroupofallnXnmatrices with (say) (ix) Theset{,pk: kreal}ofallmappings,pk' where real entries contains many subsets which form multiplicative Xc/>k =kx, groups not containing the unit matrix. We exhibit one non of2'into 2'. (x) {a+b..J2+c..J3: a, b, c rational, not all zero}with tho binary trivial specimen. operationofnumbermultiplication. Someexamplesofqroups 15 2. Whichofthe groupsinthe previousquestionare abelian? Whichof 2 the restare semigroups? 3. Showthat thenon-zeroresidueclassesmodulo6donot formagroup under multiplication. Provethat ifthenon-zeroresidueclaesesmodulonformagr\)upunder multiplication andn > 1then nisprime. Basic theorems and concepts 4. Aset ofmatricesformsagroup under matrix multiplication. Show that oithereverymemberofthesetissingularoreverymemberisnon singular. 5. Let p beany prime number. Showthat the followingset formsan additive group: {m/pn: mand nintegers}. THE theorems inthis chapter areoftwo sorts. First, wemust x4> x Prove that themapping4>forwhich =p»,where isanarbitrary deducesomefacts fromthe axioms. Second,wehave to discuss element ofthe set,isapermutation. the important concepts of homomorphism and isomorphism. 6. Let Gbeagroupofpermutations onasetX. Showthat thoseper mutations whichleavefixeda givenolemontx ofX formagroup. The facts appear rather pedantic unlessthey areviewedagainst a background of systems that are not groups, while the con 7. LetTnbothosetofn Xn uniteiangulurmatriceswithrealentries(that ismatriceswith 1foreachentry onthe principaldiagonaland0every cepts are ofan importance that cannot be over-emphasized. wherebelowthisdiagonal). Showthat Tnisamultiplicative group. Werecall our definition ofagroup asasemigroupwith right S. Let T~bethe setofnXnunitriangular matricesTw:ith residueclasses identities andright inverses,inasensemadepreciseinChapter 1. modulopforentries(pisafixedprime). Showthat isamultiplicative Sometimes a definition is given that states that each right group,and findthe number ofelementsinit. identity isalsoaleft identity and that eachright inverseisalso *9. LetSbeanysetand Gbeanygroup.' Provethat ifmultiplicationof a left inverse; but wecan deducethese facts fromour axioms. tho.mappingsiseuitablly definedthenthe setofallmappingsofS into G formsagroup. THEOREM2.01. If the element x of any group G has X-I as a 10.ThesetofallorderedtriplesofintegersisdenotedbyG;thusatypical edleefmineendtaosffGolilsow(txs,:fl,y)wheretx,fl,yare integers. MultiplicationinGis t(hreignhtx)-Iinxve=rsee,.80 thatxx-I =efor some (right)identity elemente, (tx,fl,y)(g,,),,) = (tx+(-l)llg: {3+(-l)Y7], (-I)fyH). Proof. Let ebe a (right) identity element of Gof the kind Showthat Gwiththismultiplication isagroup. mentioned inaxiom (iv). We haveto usethe associativelawin +11.Showthat there isalargest multiplicative group of2X2matrices the form (0 0) W.ithitr'ea enrrescontam..ing 5 _5 ' and find thl.Sgroup. The left-hand side of this equation reduces to x-Ie and so to +12.'AsemigroupShastheproperty that foreachelementxinS thero X-I, by axioms (iii)and (iv);therefore iSsaangeroleump?entdenoted byX-IinS suchthat y=-1 =YforallyinS. Is X-l = (x-Ix)x-I• **13.Let Sbeasemigroupwith thofollowingproperties: But, byaxiom (iv),X-I itself has aninverse, denoted by (X-I)-I, (i)There isanelementesuchthat ex= x forallx inS. such that X-I(X-I)-I =e; and wehave (ii) XFo-Irfeoarcwhhsiucchh=e-le1m=enet.eand foreachx inS there isanelement X-I(X-I)-I = {(x-IX)X-I}(X-1)-1. Prove that the set{Be: BES},whereeisfixed,formsagroup. The left-hand side is e by (iv); and the right-hand side is Some example.<! ofgroup8 15 2. Whichofthe groupsinthe previousquestionare abelian? Whichof 2 the restare semigroups? 3. Showthat the non-zeroresidueclassesmodulo6donotformagroup under multiplication. Provethat ifthenon-zeroresidueclassesmodulonformagt'(lupunder multiplication andn > 1then nisprime. Basic theorems and concepts 4. A sotofmatricesformsagroup under matrix multiplication. Show that either everymemberofthe setissingularoreverymemberisnon singular. 5. Let p beany prime number. Showthat the followingRetformsan additive group: . {rn/pn: mand nintegers}, Prove that the mapping", forwhichxrp =px,wherex isanarbitrary THE theorems inthis chapter areoftwo sons. First, wemust element ofthe set, isapermutation. deducesomefactsfromthe axioms. Second,wehaveto discuss the important concepts of homomorphism and isomorphism. 6. Let Gbeagroupofpermutations onasetX. Showthat thoseper mutations whichleavefixedagivenelementx ofX formagroup_ Thefacts appearrather pedantic unlessthey areviewedagainst a background of systems that are not groups, whilethe con 7. LetTnbethesetofnXnunitriangular matriceswithrealentries(that ismatriceswith 1foreachentry ontheprincipaldiagonaland0every cepts are ofan importance that cannot be over-emphasized. wherebelowthisdiagonal). Showthat Tnisamultiplicative group. Werecallourdefinitionof agroup as asemigroupwith right 8. Let T~bethesetofnXnunitriangular matriceswith residueclasses identities andrightinverses,inasensemadepreciseinChapter 1. modulopforentries(pisafixedprime). Showthat ~':isamultiplicative Sometimes a definition is given that states that each right group, and findthe number ofelementsinit. identity isalsoaleftidentity andthat eachright inverseisalso *9. Let Sbeanysetand Gbeanygroup.'Prove that ifmultiplicationof a leftinverse; but wecan deducethese facts fromour axioms. tho.mappingsissuitably definedthen the setofallmappingsofS into G formsagroup. THEOREM2.01. If the element x of any group Ghas X-I as a 10. ThesetofallorderedtriplesofintegersisdenotedbyG;thusatypical elementofGis(0<,p,y)where0<,p,Yareintegers. MultiplicationinGis (right) inverse, 80thatxx-l =efor some(right)identity elemente, definedasfollows: then x-Ix = e. (o<,p,y)(g,,,),O =(o<+(-l).Bg;p+(-l)Y"), (-I)fy+S). Proof. Let ebe a (right) identity element of Gof the kind Showthat G withthis multiplication isagroup. mentioned.in axiom (iv). We haveto usethe associativelawin *1.1. ShoIw' that there i.s.alar(g0est0m)ultiplicative group of2X2matrices the form Withrea entriescontammg 5 _ 5 ' and find thi.s group. The left-hand side ofthis equation reduces to x-Ie and so to *12.'AsemigroupS hastheproperty that foreachelementxin Sthere x-I,by axioms(iii)and (iv); therefore isanelementdenoted byx-1inSsuchthat y=-l = YforallyinS. Is = S agroup? X-I (x-1X)X-1. "13. Let Sbeasemigroupwith the followingproperties: But, by axiom (iv),X-I itselfhasaninverse,denoted by (X-I)-I, (i)ThereisBJlelementesuchthat ex =x forallx inS. such that X-1(X-I)-1 =e;and wehave (ii) xF-o1rfeoarcwhhsuicchh=e-le1m=ente.eand foreachx inS there isan element X-1(X-I)-1 = {(x-IX)X-1}(x-1.)-1. Prove that the set {se:eES}, whereeisfixed,formsagroup. The left-hand side is e by (iv); and the right-hand side is Basic theorems and concepts 17 lR Basictheoremsandconcepts (x-IX){X-I(X-I)-I} by the associativelaw,and clearlythis equals In allourexamples,there wasonlyoneidentity elementand X-IX. Thus wehave proved that X-IX isthe identity element e each element had only oneinverse. as required.0 ' The above theorem impliesthat Xand X-I alwayscommute. TIrEOREM 2.04. In any group G there is only one identity This isperhaps not surprising to us, for it happened in all the element, and eachelement of Ghas only oneinver8e. examples ofthe previous chapter. The reader should consider Proof. Suppose wehad two identity elements, say eland e2• ~het~er th:~e is a correspondingtheorem in a semigroupwith Sincee2isaright identity ele2 = el,andsince(byTheorem2.02) right identities and left inverses, such as a set withxy defined el is a left identity e1e2=e2• Therefore el = e2, and Gcannot to be x. (Wediscussed such objects in Chapter I. Here isa have more than oneidentity element. specificinstance. Itwill be found that ifSis Supposethat theelement aofGhasboth X and y asinverses. = = Then ax ay, and by left cancellation X y; therefore a-I is unique.0 then matrix multiplication in Sissuch that the product oftwo THEOREM 2.05. In any group, x = (X-I)-I. matrices isthe left-hand factor.) Proof. Wehavedefined(X-l)-lasasolutiony ofthe equation Another readily acceptable fact follows. x-Iy =e;weknow that eachofe,x-l, y isunique. But clearly = THEOREM2.02. If xisany elementofany group Gwithidentity y x is one solution to x-1y =e, by Theorem 2.01. Sinceit = element e,such that gg-l =efor all gin G,then ex =z. isthe onlysolution, wehave z (X-I)-I·O The next result isofa different character. Itisageneraliza Proof. The previous theorem and the axioms are used as tion ofthe associative law that appeared as axiom (ii) and it follows: ex = (=-l)X =x(x-1x) = xe = x. states, roughly, that a product of group elements can be Thus the result isproved; x and ealways commute.0 bracketed in any way onepleases. TH~OREM2.03. If a and bareelements of any group G,and if THEOREM 2.06. Letfixed elementsaI'a2,... ,anbetaken in order there ~8an element xfor whichxa = xb oran elementyfor which in any group G. Then thevalue oftheproduct ofall theseelement8 ay = by, then a =b. in thegiven orderis 1~naffectedby the 8equencein which product« PrOOf· Suppose xa = xh. The existence of inverses gives areformed. X-l(xa) =x-1(xb). Proof. Themeaningofthis theoremissubtle. Forthepurpose ofillustration letn=3.Thestatement thenisthat theelements Useofthe associative lawand Theorem 2.01gives al(a2aS) and(ala2)a3arethesame,forthereareonlytwopossible ea= eb. ways of forming products; either form a2a3 first or form ala2 Theorem 2.02nowgivesa =b. The deduction from ay =by first. The statement does not suggest that (ala2)aS = (a2aI)a3, nee~ only the ax.iomsand being thus easier isomitted.0 which is certainly false in general. When n= 4 the possible This theorem grves us left and right cancellation laws in an products ofaI'a2, aa,a4inthat orderareal{a2(aaa4)}, (ala2)(aaa4), obvious sense. The reader should consider whether there are al{(a2a3)aJ, {al(a2aS)}a4' {(a1a2)aa}a4, and 110 others. The such lawsin the set S ofmatrices mentioned above. theorem impliesthat these products are all equal. 858187 o

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