; TeX output 2010.07.06:1930nL͍color push Black color pop3WL͍Wcolor push Black color pop\$"V cmbx10FINITE b> cmmi10p-GR9OUPSINREPRESENT ATIONTHEORYOcolor push Black color popO|{Ycmr8N.XMAZZAS&ffffff(_*- cmcsc10Contents K`y cmr101. Essentialsd [2 1.1. T*oGolbagdt2 1.2. F*rattiniUUsubgroup:%5 1.3. SeriesUUofgroupsandHall'scountingprinciple)8 1.4. RegularUUp-groupsand2Aof2@groupsautomorphismsofG,i{respGectively*.Hence,a2@subgroupH?ofGis;char}'acteristic-inqvGqwif'(H)=H, forqwall'2Aut(G).%W*eqwwriteH%S-s6cmss8charGqvifHAtischaracteristic;inƵG.Inparticular,Z(G);n(G);[G;G]arecharacteristicsubgroupsofG,bforany nite;groupUUG.6color push Blackc(vii) color pop;SylHqƴpM[T(G)UUdenotesthesetofSylowp-subgroupsofG. N76color push Blackm(viii) color pop;SuppGosethatGisap-group. F*oranypGositiveintegerd,Fmwede netwocharacteristic;subgroupsWvofWuGbyconsideringelementsofGoforderp^d,WandtheelementsofGwhichare;p^d-thUUpGowers:kvW d(G)=hxUUjxprO \cmmi5d <=10ri and$w( msbm10fd(G)=hxprd 'yjUUx2G1˸iUU:32;W*eÅcallthemome}'ga-dGofFGÆandagemo-dofGG.WIntheÆliterature,wemaysometimes nd 9;theUUnotationG^prd 'yorfd(G)insteadoff^d(G).7썑Although& it& isknownthat[G;G]6=f[a;b]& j& a;b2Gg,/a& counterexample& tothisstatementishardto nd.qHere,UUwequoteexamplestakenfromRotman[Rot95@,2.43].color push BlackExample^ 1.2.GT color popThe5 rst6exampleisduetoP*.J.Cassidy(~1979).jLetk̲bGea eld,andconsiderthepGolynomialringR߲=kP[x;y[ٲ]in2vqariablesxandy[ٲ.^;RegardSZ=kP[x]andT*=kP[y[ٲ]assubringsofRDz,UUandhencesetcG=u cmex108 < : A(a;b;c):=Z0 @d 1a*&]c 01**b 00)J1Z.K1 .KAThen,UUGisagroup(forthemultiplication).qIndeed,32V/A(0;0;0)G=I3C2G}n{A(a;b;c)A(d;e;f)G=A(a8+d;b+e;c+fLo+ae)2Gu A(a;b;c)1G=A(a;b;ab8c)2G>W*eUUclaimthat [G;G]=fA(0;0;c)UUjc2RǸg:Indeed,UUbylinearity*,ifPc=X ti;j㉵ij xiTLy[ٟjY2R Ȳthen'A(0;0;c)=Y t+i;j8ޟbcA(ijxiTL;0;0)UU;A(0;y[ٟj;0)b*;whichUUshowsthat[G;G]=fA(0;0;c)UUjc2RǸg: color push Black color popnL͍color push BlacktFINITEp-GROUPSINREPRESENTZATIONTHEORYpD3 color pop37L͍W*edwantctoshowthatcA(0;0;c)isnotcacommutatorforsomecj2R+(thepreviouscargumentsays that7itisaproGduct7ofcommutators).gNotethatforeverycommutatorA(0;0;c)7in7[G;G],=therearea;a^0Q2SandUUb;b^02TsuchUUthatԱc=ab08a09b:-xNow,Hc=x^2β+Zxyu3+[y[ٟ^2d2RYYdoGesEnothavesuchaEdecompGositionab^0瓸[a^09b,HwhichsaysthatA(0;0;c)isnotacommutator,despitebGeinganelementof[G;G].T*oprovethislatterclaim,seta= iTLx^iandUUa^0Q=P USiTG ^ z0;ZiTLx^iinS.qThen,wehaveequationsP 0|sb08 z0፱0bx]=y[ٟ2,P 1|sb08 z0፱1bx]=yP 2|sb08 z0፱2bx]=1tNow,dR==ukP[x;y[ٲ]aisaanin nitedimensionalvectorspaceoverakP,dwithbasisallmonomialsx^iTLy[ٟ^j.pBut theu}threeequationsu~abGoveu}leadtoacontradiction,}sincethelinearlyindependentsetu~f1;y[;y^2Lgu}isintheUUsubspacespannedbythe2vectorsb;b^09.aObserveNthatthisexampleworksforaN niteorin nite eld,1givingthusbGoth niteandin nitegroupsUUcounterexamplesto[G;G]bGeingthesetofcommutatorsintheelementsofG.W*eealsoerecordthatusingdatabasesofsmallgroups,iRotmanfoundthatthesmallestgroupgivinganother1.6color push Black򪨲(ii) color pop;Everynontrivialnormalsub}'groupofGinterse}'ctsZ(G)nontrivially.6color push Black㋲(iii) color pop;IfjGjp^2|s,thenGisab}'elian.6color push Black*(iv) color pop;Everymaximalsub}'groupofGhasindexpandisnormalinG.6color push BlackIJ(v) color pop;Gisnilp}'otent.Inparticular,H<NGڲ(H)foranypropersubgroupHcofG.6color push Black*(vi) color pop;Hc3 Rcmssi8char+K~4EG=UX)HEGandHccharKKcharIG=UX)HccharG.6color push Blackc(vii) color pop;F;oranyHG,themappingG\mg cZ:NGڲ(H)=CG(H)Ψ ! Aut(H) g[CGڲ(H)Ψ7v! ^d!cg :ZH |!4}Hh |7!4}cg(h)=^gwhfS^;isaninje}'ctivegrouphomomorphism.LetUUuspGointoutthatproperties(vi)and(vii)holdforarbitrary nitegroups. color push Black color pop#)nL͍color push Black48N.MAZZA color pop37L͍ThehproGofgofthe rstitemabGovehisaconsequenceoftheclass!e}'quation.RecallthatifagroupG actsUUona nitesetX,thenforanyx2X,Q0vϸjG:CGڲ(x)j=jG8xj whereUUCG(x)=fg"2GUUjg8x=xg;andGx=fg[xUUjg"2GgdenotevrespGectivelythevstabiliserofxinG,QandvtheG-orbitof͵x.֊SinceCGڲ(x)isavsubgroupofG,wechavecthattheorbitshavecsizedividingtheorderofG.Now,gtakingXd=߁Ga nitep-groupandthejKjthenHom߀GZ(HA;K)=;,0whereasifjHj=jKj,1theneitherHom߀GZ(HA;K)=;,QjorPnHom%ǟGߡ(HA;K)PoisasetofgroupPnisomorphisms(i.p%e.H mandPoKareG-conjugate).NotePoalsothatAut%ğG,g(G)UUisthesetofinnerautomorphismsofG.InSection6,+welistseveral\well-knownfacts"abGout nitegroups.twHereisaselectionofresultswhichUUwerecordapart.color push BlackLemmaT1.6. color pop[Isa08<,UULemma4.6]L}'etMGbea niteLgroupandAanabeliannormalsubgroupofGLsuchthatG=qAiscyclic.Letxݍ2ݎGsuchthatxAgener}'atesG=qA.ThenT[G;G]=f[a;x]ja2AgT͍+3= UNA=qA8\Z(G):Inwparticular,:theproGductoftwowcommutatorsofGcanbGeexpressedasacommutatorofweight2(seeUUDe nition1.18bGelow)oftheform[a;x]forsomea2A.*color push BlackPr}'oof. color pop4CChoGoseUUx2GsuchthatG=qA=hxA 7i,andde neSеc:mA!A by=G(a)=[x;a] forUUalla2A.Notethatiswell-de nedsince[x;a]>=> (a^1 t)^xa2Aforalla>2A.Moreover,.isagrouphomomorphism: 荑~1G(ab)=[x;ab]=[x;b][x;a]b\=[x;b][x;a]=G(a)(b)7wherethesecondequalityisgivenbyacommutatoridentity(seeSection6),andthethirdoneholds bGecauseB;bothbBshowthatim(G)=G^09.Since=clearlyim(G)G^09,wit=is>enoughtoprove=thatimn(G)EGUUandthatthefactorgroupG=im ()isabelian.qW*eobservethatJimUd(G)ANGڲ(im t()) and$wx'z((a))D=x%T[x;a]=[x^^?[x;y[;zp]=[[x;y];zp] forUUanyx;y;z72G;forUUanygroupG (leftassoGciativity).color push BlackLemmaT1.7. color popL}'etGbea nitegroup,letHA;K(;LG,andletx;y[;z72G.6color push BlackqŲ(i) color pop;(Hall-Wittidentity) 8[x;y[ٟ^1 M;zp]^y·[y[;z^1 - ;x]^z[z;x^1 t;y[ٲ]^x=1.6color push Black򪨲(ii) color pop;(Thr}'ee-Subgrouplemma) 8If[HA;K(;L]=1and[K;L;H]=1,then[L;HA;K]=1.}color push BlackPr}'oof. color pop4CTheeHall-Wittidentityeisdaroutinecalculationleftasanexercise.F*ortheThree-Subgrouplemma,suppGosethat[HA;K(;L]=[K;L;H]=1.qPickanyh2ɵHA;a5kd`2KmandlF)2L.pThen,[h;kP^1 ;l2`]^k=[kP;l^1 Ե;h]^l=1@ebyassumption.jThus,DtheHall-Wittidentityyields[l2`;h^1 t;kP]^h=1,Dandthereferore[l2`;h^1 t;kP];=;1toGo.BSincethisholdsforanyh2;HA;Ek2KRandlmq2L,Pweconcludethat[L;HA;K]=1.m}Infact,*mtheThree-SubgrouplemmaasstatedabGoveisequivqalenttoaresultofP*.Hall(byreplacingGUUwithG=qN).color push BlackLemmaT1.8. color pop[Hup67w,UUHilfsatzIGII.1.10UU(b)]Supp}'ose thatNE{G.AIf[[A;Bq];C]|N;and [[Bq;C];A]N,/then[[C(;A];Bq]|N.AIn p}'articular,.ifA;Bq;C~4EGthen[[C(;A];B][[A;B];C][[B;C];A].W*eUUendthissectionwiththefollowingobservqation.color push BlackLemma1.9. color popL}'etTpbeTanoddTprime.wAssumeATandBaresubgroupsTofexponentpTofap-gr}'oupG,suchthatANGڲ(Bq)andsuchthat[B;A]CB (hA;Bqi).ThenhA;Bihasexp}'onentp.}color push BlackPr}'oof. color pop4CLetha62Agandb2Bq.SetgzͲ=[b;a],sothatgabzͲ=baandz͸2CB (hA;Bi).Then,forany nonnegative[integern,wehavethat(ab)^n8=a^nq~b^nzp^dI,withd=&hKn(n1)Kʉfe_r 12.AHence,forn=p,theequalityVZreads\(ab)^p=Eεa^pRb^pzqʳp(p1)ʟW %P 2>=E1.USinceANGڲ(Bq),^anyelement[ofhA;BicanbGewrittenasxEβ=E͵abforUUsomea2AUUandb2Bq.qConsequently*,UUx^pfj=1forallx2hA;Bi.4x/color push Black color popExerciseT1.4.qDzProveUUthattheassumption[Bq;A]CB (hA;Bqi)UUinLemma1.9isnecessary*.X1.2.F rattiniTsubgroup.ThekF*rattinisubgroupislacharacteristicsubgroupde nedforany nitegroup. TherelationshipbGetween˯agroupanditsF*rattinisubgrouparecomparableˮtothatofanalgebraanditsJacobsonradical.oInOthissection,PUwereviewtheOtipoftheicebGergofitsproperties,PUandOreferto[Hal59G!,12.2],toUU[Gor80Ò,5.1,6.1],to[GLS96!,2],andto[Hup67w,IGII.3]UUforfurtherdevelopments.color push BlackDe nitionP1.10.8 color popLetGbGea nitegroup.KXTheF;r}'attini)subgroupofGistheintersection(G)ofallthesmaximalssubgroupsofG.&W*eset(1)=1.&ThesF;r}'attini[factorgroupsissthefactorgroupG=(G).TheUUF*rattinisubgroupprovidesusefulinformation,orrather\non-information".color push BlackTheoremT1.11. color popL}'etGbea nitegroupandlet'εx1|s;:::;xn82ZG.Then,>^otG=hx1|s;:::;xn+i0(UX)G=h(G);x1|s;:::;xnF㕸i:InUUotherwords,theF*rattinisubgroupofGisspannedbythe\non-gener}'ators!"ofG. color push Black color popUnL͍color push Black68N.MAZZA color pop37L͍color push BlackPr}'oof. color pop4CSuppGose;hthatGF=Fh(G);x1|s;:::;xnF㕸i.$LetG0=hx1|s;:::;xn+i,tandsuppGosethatG0a nitegroupandNZasubgroupofG,orQahomomorphicimageofG.W*eUUwanttocompare(G);(N)and(Q).qTheanswerwegiveistakenfrom[Hup67w,IGII.3].qcolor push BlackTheoremT1.12. color popL}'etGbea nitegroup,andletN;HGwithN3EG.z6color push BlackqŲ(i) color pop;Supp}'osethatNandHcarepropersubgroupsofG.IfG=HN,thenN36(G):6color push Black򪨲(ii) color pop;Supp}'osethatN3(H).ThenN(G).Inp}'articular,(N)(G). 6color push Black㋲(iii) color pop;L}'et'beagrouphomomorphismwithdomainG.MThen,ݓ'bW(G)b\ob'(G)b.Inp}'articular, ;ifq'qisthepr}'ojectionqmapG!G=qN,x~thenq(G)NA=N3(G=N).+Mor}'eover,x~ifqN3(G),;then(G)NA=qN3=(G=N).color push BlackRemark 1.13.i color popWithZtheYsameassumptionsasinTheorem1.12,wepGointoutYthatifH<Gthen(H),neednot,bGecontainedin(G).Also,atheinclusion(G)NA=qND,(G=N)may,bGeproper.Indeed,UUletusconsidertheexampleofasemi-directproGduct;G=hx;y.jUUx5C=y[ٟ4d=1;y#ܵx=x2w~iT͍+3= UNC5So8C4ȵ:T*akeUUN3=hxiT͍+3= UNC5ȲandH=hy[ٸiT͍+3= UNC4|s.qThen@֋(G)NO\8H=1 andUUso0c(G)=1 and$w(G=qN)T͍+3= UN(H)=hy[ٟ2 kiT͍+3=C2ȵ:CuyThus,@.(H)6(G) and$w(G)NA=qN3=1<(G=N)UU:𠍍color push BlackPr}'oof. color pop4CF*or/(i), suppGosethatNKisanormalsubgroup0ofGcontainedin(G)andthatG.=-HN. Then,UUwehaveXndG=HN3=H(G); whichUUimpliesP㔵G=H byUUTheorem1.11.:ByUUcontrapGositive,ifG=HNlpwithH+3>=`CpR.3Thus, IM[G;G]f^1|s(G),forallmaximalsubgroupsofG,whence(G)[G;G]f^1|s(G)toGo.Conversely*,we needtoshow thatifN7isanormalsubgroupofGsuchthatthefactorgroupG=qN7iselementary UMabGelian,3then+;(G)N.cHence,let+;us+:picksuch+;anNBVandregardthefactorgroup}fe̟G =G=qNBVasanFpR-vectorUUspace.qLetf~fe 3gx1 3;:::;~fe (gxn BgbGeabasisof}fe̟G 2!.Hence,J+],G=hx1|s;:::;xnq~;N9&Ÿi>hx1|s;:::;MSxi ;:::;xnq~;NXki forUUall-x1inUU;J,wherethesymbGol>xiCmeansthatweremovetheelementxiafromtheset.F Thestrictinequality impliesythatxif=2.I(G)foralli,bythecontrapGositiveofTheorem1.11.3Therefore,wemusthavethatUU(G)N,asrequired.qThestatementfollowsbytakingN3=[G;G]f^1|s(G).F*orUU(ii),weneedtoprovethateverycommutatorisaproGductofsquares.qThus,writenqb#[x;y[ٲ]=x1 ty1 Mxy"=x1 ty2 Mxx2xyxy"=(x^YseriesofGisthenormalseriesofGde nedrecursivelyby0C=G;1=[G;G];and1k=[k+B1(;k+B1],kforall0k2.PAgroupwhosederivedseriesconvergesto10issolvable.;TheUUcompGositionfactorsofasolvqablegrouparethusallofprimeorder.6color push Black*(vi) color pop;Ac-c}'entralseriesc0isc1anormalserieswhereHiTL=Hi1n0ZbLsG=Hi1 bHforall10i1n.ZNotc0all ; nite9groupshave9acentralseries(e.ganonabGeliansimplegroup).hA9groupGwhichhasa;centralUUseriesisnilp}'otent.qW*edistinguishtwoparticularcentralseriesofG:;color push BlackUV(a)u color popP<The>2lowerj#c}'entralseries,xior>2,m#R cmss10LCSforshort,isthecentralseriesofGde nedbytheP<commutatorUUsubgroups: EcG1C=G ,G2=[G;G] and$wGk=[Gk+B1(;G]=[G;G;kP] forUUall-xk2UU:;color push Black(b)u color popP<Theفupp}'er!centralseries,EorفUCS,de nedastheseriesقofGstartingwithZ0C=1;qZ1=P<Z(G)andsuccessively*,Zk됲(G)istheuniquenormalsubgroupofGcontainingZk+B1((G)P<andsuchthatZk됲(G)=qZk+B1((G)W=Z(G=Zk+B1((G)).uIfthereisnoconfusion,wesimplyP<writeUUZk@insteadofZk됲(G),andwecallitthekP-thc}'entreUUofG.ՍAnhessentialfactabGoutanynilpGotentgroupGisthatthehLCSofGconvergesto1ifandonlyifsodoGesr/itsr0UCS.ThenbGothserieshaver/samelength,yfwhichr/wecallr/thenilp}'otencesclassDP(orsimplyr/theclass)VwofVxG.u.Thatis,VtheclassofGistheintegernsuchVxthatGn:z6=1=Gn+1.u.ThenilpGotenceVwclassofa nitegroupisthesmallestlengthofanycentralseries,whichalsomeansthatbGoththeLCSandthe^QUCSare^Pthemore\ecient"centralseries.Notealsothata^PgroupisabGelianifandonlyifitisnilpGotentUUofclass1.RoughlyspGeaking,nilpotentgroupsare\the*e}'asiestgroups"toconsider,aftertheabGelianones,intheasensethatatheyarethedirectproGductsoftheirSylowp-subgroups,dandtheirpropGersubgroupsare3propGerlycontained3intheirnormalisers.f]ThemotivqationformentioningaparttheLCSandUCSofanilpGotentgroupisthatthegroupsintheseriesarecharacteristicsubgroups.EWW*ereferto[Gor80Ò,xUU2.3]and[Khu98w,x2.3]forfurtherdevelopmentsUUonnilpGotentgroups(notnecessarily nite).Inxparticular,`all nitep-groupsxarenilpGotent,andtherefore,theLCSofap-groupxGisstrictlydecreasing.oItfollowsthat,ifjGju=p^nq~,thenGhasatmostclassnY1,inwhichcase,wesaythattheUUgrouphasmaximalclass.qThesegroupsarediscussedin[Hup67w,xIGII.14].NextUUareHall'sc}'ountingprincipleUUandtheresultsin[JK75]thathaveUUbGeenobtainedfromit.֍color push BlackTheoremT1.19. color pop[Hal34G!,UUTheorems1.41and1.51]L}'et:Cbea:classofproper:subgroupsof:ap-groupG.F;oreach:subgroupH of:G,dTletn(H)b}'ethenumb}'erofelementsofC)>containedinH.Then,~t n(H) ~XN]鍍JPM,beanyofthefollowingclassesofp-groups:~6color push Black color pop;elementaryab}'eliansubgroupsoforderp^kwforsome xedk5;6color push Black color pop;ab}'eliansubgroupsoforderp^kwforsome xedk5;6color push Black color pop;ab}'eliansubgroupsof xedindexporp^2|s.Supp}'osethatC]6=L;.ThenthenumberofelementsofC)(inG)isc}'ongruentto1q(moGdp),exc}'eptintheindexp^2Zc}'ase,wherethenumbercanalsobeexactly2.Itfollowsthat:~6color push BlackqŲ(i) color pop;ifGEX\forap-gr}'oupX,thenthereareelementsofC)>whicharenormalinX;and6color push Black򪨲(ii) color pop;ifpt>u2andXZisap-gr}'oupcontaininganabeliansubgroupofindexp^3|s,яthenXZhasa;normalsub}'groupofindexp^3|s.r׍1.4.RegularTp-groups.AnotherUUusefulconceptfor nitep-groupsduetoP*.Hall:qr}'egularity.r֍color push BlackDe nitionT1.21. color popLetUUGbGea nitep-group.qW*esaythatGisr}'egularhifؠFC8x;y"2G9UUk2Nands1|s;:::;sk2b\ohx;y[ٸibW߷0 *suchUUthat5S(xy)pfj=xpRyp+s1ɍpxݍ1I:::Is1ɍpvk;whereUU(hx;ykٸi)^0#isthederivedsubgroupofhx;yi.r֍Oneathingtokeepinamindisthatmostp-groupsareregular.Aninstanceofanon-regularp-groupisУtheФwreathproGductCp*foCpS(seeSection6).T*osupportthisФclaim,vthenexttheoremsuggeststhatUUweneedtoloGokforirregularp-groupsamong\verylarge"and\complicated"p-groups.color push BlackTheoremT1.22. color pop[Hup67w,UUxIGII.10]Thep-gr}'oupGisregularwheneveratleastoneofthefollowingconditionsholds:~6color push Black8(a) color pop;Ghasclasslessthanp;6color push Black򪨲(b) color pop;jGjp^pR;6color push Black(c) color pop;G^0b iscyclic;6color push Black򪨲(d) color pop;Ghasexp}'onentp;6color push Black(e) color pop;Ghasnonormalsub}'groupofexp}'onentpandorderp^p1.AssumenowthatGisr}'egular.6color push BlackqŲ(i) color pop;Ifx;y"2Ghaveor}'deratmostp^nq~,then(xy[ٲ)^prn Mֲ=1.6color push Black򪨲(ii) color pop;F;or,all+x;yZ2XGandforallnonne}'gativeinteger,n,=thereis,z2XG,=dependingon,xandy[, L;suchthatx^prn y[ٟ^prn =zp^prn U.6color push Black㋲(iii) color pop;jG= nq~(G)j=jf^n(G)j,andmor}'eoverԍ.͟ nq~(G)Z=fx2Gjxprn Mֲ=1g 8and$Ifn(G)Z=fxprn jx2Gg: 8(Se}'eNotation1.1).r֍T*otheabGovelist(a)-(e),letusaddthefollowingobservqations,takenfrom[Hal59G!,x12.4](sectioninUUwhichregularp-groupsarethoroughlydiscussed).color push Black(f)!c color pop;GUUisregularifanysubgroupofGgeneratedbytwoelementsisregular.color push Blackt(g)!c color pop;IfUUGisregular,theneverysubgroupandfactorgroupofGisregular.color push Black;(h)!c color pop;IfGisregular,G^thentheorderofaproGductx1':::|jxl(ofelementsofGcannotexceedthe;orderUUofallofthexiTL's.NoteUUthatitem(h)istheiterativeversionof(i).color push Black color popExerciseT1.7.qDzProveUUthatCp2o8Cpisirregular.m2.AUutomorphismsofp-groupsMostofthereferencesfortheresultsonautomorphismsofp-groupsstatedbGelowcanbefoundinUU[Gor80Ò,Hup67,Khu98]. color push Black color pop nL͍color push Black108N.MAZZA color pop37L͍2.1.FiniteTgroupsandtheirautomorphismgroups.As>noted=intheconstructionoftheexternalsemi-directproGduct(De nition6.7),wwheneverGisa subgrouphofithegroupofautomorphismsofagroupV8,thenwemayiformtheexternalsemi-directproGductHa=VoG.gExplicitly*,CtheelementsofH޲havetheform(x;'),Bwithx2VIJand'2G.Given(x;')and(y[; )B2BH,BtheproGductis(x^'y {;' [ٲ)inH.PHence,Bweset^'y=y=B'(y),Basitisthenatural)way)tode nethisconjugation,2_accordingtotheprinciplesofthesemi-directproGduct.c6Now,letUUusconsiderwhathappGenswiththese\mixed"commutators:qforx2V9andUU'; "2G,wehave0[x;']=x1 tx'=x1'1(x) asʵx'='1(x)UU:Then,UUwemayforinstancecheckthattheequality[x;' [ٲ]=[x; [ٲ][x;']^ `NholdsUUinH%S:#[x; [ٲ][x;'] =x1 t 1 M(x)UU 1bx1ɵ'1 t(x)b\o=x1 1 Mb'1(x)b\o=x1(' )1*(x)=[x;' ]UU:7This\passagetotheinverse"ofthemap,`i.e.67x^'='^1 t(x)explainswhyitissensibletousetherightactionofmapsinsteadofleftone(withthencompGositionreadinglefttoright,e.g.[Gor80Ò]).Theconvention inournotesisthatmapsactonFtheEleftز,asabGove,so thattheorderofacompGositionreads;righttoleftwith;themapapplied rsttoanelementadjacentto;theleftofthatelement.Onntheootherhand,uwhenconsideringasemi-directproGductV̀oGwithVRanelementaryabGelianp-group,UUwecanalsoregardV9asarightjFpRG-mo}'dule[.J color push Black color popExerciseT2.8.qDzCheckUUtheequality[xy[;']=[x;']^y#ܲ[y[;'],UUinH=Vqo8GasabGove.э2.2.LinearTtransformations.SinceYelementaryXabGelianp-groupsarealsoFpR-vectorspaces,Ιthegroupsofautomorphismsofsuchp-groupsqaregroupsofqinvertibleqlineartransformations,yi.e.oAut9(V8)T͍+3= yGL^n!>ܲ(p),yforanyelementaryabGelianXp-groupVQ1,and3by4Lemma2.2,CVɲ(M)isaFpRG-moGdule.Notethatg[ٟ^pY2/Mforallg 2.G.Since(X^1)^pN=/X^p12/FpR[X],gwemusthavethatGtheminimalpGolynomialoftheactionofGanyg"2GonCVɲ(M)divides(Xܸ1)^pR.mLSo,Jifwepickanyܵg{2sG}}M,ֽthereisaneigenvectorv{2sCVɲ(M)fortheeigenvqalue1determinedbythelineartransformationVofg8.onCVɲ(M).SinceMpismaximalinGandUg d=2VM,wehavethatUG=hM;gi.Moreover,UU06=v"2CVɲ(M)is xedbyg[ٲ,andthusv.is xedbyG.G*2.4.Automorphisms.InECthissection,HzweEDdiscussresultswhichrelatetheEDstructureofa nitegroupanditsautomorphismgroup. 0AdditionalZusefulconsequencesarisein[thecaseof nitegroupsofprimepGowerZorder.F*urthermore,someIofthefollowingclassicaltheoremsinvolvetheHF*rattinisubgroupofa nitegroup,complementingUUinsomesenseSection1.2..color push BlackTheoremT2.4. color pop[Hup67w,UUSatzIGII.3.17]L}'et1Gbe0a nitegroup,UandsupposethatGcan0begeneratedbydelements.ThenjAutr(G)jdividesthepr}'oductjAutr(G=(G))jj(G)j^d.rcolor push BlackPr}'oof. color pop4CFixasetfx1|s;:::;xdgofgeneratorsofG.|ConsiderthemapfԲ:Aut(G)i&!AutZ(G=(G))whichsendseveryautomorphism 2 ,Aut(G)totheinducedautomorphism~feog ofG=(G).NotethatUUriswell-de nedsince(G)char G.qSetK~4=ker#(G).qThatis,e\K~4=f В2Aut(G)UUjg[ٟ1 M z(g[ٲ)2(G)8g"2GgAConsiderxallthexorderedsetsM= fy1|s;:::;ydgwithyiZ=xiTLzi6forsomeziY2(G).܅NotethatjMj=j(G)j^d,+4and any elementof MisasetofgeneratorsofG.`:Now,+4themorphismsinKȲactonMjdividingiitintodisjointorbits,sayT1|s;:::;Tk됲.=#Bytheclassequation,eachorbitTi hascardinalityjK~4:KiTLj,0where'Ki{isthestabiliser'ofTi{inK.bSincetheelementsofTi{areorderedsets,0thismeans[PKid=f В2K qjUU z(yj6)=yjĸ81jYd and$w8fy1|s;:::;ydg2TiTLg:Thus,every=morphismin2fxN.FTThatis,y2eCGڲ(')andtheassertionholds.FUAssumenowthattheorder ;ofx'isycompGosite,saymnxwithbothm;n>1,thenxbyyinduction,ןLщfe!џ/CGڲ('rnq~)'Y=Csetfy1|s;:::;yrmgcontainedinOG.˃Now,zeveryelementofGisawords?intheyiTL'sandthuseveryelement7ofGisalso xed8by'^sF:.mSinceweassumethat'isnotthe8identity*,Koitforcess:=o('),as$claimed.-Itfollowsthat$thepGermutationonMde ned$by'dividesM$Ͳintodisjointsubsets,eachYofcardinalityo(').pButMcontainsp^mr elements,[whereaso(')iscoprimetop,whichforceso(')=1,UUi.e.qǵ'=Id .M0ëThereismuchtosayabGoutp^09-automorphismsofp-groups,andwetakeabriefglanceattheseresults.dF*or:*reference,rpropGertiesofautomorphismsofp-groupsarethoroughlyinvestigated:*in[Gor80Ò,5.2,5.3],UUin[GLS96!,11],in[Hup67w,IGII.19],UUandin[Khu98].W*eUUstartwiththefollowingobservqation,adaptedfrom[GLS96!,Lemma11.2].NDcolor push BlackTheoremT2.7. color popL}'etAbeap^09-groupofautomorphismsofanabelianp-groupG.^6color push BlackqŲ(i) color pop;TheF;r}'attinifactorgroupG=(G)isisomorphicto 1|s(G)asFpRA-modules.6color push Black򪨲(ii) color pop;G=[G;A]8CGڲ(A).NDRecallUUthatanormalseriesofGisachain)e$G=K0C8Kn8=1 withUUKidEKi1forall1in.*W*eUUsaythatasubgroupAofAutG(G);stabilisestheseries'vif:color push BlackX(i)!c color pop;'(KiTL)=KiforUUall0in,UUandforall'2A,UUandifcolor push Black;(ii)!c color pop;AUUactstriviallyoneachfactorKiTL=Ki+1 tO.^InMparticular, ALisalsoasubgroupofautomorphismsofeachfactorKiTL=Ki+1 tO.Now, regardingthisinUUthesemi-directproGductH=G8oAUU(seeDe nition6.7),wecanconsidercommutatorsR[x;']=x1 t'1(x) asUUinSection2.1.)SuppGose0that/AstabilisesanormalseriesofGasabGove.WLet1in0andset|fe XUUrՌ:.Ki1 =Ki forthepro8jectionmap.F*oranygѸ2Ki1kϲand'2A,}wehave~fe gg ='(~fe gg )=Lщfer/'(g[ٲ),}orequivqalently*,Lщfe/[g[;']-.=[ fe1, i.e.R[g[;']2KiTL.RThus, ifnAstabilisesthenormalseriesG=K0C8Kn8=1mofnG, then AʸAut̔(KiTL)a&and[Ki;A]ʸɵKi+1ufora'all0ʵi]de ne}'daboveisaLiebracketonL(G).color push BlackPr}'oof. color pop4CW*eDproveDtheassertionmostlyusingthemultiplicativenotationDofthegroupG,H>andleavingthetranslationintotheadditivenotationofL(G)asanexercise.SLemma6.2(iv)givestheinclusion[GiTL;Gj6])Gi+jfor!alli;j."*Itfollows that[;:ɲ]iswell-de nedinthesensethatitdoGesnotdepGendonRthechoiceQofrepresentativesofeachfactorQgroup:sayx;x^0J2GiTL,andQy[;y^0$2Gj6,withRxx^0(moGdGi+1 tO)|andydhy[ٟ^0Ҳ(moGdGjg+1V).W*ritex^0Ȳ=xuandy[ٟ^02=y[vvforsomeu2Gi+1andvdh2Gjg+1V.UsingUUthe rsttwoUUidentitiesinLemma6.2,wehavemH[x09;y[ٲ]=[xu;y[ٲ]=[x;y[ٲ][x;y;u][u;y][x;y[ٲ] (moGdGi+jg+1N)sinceUUbGoth[x;y[;u],UUand[u;y]areinGi+jg+1N.qSimilarly*,wehaveN=[x;y[ٟ0*][x;y[ٲ] (moGdGi+jg+1N) soUUthat[x^09;y[ٟ^0*][x;y[ٲ] (moGdGi+jg+1N).Next,UUweneedtoprovethebilinearityof[;],i.e.M[xx09;y[ٲ][x;y[ٲ][x09;y] (moGdGi+jg+1N) forUUallx;x^0Q2Giandally"2Gj6,andlikewiseforthesecondvqariable(whichisleftasanexercise).JThisisagainaconsequenceof LemmaUU6.2,as ƍu[xx09;y[ٲ]=[x;y[ٲ][x;y;x09][x0;y[ٲ] andUU[x;y;x^09]2Gi+jg+1N.`Finally*,AIwe1and thecongruencetrueforany pGositiveintegermThen,)2usingthesamecommutatorUUidentity*,wehaveFh1=[x;1]=[x;y[ٟ1 My[ٲ]=[x;y[ٲ][x;y1 M][x;y1 M;y][x;y[ٲ][x;y1 M] (moGdGi+jg+1N)Bsince[x;y[ٟ^1 M;y[ٲ]K52Gi+jg+1N,kandlikewiseweobtain1K5[x;y[ٟ^1 M][x;y[ٲ]q(moGdGi+jg+1N),kwhichsaysthat ff[x;y[ٟ^1 M]=[x;y[ٲ]+1%asUUclaimed.)\U color push Black color pop{nL͍color push BlacktFINITEp-GROUPSINREPRESENTZATIONTHEORYl17 color pop37L͍color push BlackRemarky3.9. color popF*romLemma3.7,(ifeveryfactorgroupLiTL(G)=Gi=Gi+1ykintheLCShasexpGonentp, thenVL(G)VisanalgebraoverFpR.Indeed,byVDe nition3.5,weVhaveߵp(xGiTL)=(x^pR)Gid=0inVL(G),forallMxGid2L(G).o0W*ealsorecordMfromLemma3.8,OthatthepassagefromthemultiplicativegroupMGtogroupsandLiealgebras([Laz54]).ItturnsoutthatMichelLazard'sresultsalsoсapplyрinthecaseof nitep-groups,andestablishacorrespGondencebetweenрmultiplicativeс nitep-groupsY7ofnilpGotenceclassY8lessthanpandLiealgebrasalsoofnilpGotenceclasslessthanp.}nLetuspGointoutthatthisisastrongerrequirementthantheequality\xiTLGi+1+eyiGi+1=d(xiyi)Gi+1KlinsomeʵLiTL(G)forhomogeneouselementsxiGi+1 tO;yiGi+1)2ڵLi(G)"giveninDe nition3.5,gandthereare>demanding=computationswhichwedo=notshowinthe=lecturenotes,vsincetheyarenotnecessaryformourpurpGoses.Nevertheless,ronpage22,wegiveanexamplemoftheLazardcorrespGondencewithaUUp-groupofnilpGotenceclass2(andwithpoddforreasonsthatwillshortlybecomeobvious).AKmotivqationkforlusingthislinearisationproGcessisthataseveralquestionsonp-groupsareeasierto.OhandlewhentackledfromthelinearpGerspective,6and.OinSection4wepresentusefulresultsusingthis approach.XNow,inordertostudyit,we rstneed todiscusstheMal'c}'evNcorrespondence[,whichreliesontheBaker-Hausdor Eformulac.EYAsthisrepresentsaconsiderableamountof material,wewillonlyUUsurveythefundamentalsnecessarytous.W*eUUthusstartwithafewnotionsandwe xthenotationwhichweuseuntiltheendofSection3.LetAbGeanonunitalfreeassociativenilpotentQ-algebraofclassc.^!W*ede netheassociatedLieringUUA^ VonthesamesetofelementsasA,andweset ܋[a;b]=ab8ba inUYA,UUforall:a;b2A: Then,}A^ 5&containsu&au%freeLiesubring,}describGedasfollows.8Fixasetofnon-commutinggeneratorsfx1|s;x2;:::g7of6A(wemay7suppGosethenumbGerof7xir niteorin nite)._hHence,)=Ahasabasisformedbyallthemonomialsoftheformxi1;lxi2:::;cxikk,5forsomekK1.hW*esetLfortheLiesubringgeneratedbyalltheb}'asicmonomialsxiTL.A(Inparticular,x1|sx2 a=ED2L,but[x1;x2]Ѹ2L.)AFinally*,wewrite ؍L8QL=haxUUja2QUU;x2LQ޸i forUUtheLieQ-algebraspannedbythexiTL's.qThen,UUQLisafr}'eeLieUUsubalgebraofA^ Vby[Khu98w,Theorem5.39].LetyusxturntoA,whichisnonunital,andaddamultiplicativeyidentity1xtoit.QFixasetofelementsfy1|s;y2;:::gUUinA2C=[A;A],andconsidertheset S0a18+A=f18+aUUja2AgUU: Hence,18+AUUisa(multiplicative)UUgroup. SinceAisnilpGotent(ofclassc),)wecande nethesubgroupGof1+AasthesubgroupspannedbytheUUsubset ؍۫f18+xi,+yijUUyid2A2ȵ;xi2Ag ofUU18+A,pwhereUUthexiTL'sandtheyi'sarethosechosenabGove.Bcolor push Black color popExerciseT3.13.qDzProveUUthat18+AUUisagroup.Inparticular,foranya2A,UU nd(18+a)!1/"218+Aexplicitly*.Wcolor push BlackTheoremT3.13. color pop[Khu98w,UUTheorem9.2]Thegr}'oupGisfreenilpotentofsameclasscasA,withfreegenerators18+xi,+yiTL.VThe0relationshipbGetween0bothconstructions0ofLandofGisthatListheLieringassoGciatedtoGinUUthesenseofDe nition3.5. color push Black color pop nL͍color push BlacktFINITEp-GROUPSINREPRESENTZATIONTHEORYl19 color pop37L͍NoteXNalsoXMthat,usingtheLCS(of nitelength)ofA,weobtainagradingonA,whichinducesgradings onG1+AandonL.mMThatis,JwecanGwriteeachelementintheseLieringsasGasumofhomogeneousterms.[Thatis, A=nq~An, whereAnSisthesubsetofAformedbyalltheLiemonomialsinthexiTL'sofUUdegreen.F3.3.Bak9er-Hausdor Tformula.ThisformulaisalsocalledtheBaker-CampbGell-Hausdor formula. DThepurpGoseistolinkthemultiplicative`groupastructurewiththeadditivestructureofthecorrespGondingLiealgebra,#whichworksDinEourcontextE(amongothers). T*opresentit,CletusrecalltheMacL}'aurin>seriesoftheexpGonentialUUand(natural)logarithmfunctions.qF*orx2RUUinaneighbGorhoodUUoftheorigin:% exz=%-X n0<$x^nwfe ( (֍0rn!#=18+x+<$lx^2lwfe 3 (֍692! +<$lx^3lwfe 3 (֍693!+8:::0and!+x6log(18+x)z=%-X n1 (1)n1<$%x^n%wfe ( (֍n  =x8<$lx^2lwfe 3 (֍Dz2 +<$lx^3lwfe 3 (֍Dz3+:::򂍑In^'asimilarfashion,wede neformal^&exp}'onentialandlogarithmfunctionsforournilpGotentassociativealgebraUUAofnilpGotenceclasscasfollows:qsincea^c+1=0foralla2A,wesetэKa*ea=18+a+<$la^2lwfe ţ (֍bѲ2+g+<$la^clwfe \ (֍ c!Xand3vlog@"(1+a)=a<$la^2lwfe ţ (֍bѲ2+g+(1)c1<$a^cwfe \ (֍c{color push Black color popExercise3.14.gLet7#A7"bGeanilpotentassociativealgebraofclass7"c,=-andleta2A7#andk2Z.gProvetheUUfollowingequalities.saJelog X(1+a)"U=18+a ,qlog%\s(eap)=a; and%(ea)k=e(k+Ba)#ٍConsideragainthesetsoffreegeneratorsfx1|s;x2;:::gͲof̵A./W*enowde nethecorrespGondingelements|y1|s;y2;^2 2A2rwhich|are|neededfortheremainderofthissection.Recallthatsofar,theonlywconstraintxwehaveonthexyiTL'sisthattheyareinA2|s,i.e.h.theyarelinearcombinationsofLiemonomials.0ofdegreeatleast2.dHence,6weusetheformalde nitionoftheexpGonential,6andthefactthatAhasnilpGotenceclassc,andsete^xi W=$1^+xi\+yix\2Gforalli. TheyiTL'sare\hidden"inthestatementoftheBaker-Hausdor formula,Zanditderivqatives.ޢThus,Zweshallnotrefertotheminfuture,UUalsobGecausecomputingthemexplicitlyrequiresaconsiderablee ort.8color push BlackDe nition3.14.J color popTheBaker-Hausdor ڮformulaPc(BHformulaforshort)isthepGolynomialexpressionH(x1|s;x2)UUinQLde nedbytheequationVz+eH(x1 ;x2)#Ӳ=ex1ex2forUUanygeneratorsx1|s;x2ȲofA.UW*eUUwriteHnq~(x1|s;x2)UUforthehomogeneouscompGonentofH(x1|s;x2)UUofdegreeninAnq~.7color push BlackRemarkQ3.15.٥ color popIt*must*bGepointed*outthattheclaimH(x1|s;x2)*inQLisastatementthatrequiresaproGof,anditispreciselywhatiscalledtheBHformula.W*ereferto[Khu98w,Theorem9.11]forthe{demonstration,cwhich{usesLemma3.16bGelow.Also,cthede nitiononlyusesgeneratorsx1|s;x2,butUUitgeneralisestoanypairofdistinctfreegeneratorsinfx1|s;x2;:::g.AsUUaconsequenceoftheBHformula,weobtainthatthesubsetV7̵e)qymsbm7QL5P=felgjUUlx2QLg isanilpGotentgroupofclassc(cf.[Khu98w,Corollary9.14]).UA0sketchofproGofconsistsinobservingthatwehave1=e^0 p2e^QL n8,sothate^QLisnotempty*.Moreover,2for)l1|s;l2C2QL)wehavee^l12e^QL.ande^l1 +l2G=e^H(l1 ;l2)۸2e^QL.bytheBH)formula.cRThus,e^QL"ڲisasubgroupof1xi+xhA.Finally*,uwededucethate^QL"ڲhasnilpGotenceclasscfromthefactthat18+AUUhasnilpGotenceclassc.7F*orUUn=1;2,weobtaintheformulae俍QiH1|s(x1;x2)=x1S+8x2|tand(H2|s(x1;x2)=<$K1Kwfe (֍2 -(x1x2S8x2x1)=<$K1Kwfe (֍2 -[x1;x2]UU:nMoreover, 덑Y[ex1;ex2]=18+[x1|s;x2]8+ 8...(termsUUofdegree3)...c=e[x1 ;x2]G&:?color push Black color popExerciseT3.15.qDzProveUUtheequality[e^x1;e^x2]=e^[x1 ;x2]{inUUthecasewhenc=2. color push Black color popenL͍color push Black208N.MAZZA color pop37L͍AssminslthecaseofthetraditionalLietheory*,zitisalwayssmusefultoconsidertheadjointmap.Inthis particularUUcase,itisusedintheproGofoftheBHformula.32\xadm :mA!EndƟQ(A) with)tad3(a)(b)=[b;a] forUUalla;b2A. color push BlackLemmaT3.16. color pop[Khu98w,UULemma9.10]F;oranygener}'atorsx1|s;x2ZofA,s)add(H(x1|s;x2))=H(ad ;(x1|s);ad 8(x2)): The.termsHnq~(x1|s;x2).for.n1>1IJ2arealsoexpressibleintermsofcommutatorsinx1|s;x2cofhigherweight,UUforwhichwereferto[Khu98w,Lemma9.15].3.4.Mal'cevTcorrespQondence.Both,theTFMal'cevTEandtheLazardcorrespGondenceareconcernedwithdivisibilitypropGertiesofin nitegroups.Thatis,AgivenagroupGandanelementg2G,Awetrytosolveanequationoftheformx^q1=g[ٲ,forosomegivenqIinagivensubringpofQ.6{ItisthennaturaltoaskwhatarethepropGertiesofagroupƿGinwhichanysuchequationalwayspGossessesauniquesolution(seeƾ[Khu98w,DExample1.40]). color push BlackDe nitionA3.17.8 color popLetbGeasetofprimenumbers.Anintegernisa[-numb}'er6ifnisaproGductofbelementscof[ٲ.-Hence,fwewritecQ VѲforthebsetofallrationalnumbGerswhosecdenominatorsare[ٲ-numbGers.AnarbitrarygroupGis-divisibleEifforanygԓ2xGandany[ٲ-numbGerrthereexistsh2GUUwithh^r4=g[ٲ,i.e.qǵhisarG-throotofg[ٲ.F*or instance,$wif7=Ff2g,andG isagroupofoGddorder,$xthenGis2-divisible:2indeed,$xletg72EGwithUUo(g[ٲ)=2a81anda1.qThen(g[ٟ^a2I)^2C=g[ٟ^2a Բ=g[ٲ.color push Black color popExerciseT3.16.qDzProveUUthatanyp^09-groupisp-divisible,foranyprimep.qGeneralisetoanarbitrarysetUU.ofprimes. color push BlackDe nitionT3.18. color popLetUU.bGeasetofprimenumbers.6color push Blackq(i) color pop;AfgroupfѵGfвistorsion-fr}'ee'+ifftheonlyelementof niteorderinGistheidentity*,.which;impliesʔofʕcoursethatGisin niteortrivial.хHence,wesayʕthatGis[-torsion-fr}'eeifʔG;hasUUnononidentityUUelementswhoseordersare[ٲ-numbGers.6color push Black(ii) color pop;AagroupaGisQ-p}'oweredaifaforeverypGositiveintegernaandeveryg⡸2ȵG,thereexistsa;uniquelh2Gsuchmthatg =h^nq~. Thatis,Gismtorsion-freeanddivisible(orequivqalently*,;everyelementofGpGossessesauniqueеn-throot,Rforeachn2N).EByanalogy*,Rwesaythata;group#4Gis#5Qmo-p}'owered#4ifGis[ٲ-divisibleand[ٲ-torsion-free(orequivqalently*,-;every#4element;ofUUGpGossessesauniquen-throot,foreach[ٲ-numbGern). @6color push Black(iii) color pop;SuppGoseuthatuGisanilpGotentgroupandletHEbGeasubgroupofG.|W*ewritep fe wvH`xforthe;setUUfg"2Gj9n2Nm:g[ٟ^no2HgUUofallr}'oots'vofUUtheelementsofH%SinG.qLikewise,wesettmv pg fe qH}=fg"2GUUj9 a[ٲ-numbGerF%nm:g[ٟno2Hg: ObserveUUthatin(iii)theambientgroupGmustbGeexplicit.color push Black color popExerciseT3.17.6color push BlackqŲ(i) color pop;ProveUUthata nite2-torsion-freegroupissolvqable. & 6color push Black(ii) color pop;LetUU!"=e 33-i33xW ŸPѰ8 @2CandG=Q[![ٲ]C.qFindmϔp $fe̟wvG,where"=f2g.qWhatifG=R?zQmo-pGoweredgroupssatisfythepowerrule,i.e.<(x^rm)^s}=7ŵx^(r7s) βforanyxinaQmo-poweredgroupandfor,all[ٲ-numbGers,r;s.d?Also,4wecanseeQ-pGoweredgroupsasthespGecialcaseofQmo-poweredgroupsforUUtheset.ofallprimenumbGers.aW*e:now:listtheadditionaltechnicalresultswe:needtostatetheMal'cevcorrespGondence.hTheyarepresentedUUandprovedin[Khu98w,Sections9and10].Theorem3.19givestherelationshipbGetweenasubgroupofGanditsradical.JItiscomparabletothatUUofanidealinaringanditsradical. color push BlackTheoremT3.19. color pop[Khu98w,UUTheorem9.18]@L}'etsHG.Thenp ڟfe wvH\[isaQ-poweredsubgroupsofGcontainingH.Moreover,yifHEK~4G,ythen 6pQfe wvH(gEp ofe 5UwvKQ. color push Black color popnL͍color push BlacktFINITEp-GROUPSINREPRESENTZATIONTHEORYl21 color pop37L͍TheoremUU3.20describGessubgroupsofe^QLÍwhichhaveradicalequaltoe^QL n8.Ucolor push BlackTheoremT3.20. color pop[Khu98w,UUTheorem10.4andCorollary9.22]L}'etpFӥbethesubgroupFƵF=Uhe^xi øji1, iofp1l+A,!identi e}'daspsuchviatheformalexp}'onential. Then,=FBis`fr}'ee_nilpotent,of_sameclassc`asA,andFBis`fr}'eely_generatedbythesetfe^xi feȟwvF=e^QL n8.UACwconsequenceC|ofTheorem3.20isC{thatitenablesustoconsiderH(;$)asanopGeratorfromtheLiealgebraQLtotheQ-pGoweredgroupe^QL n8.Indeed,4sincep ]DfeȟwvFʲ=e^QL,4wehavethate^x+yande^[x;y@L] 6areelementsofp  feȟwvFforeveryx;y:2_aQL.BydevelopingtheseexpGonentials,ǍwethusobtainformalexpressionsUUgivingrisetotheinverseUUBaker-Hausdor formulae.Ucolor push BlackDe nitionT3.21. color popLetUUa;b2QL.qDe netheinverseBaker-Hausdor formulaewh1|s(eap;ebD)=ea+b~Eand2h2(eap;ebD)=e[a;b]AӲinUUe^QLÍ.UIntuitively*,\theIfunctionsh1|s;h2Pde neanadditionandaILiebracketIone^QL makingthismultiplicativeQ-pGowered.group.intoa.Liering.dHence,6hwe.usetheexpGonentialinordertoidentifytheLiealgebraQLUUwiththeexpGonentsinthegroupe^QL n8:7_QL!eQLn9,"5Wa7v!eaq,6andS1thende neasumh1|s(a;b)ine^QL n8,SbytakingthesumoftheexpGonentsinQL,Sandlikewise,Sthe aLieLbracketKinQLinducesaLiebracketh2|s(a;b)=e^[a;b]@ine^QL n8.lInfact,(itisenoughKtoconsiderthecasewhenaandbarefreegenerators,@sayx1|s;x2RofA,?sincethealgebratheygenerateisalsofreenilpGotentofclassc.Explicitly*,ֵhiTL(x1|s;x2)arewordsinrationalpGowersofgroupcommutatorsin޵e^x1;6e^x2 {forߵi̥=1;2.cF*romthisviewpGoint,فwecanwriteequalitiesoftheform(see[Khu98w,LemmaUU10.7]):8Lex1 +x2J=h1|s(ex1;ex2)=ex1ex2 {and$%e[x1 ;x2]>=h2|s(ex1;ex2)=[ex1;ex2] where]d f߲and 䀲are]erationalpGowers]dofgroupcommutatorsine^x1; e^x2 Mhofweight]eatleast2for f޲andatUUleast3for .The;upshotisthat:theBH,formulaH(;)transformstheLiealgebraQLintotheQ-pGoweredgroup @pQfeȟwvF'1=e^QL n8,($whileitsinverseh(;ǀ)pGerformstheconverseopGeration,($fromthegroupe^QLbacktotheLiealgebraQL.U8Thismotivqatesthefollowingnotation.U8SuppGosethatagroupGandQLareinsuch 6a[correspGondence,i.e.ڟp 1fe̟wvGG=q"e^QL n8.W*ewrite\Lq!=LG u5tomeanthat\ListheLiealgebr}'aassociatedtoG,UUandsimilarly*,wesetG=GL QɲtoUUmeanthatGistheQ-p}'oweredgroupassociatedtoL.TheUUMal'c}'evcorrespondencestatesthefollowing.Ucolor push BlackTheoremT3.22. color pop[Khu98w,UUTheorem10.11]L}'etIεGbeIanilpotentQ-poweredgroup.LThecorrespondingLiealgebraLG isIde nedonthesameunderlyingsetofelementsG,andise}'quippedwiththeop}'erations:62a8+b=h1|s(a;b) 8,в[a;b]=h2(a;b) 8and$Iq[a=aquforalla;b2Landallq"2Q.7Conversely, ;ifеLisanilp}'otentQ-algebra, ;itsassociatedQ-poweredgroupGLDhasthesameunderlyingsetofelementsL,andthemultiplic}'ationisgivenbycg[h=H(g;h) 8and$Igq=qg forallg;h2Gandallq"2Q._;Mor}'eover,LGmL=L 8and$IGLmG=G:ULetUUusoutlinethemainpropGertiesofthecorrespondence.Ucolor push BlackTheoremT3.23. color pop[Khu98w,UUTheorem10.13]Supp}'oseKthatKanilpotentKQ-poweredgroupGKandanilpotentKLiealgebraKLareinKMal'cevcorrespon-denc}'e.d6color push BlackqŲ(i) color pop;AsubsetٵHofصGisaQ-p}'oweredsubgroupofٵGifandonlyifHL MisaLieQ-sub}'algebraofL,;wher}'eHL isthesetHonwhichtheaddition,ALLiebracket,AMandmultiplicationbyrational;numb}'ersarede nedbytheinverseBaker-Hausdor formulae,asabove.6color push Black򪨲(ii) color pop;H1CEH2GE asE Q-p}'oweredgroupsifE andonlyif(H1|s)L Aisanide}'alof(H2|s)Lt.MF;urthermor}'e,;theXfactorgr}'oupXH2|s=H14isabelian(resp.centralXinG)ifandonlyifthequotientQ-algebr}'a;isc}'ommutative(resp.centralinL). color push Black color popԠnL͍color push Black228N.MAZZA color pop37L͍6color push Black㋲(iii) color pop;GandLhavethesamenilp}'otenceclass,andthesamederive}'dlength. 6color push Black*(iv) color pop;Aamapaf ²:2Ge^QʴLղisaQmo-pGoweredAgroupcorrespGondingtotheQmo-LiealgebraQmoL.P:Indeed,inthiscase,thedenominatorsappGearingintheBHformulaareall[ٲ-numbGers(recallthatn-throGotsforany[ٲ-numbGernarewell-de nedinaQmo-pGoweredgroup,seealso[Khu98w,Corollary /10.21]).UThisis .preciselytheuseoftheL}'azard;5correspondence[;fnamely /itistheexten-sionofMal'cev'sresultswhichsubstitutesQwithQmo,EandwiththeadditionalassumptionthatcontainsUUalltheprimeswhicharelessthanthenilpGotenceclass.color push BlackTheoremT3.25. color pop(Lazard)L}'etc2Nandasetofprimeswhichc}'ontainsalltheprimeslessthanc.6color push BlackqŲ(i) color pop;Supp}'ose Pthat QGisanilp}'otentQmo-powered Pgroupof Pnilpotenceclass Pc.>Thec}'orresponding;LieQmo-algebr}'aLG Misde nedonthesameunderlyingsetofelementsG.Moreover,32/a8+b=h1|s(a;b) 8,в[a;b]=h2(a;b) 8and$Iq[a=aquforalla;b2Landallq"2Qmo.6color push Black򪨲(ii) color pop;Conversely,#if͵Lisanilp}'otentQmo-algebraofclassc,#itsasso}'ciatedQmo-poweredgroupGL ;hasthesameunderlyingsetofelementsL,andthemultiplic}'ationisgivenby`^*g[h=H(g;h) 8and$Igq=qg forallg;h2Gandallq"2Qmo.6color push Black㋲(iii) color pop;Mor}'eover, ǵLGmL=L 8and$IGLmG=G:"2W*e1now0applytheLazardcorrespGondenceto nitep-groups.ZNoticethatLazard'stheoremappliestoB nitep-groupsofsmallnilpGotenceclass,FOi.e.ksuchthatpisgreaterthantheclass.Indeed,FOsuchagroupUUisnilpGotentandQmo-powered.color push BlackExampleT3.26. color pop[Khu98w,UUExample10.24]LetȔGbGeaȓ nitep-groupofnilpotenceclasslessthanp.˃Set$mforthesetofȓprimedivisorsofp!.Thus,GVis[ٲ-divisibleand-torsion-free,i.e.MȵGisaQmo-pGoweredVgroup.Therefore,theabGoveVresultsinSections3.3,3.4,nand3.5apply*,mandsaythatwemayde neopGerations+and[;:]onG,minordertomakeGintoaLieringGL  suchthateveryautomorphismofthegroupGinducesanautomorphismof9theLieringGLt,?@andeachelementofGinGL 6/has9thesameorderunder+asits(multiplicative)order0$in0#thegroupG.eaMoreover,7each0$subgroupofthegroupGisaLiesubringofGLt.eaF*orinstance(see,[Gla07,bp.423]),suppGosethat,pisoddand,Ghasclass2.Then,binGL ),andwiththesamenotationUUasabGove,khݵx1S+8x2C=x%eP 33133xW g P2A1Mx2|sx%eP 33133xW g P2A1 =x1x2(x2;x1)/ 33133xW g P2Mand*i[x1;x2]=(x1;x2) color push Black color popBnL͍color push BlacktFINITEp-GROUPSINREPRESENTZATIONTHEORYl23 color pop37L͍whereϒforϓu={2=|G,.!weletu 33133xW g P2 fortheelementofGwhichsquarestou(sincepisoGdd,.!Gis2- divisible).Also,(toddavoidanydeconfusionwiththeLiebrackets,(the(; )dedenotetheusualgroupcommutators.oWTheseNtwoequalitiesNrefertoacertainconstructionwhichworksNwhenGhasclass2,bGecauseUUotherwise[x1|s;x2]6=(x1|s;x2).,acolor push Black color popExerciseT3.18.6color push BlackqŲ(i) color pop;V*erifyUUtheequality{}ex Hey·ex=ey@L+[y;x]+}r33[y-;x;x]33W \1Pz2—+}r33[y-;x;x;x]33W 4P 3!+:::jǙ=eS%#cmex7PgӸn1'-[y-;x;n]-W *Pn!2@ߵ:XÍ6color push Black򪨲(ii) color pop;SuppGoseUUthatGisap-groupofclass2andthatpisodd.qUsingtheaboveUUde nitionsof ;additionUUandmultiplicationinGL QɲverifythatadditionandmultiplicationarebGoth;assoGciative,5and.that.theadditioniscommutative...ػand.fortherainySunday.afternoGon,;proveUUtheremainingaxiomsde ningaLiering.,aA1ccase1ofparticularinteresttousis1whenHisasubgroupofexpGonentpofa nitep-groupG,andkxθ2GhaskorderpandnormalisesH.ThentheLazardcorrespGondencegivesusalowerbGoundonUUtherankofCH(x).wcolor push BlackLemma33.27.kj color popL}'et8G8bea8pgr}'oupandHasub}'groupof8G.Supposethat8Hhasexp}'onentpandnilp}'otence classlessthanp. @SetjHj=p^nq~.F;or anyx2NGڲ(H) ofor}'derp,6therankofCH(x)isgr}'eaterthanore}'qualton=p.5color push BlackPr}'oof. color pop4CByDtheDLazardcorrespGondence(Theorem3.24),asregardedinExample3.26,conjugationbyG7xonH5inducesanautomorphismcx PofG8theadditivegroupofthecorrespGondingLieringHLofӵH.HAMoreover,cxhasorderdividingjHj>?=>>p^n RandtheadditivegroupHL GiselementaryabGelian,whenceFanFpR-vectorFspaceofdimensionn.FLWithoutloss,7wemayFsuppGosethatcx OisgivenasamatrixinGL7nv(p)inJordanform._Inparticular,)eachJordanbloGckhassizeatmostp,)sincex^pfj=1,andsothereareatleastn=pbloGcks.Because1istheuniquep-throotof1, thenumberofJordanbloGcksGequalstheGnumbGeroflinearlyindependentGeigenvectors,Ji.e.mCvectorsG xedbyGɵcx.mCBy\lifting"thesexeigenvectorsinthe(multiplicative)groupH,wegetelementsoforderpbGelongingtoCH(x),andwhichspananelementaryabGelianp-groupofrankequaltothedimensionoftheeigenspace(fortheUUeigenvqalue1),andalsoequaltotherankofCH(x).ѷTղ4.aElementUTaryabeliansubgroups:resulUTtsandconjecturesThroughout,epbbdenotesabcprimenumbGer.Letusbbstartbybbabriefrecallofthenotionstreatedinthissection.K 4.1.Elemen9taryTabQelian'sessentials.W*eUUstartwitharecallofpartofDe nition1.3.color push BlackDe nitionT4.1. color popLetUUGbGea nitegroup.,a6color push Blackq(i) color pop;An=hxiZ(NG(E)),P<wher}'eZ(NGڲ(E))iscyclic.color push Black color popExerciseT4.20.qDzFindUUnG /forthefollowingp-groupsG(seeSection6forthede nitions).m6color push Blackq(i) color pop;G=Cp2o8Cp./6color push Black(ii) color pop;GUUisextraspGecialoforderp^3Ȳandexponentp.捍6color push Black(iii) color pop;GUUisaSylowp-subgroupofSL#4(p).KClearly*,mifhYZ(G)hZisnotcyclicandGhasrankatleast2,mthenE2k(G)isconnectedbyTheorem4.3.Thus,*Othe\interestingcases"arisewhenE2#(G)isnon-emptyanddisconnected._Hence,*OweaddresstheUUfollowingtwoquestions:color push Black ~Q.1.!c color pop;CanwebGoundthenumbGer͵nG ofconjugacyclassesofconnectedcompGonentsofE2^(G)?;(i.e.XtheLnumbGerofLconjugacyclassesofmaximalelementaryabGeliansubgroupsofGof;rankUU2.)color push Black ~Q.2.!c color pop;CanUUwebGoundtherankofGbeyondwhichE2X(G)isconnected?W*enextprovethatbGothquestionshaveanarmativeanswer,andworkthroughtheproGofsforpoGdd.qF*orUUp=2,westatetheadhoGcresultsandgiveexamples.As0we0wantto0analysethesituationwhenE24j(G)hasmorethanonecompGonent,gwehenceforthsuppGosethatZ(G)iscyclic.Thus,weletZf=t 1|s(Z(G))bGetheuniquecentralsubgroupofGoforderGp.By[Hup67w,HilfsatzIGII-7.5],GGcontainssomenormalelementaryabGeliansubgroupE07ofranko2.SetG0oY=CGڲ(E0|s).NotethatG0isaomaximalsubgroupofG,v0sinceG=G0isisomorphictoa8nontrivialp-subgroupofAut*i(E0|s)T͍+3= UNGLn32ꦲ(p),>]byPropGosition1.4.h5InviewofTheorem4.3,>]ifGhasrankV3ormore,W%thenVallthesubgroupsinE2ZZ(G)ofrankatleast3lieinthebig1ifandonlyifeithercolor push Blackc!c color pop;GUUhasrankatleast3andanisolatedsubgroupE,orcolor push Blackc!c color pop;GUUhasrank2andisnotmetacyclic. color push Black color popNnnL͍color push BlacktFINITEp-GROUPSINREPRESENTZATIONTHEORYl25 color pop37L͍RecallUUthatagroupGismetacyclicifGsitsinashortexactsequenceϢi 1 5D xyatip10/5D xybtip10/q 2fd *Cn://:2fd:ꪵGق//2fd܂*Cm@P//@Q2fd @P1@Rwith8n;m1 ,]ۍthatuis,}Gisanuextensionofacyclicgroupbyacyclicgroup.DW*eknow(cf.Cforinstance[Maz03a!w]) that[ifG\isametacyclicp-groupthenGhasrankatmost2andhasatmostoneelementaryabGeliansubgroupUUofrank2(whichisthusalsomaximalandcharacteristicinG).W*rite+SE2.(G)+Ras+SadisjointunionofconnectedcompGonentsBd[LjfE1|sg[lj9[ljfEnq~g,`whereBy.maycontain)more)thanonesubgroup.LIfGhasrank2,^thenBv=(fE0|sg,forsome)elementaryabGeliansubgroupZxE0EzPGZwofrank2./Inthiscase,E0neednotbGeunique,sothatweneedtomakeanarbitrary choice.Y_Nevertheless,once thisE0is xed,wedo notconsiderE0asisolated.Y^W*eshallseebGelowUUthatthisisirrelevqantinthe\interesting"situations.TheȇthirdȈpartinTheorem4.3isduetoN.Blackburn([Bla85w]),TwhousesHGethelyi'swork,TwhichweUUnowpresent.4.3.SoftTsubgroups{theansw9ertoQ.1.color push BlackDe nition4.4.& color popLetdGbGedap-group.!_Acsoftsub}'groupofGdisdasubgroupAofGsuchthatA=CGڲ(A)andUUjNGڲ(A):Aj=p.qAsoftsubgroupAofGisade}'epsoftsubgroupofGUUifjG:Ajp^2|s.Inotherwords,asoftsubgroupofGisamaximalabGeliansubgroupofGwhichhasindexpinitsnormaliser.QClearly*,9ifٵGزisabGelian,:thenGhasnosoftsubgroups.QMost(ifnotall)oftheresultsregardingp-groupswithsoftsubgroups(panyprime)havebGeenobtainedbyHGethelyiin[HGet84Y,HGet90.S].qTheyUUaresummarisedinthenexttheorem.color push BlackTheoremT4.5. color popUU(HGethelyi) 8L}'etGbea nitep-group,andletAbeasoftsubgroupofG.6color push BlackqŲ(i) color pop;Thesub}'groupscontainingAformachainA=N0|s;N1;:::;Nr4=M,wher}'eNid=NGڲ(Ni1 ),;for1iMoreover,fromTheorem4.3,wehavethatCGڲ(E)isabGelianandthatEcharKCG(E).AThislastfactimpliesthecontainmentsCGڲ(E)q1andsuchthatthecentraliserof#anisolatedsub- group6NEј2> E2A(G)6Oisdeep.Then,nthepictureofGthatspringstomindisasfollows:3writeNiforthesuccessivenormalisersofthesoftsubgroupCGڲ(E)X=YN0|s,asinTheorem4.5. F*orshort,setM3=Nrand1L=Z(N1|s),8and xanormalelementaryabGeliansubgroup1E0XofGofrank2.eFinally*,writeUUG0C=CGڲ(E0|s)andH=G^09L.qThen,GloGoksasfollows. *\eiHꪵGH_Ο66δ xydash10}K}Nl}Q;1}U+ *}X]ٟ 7}[ D,}^wP}~JLJ*5JL{Jn¶J J|r PRJxcԟ JtTJpF$J:nq%G0^˟?pA[a;yG(ļG$:G!/?Gn-N0PRo{Sȣ{W?C{Z{^,Ѻ3{a#Ϊd{eà˚{hcȊ{+3$A0A-A*A'tAh$"HAA!1A|?AFɠDLi皠aƹCf?6CbL C^C[HCWWCS_CPR 4CDE{5E]|~v֠Jo|O|^T|:Y|{^|*c|nh|p ꘵Z Twhereeachedgeindicatesaninclusionofasubgroupofindexp,exceptpGossiblyfortheinclusionsEZN0ȲandUUZ~4L.aAsqanreasycorollaryofTheorem4.5,wegetthefollowingresult.YThis,andtherestoftheanswertoUUQ.1.qDzistakenfrom[Maz08v].~color push BlackPropQosition4.8. color popAssumeqthatqGhassomeisolate}'dsubgroupqwhichisnotnormalinG.3Then,2ӆnG `p=ٲ+=1andõGhasatmostpc}'onjugacyclassesofmaximalelementaryab}'eliansubgroupsofr}'ank2.;color push BlackPr}'oof. color pop4CClearly*,nG G>B1.KLeteֵE и2E2Բ(G)bGeeisolated.JSetAC=CGڲ(E)andH]@=G^09Z(NGڲ(A)).Then,Absatis esbthehypGothesisofTheorem4.5part(v).Thus,H2isanormalsubgroupofG,indepGendentJofJthechoiceJofE,JandG=HiselementaryabGelianofrank2.PInparticular,KGhasp?+1_maximalsubgroups_containingH/ֲandhencemaycontainsomeisolatedsubgroup_ofG.PNow,H? <o G0|s,MsinceSZ(NGڲ(A))ScentralisesE0((recallthatE0~NGڲ(A))andsinceSG^0!iscontainedinanymaximalsubgroupofG.DSinceG0 ,doGesnotcontainanyisolatedsubgroupofG,QthereareatM~mostpmaximalsubgroupscontainingisolatedsubgroups,whicharethenallG-conjugate,byTheoremUU4.5(iv)and(iii).qThepropGositionfollows.In ordertoanswerthequestiononthebGoundfornGڲ,{westillneedtoanalysethecasewhenthereisUUanisolatedsubgroupwhichisnormalinG.qThesolutionisimmediatefromPropGosition4.7.}color push BlackPropQositionQ4.9.c color popSupp}'ose~thatE& 2}E2(G)isnormalandisolatedinG.[ThenGisacentralpr}'oductX4ZZ(G),withXextr}'aspecialofor}'derp^3Vandexponentp,andwhereZ(G)iscyclic.l7Inp}'articular,Ghasrank2andnG =p8+1.Ǎcolor push Black color popExerciseT4.21.qDzProveUUPropGosition4.9.Inconclusion,ifGhasrank2,thenthereareatmostpiز+1conjugacyclassesofmaximalelementary abGelian|subgroups}ofrank2,AandifGhasrankatleast3,AthenthereareatmostpconjugacyclassesofUUmaximalelementaryabGeliansubgroupsofrank2.qThisanswersQ.1.. color push Black color popnL͍color push BlacktFINITEp-GROUPSINREPRESENTZATIONTHEORYl27 color pop37L͍color push Black color popExerciseT4.22.qDzDetermineUUnG /inallthecaseswhenGhasrank2.(Hint:see[Maz08v])״4.4.AlpQerin-Glauberman'sTexperience{theansw9ertoQ.2.T*otackleQ.2., weobservethatthewreathproGductCp ̸oyCphasrankpandauniqueconjugacyclass of_maximalelementaryabGelian^subgroupsofrank2.Whatwearegoingtoprove^nowisthatthisiss!the\extremal"situation;inthesensethat,zforpanoGddprime,ifGisap-groupofrankgreaterthankp,pthenGhasnomaximalelementaryk abGeliansubgroupsofrank2,i.e.nG C=h1.&WhatfollowsinUUtheanswertoQ.2.qDzistakenfrom[GM10].SuppGosethatpisanoddprimeandthatGisa nitep-groupwithnG >1.QByPropGosition4.9,ifGhasClsomeCmisolatednormalsubgroup,GthenGhasrank2andallthesubgroupsinE2F(G)arenormalinUUG.qHence,wealsomaydiscardthiscase.qW*estartwithaclassiclemma.鍑color push BlackLemmaT4.10., color popL}'etdH4beadnormalsubgroupdofGofor}'derp^nq~,forsomenAYAX1. F;orallinte}'gers0kn,jHcc}'ontainsanormalsubgroupHkwofGoforderp^k.ˍcolor push BlackPr}'oof. color pop4CW*e{eproGceed{dbyinductiononkP.IfkDz=00,theclaimtriviallyholds.AssumekǸ/1.ByPropGosition 1.4(ii),90anynontrivialnormalsubgroup ofGintersectsZ(G)nontrivially*,90andsowemay%pickasubgroupH1 rHYi\jZ(G)withjH1|sj=p.7Then,YH1 EG.Set%K:G!G=H1Jforthe UMnaturalpro8jectionmapandwrite}fe 5UK&=[ٲ(K)fortheimageofasubgroupKղofGunder[ٲ.]In}fe̟G ,$wehavebyinductionhypGothesisthat}fe H%3containsanormalsubgroup}fe ;Hk@Ųof}fe̟G oforderp^k+B1(.V4Therefore, Hk=[ٟ^1 M(}fe ;Hk ;)UUisanormalsubgroupofGcontainedinH%SandjHk됸j=p^k.j!ʍW*eUUnowspGecialisetoacasewhichwillbGerelevqantforourpurpGoses.color push BlackLemmaQs4.11. color popAssume*fthat*gGhasanon-normalmaximalelementaryab}'eliansubgroup*gEofrank*f2,andthatHcisasub}'groupofexp}'onentpinGthatisnormalisedbyE.LetjHj=p^nq~.φ6color push Black8(a) color pop;F;ore}'achpositiveintegerkAnlessthann,HHcontainsasubgroupHk goforderp^k gthatis;normalise}'dbyE.6color push Black򪨲(b) color pop;np.6color push Black(c) color pop;IfE'tisnotc}'ontainedinH,thenthesub}'groupHkwinp}'art(a)isunique,foreachkP.ˍcolor push BlackPr}'oof. color pop4CLetUUEZ=hzp;x?i,withz72Z(G).qNotethatH%SisnormalinHE.EachppartpofthelemmaisvqacuousorobviousifjHj%pporifH#=$E.īSoweassumethatjHj$%p^2andUUthatH6=E.qThenCEm(H)=hzpiandX6color push Black(A) color pop{DCH(x)=CH(E)=H޸\8EZ:XPart(a)isLemma4.10appliedtoG K=HEhandHI=H.ZThus,H|foreach0 L Kk[p^2|s,weseethat "hxip^k됲.SnEInviewof[GLS96!,DPropGosition10.17](or[JK75,Theorem]),ifp=3andGhasrankatleast4,Dthen G_has`normalrank4.?Consequently*,]TheoremA9holdsforp=3,]asalso_observedin[Maz08v].?Hence,weUmayTinadditionsuppGosethatpVl5fromnowon.sRecallTthatthenormalr}'ankofap-groupGisthemaximumoftheranksofthenormalelementaryabGeliansubgroupsofG,MandisingeneralsmallerUUthantherankofG.獑color push BlackTheoremT4.14. color pop[AG98,UUTheoremD]Ifpisatle}'ast5,iandAisanelementaryab}'eliansubgroupoforderp^n 3ofthep-gr}'oupPc,ithenthereisanelementaryab}'eliansubgroupBXofPc,alsooforderp^nq~,satisfyingthefollowingconditions:i6color push Black8(a) color pop;BXisnormalinitsnormalclosur}'einPc;6color push Black򪨲(b) color pop;[PG;Bq;3]=1; color push Black color popnL͍color push BlacktFINITEp-GROUPSINREPRESENTZATIONTHEORYl29 color pop37L͍6color push Black(c) color pop;BXisnormalise}'dbyanyelementxofPvsatisfying ^QBq;x;<$۵p8+1۟wfe@ߟ (֍o2Q^h=1:Սcolor push BlackTheorem4.15. color popL}'etspsbeaprimesgreaterthans3,wandassumethatGhasor}'derp^nq~.8IfGhasa non-normalmaximalelementaryab}'eliansubgroupofrank2,thenGhasrankatmostp.color push BlackPr}'oof. color pop4CW*eUUassumethatGhasrankgreaterthanpandworktowardsacontradiction.Let8ÊbGea8non-normalmaximalelementaryabGeliansubgroupofrank2inG.hRByhypGothesis,>p5andUUGcontainsanelementaryabGeliansubgroupAofrankp8+1.By\e[AG98,)Theorem\fD],wemay\fchoGoseAto\fbenormalin\fitsnormalclosure,*hA^Gڸi,)inG.LetN05=hA^Gڸi.IfN=G,thenwearedone,byLemma3.27,aseveryelementoforderphasacentraliser'2of'1rankatleast3.bfLetusthenassumethatN3<G.bfSinceGisnilpGotent,0lwehave'2Aܚ/N,andUULemma4.10saysthatAcontainsanormalsubgroupBƲofNlphavingorderp^p1Ʋ.LetM3= 1|s(Zp1Ʋ(N))._ThenMw/\GandBGM6-byLemma4.13._SinceZp1Ʋ(N)hasclassatmostpxtxs1,]itisaregularp-group.QTherefore,M 9hasexpGonentpbecauseitisaregularp-groupgeneratedbyH$elementsH%oforderp(seeTheorem1.22).mbSinceM5/G,JLemma4.11yieldsthatjMjp^pR.mbHence,jM3:Bqjp.LetUUY= 1|s(Z2(N))andW*= 1(Z(N)).qThen}W*YM and%3W;Yq/8G:AssumeE rstEthatYxWA.BThenCGڲ(Y8)/GandAWCG(Y8).BTherefore,Nn=hA^GiWCG(Y8), andYQ;VZ(N).Moregenerally*,DobservesimilarlythatanynormalabGeliansubgroupofGwhichiscontainedUUinanyconjugateofAisnecessarilycontainedinZ(N).qThen},IA8\Z2|s(N)=A8\Y=A\Z(N);andUUAZ(N),byLemma4.13.qButthen,VsA 1|s(Zp1Ʋ(N))=M and%3pp+1m=jAjjMjppfj;|aUUcontradiction.qThus,Y9isnotcontainedinA.qTherefore,BG<BqYM.SincecjM-:Bqjp,fwehavecM.=BqY8.$Moreover,fYq=WAZ(NA=Wc).$Therefore,M=Wƭisccentralised byX=AWG=Wc.zAsMQ/:G,XitX>followsthatX>M=W̲iscentralisedX>byhA^GڸiWG=Wc,Xi.e.,XbyX>NA=W.zTherefore,M3Z2|s(N).qButUUnow,}|A8\Z3|s(N)A8\Zp1Ʋ(N)=A8\M3=A\Z2|s(N):SoƵAԸ\յZ3|s(N)|=A\յZ2|s(N).ByLemma4.13,!AZ2(N).Hence,!AM,!andweobtaina contradictionUUasinthepreviousparagraph.!Now,UUwecananswerQ.2.ocolor push BlackTheorem4.16. color popL}'etȵpbeanoddprimeandletGbea nitep-group.9IfGhasrankatleastpIt+1,thenGhasnomaximalelementaryab}'eliansubgroupoforderp^2|s.o4.5.TheTcasep=2.The/answers/toQ.1.e andQ.2.in/thecasewhenp=2/areasfollows.e ThepGosetE22(G)fora nite2-groupGhasatmost5compGonents,0by[Car07#].{By[Mac70v],a2-groupGwhoserankisgreaterthan^4_cannothavemaximal^elementaryabGelian_subgroupsofrank2.pIndeed,ifGhasamaximalelementaryabGeliansubgroupofrank2,thenGhasnonormalelementaryabGeliansubgroupofrank3by͡Lemma3.27.DNow,AnneMacWilliamsproves͢thatinsucha2-groupevery͢subgroupisgeneratedbyUUatmost4elements.InsteadUUoftheproGofs,wegiveexamplesinforp=2UUof2-groupsreachingthesemaximalbGounds.A0zgroup0G0withE24E(G)having5conjugacyclassesofcompGonentsistheextraspGecial2-groupoforder32whichisthecentralproGductG}=}D8qQ8?uofadihedralandaquaterniongroup,nbothoforderUU8.qHence,eachcompGonentcontainsauniquesubgroup.zcolor push Black color popExerciseT4.23.qDzGivenUUG=hx;y[;s;t ոiwithhx;y.jx^2C=y[ٟ^2d=(xy[ٲ)^4=1q*iT͍+3= UND8Ȳandhs;tUUjs^4C=1;s^2=t^2ȵ;^tګsQò=s^1ѴiT͍+3= UNQ8|s,UUdeterminethe5compGonentsofE2X(G). color push Black color popnL͍color push Black308N.MAZZA color pop37L͍ExamplesUUin[GLS96!,p.q68]showthatGmayhaverank3.qHere,wegiveanexampleofrank4.Gtcolor push BlackExample4.17.t color popLetQeFbGeQdthe nite eldoforder4.pwF*oreacha;b;cQeinF,R/letM(a;b;c)bGeQethe3013 matrixUUoverFgivenbyߍ͒M(a;b;c)=Z0 @d 1a*&]c 01**b 00)J1Z.K1 .KA"VLetyUbGethesetofallmatricesM(a;b;c).2ThenyUisagroupundermultiplicationandisaSylow2-subgroupUUofGLn:3ꭲ(4).F*or9leachainF,?let~feI0ga I=a^2|s;Bthus,?weobtaintheuniquenontrivial eldautomorphismofF.hzLettbGetheUUmappingonUlpgivenby= hM(a;b;c)tLn=M( feJ$bJ;~feI0ga;~feI0ga feJ$b wb+8~feS۟gc)UU:VThenwtisanautomorphismvofordertwowofU(andcomesfromaunitaryautomorphismofordertwoofUUGLn:3ꭲ(4)).qLetUUGbGethesemi-directproductofUlpbyhti(seeDe nition6.7).Note%\thatCU(t)isthegroupofallmatricesoftheformM(a;~feI0ga;c)%[suchthatc+~feS۟gc=a~feI0gaI0.aThis%\groupisUUaquaterniongroupoforder8,andCGڲ(t)=CU(t)8hti.8color push Black color popExercise4.24.mInIExampleI4.17,L ndamaximalelementaryabGeliansubgroupofGofrank2,LandoneUUofrank4.z4.6.Conjectures.4.6.1.Theclass-br}'eadthconjecture.Gtcolor push BlackDe nition#4.18.Ѯ color popLet!ݵG!ܲbGea nitep-group.`F*orxinG,,(thebr}'eadthdofx!ݲistheintegerb(x)de ned abyvVtheequalityvUp^b(x)D=jG:CGڲ(x)j.InvVparticular,~b(x)=0ifandonlyvUifxliesinZ(G).Thus,~the br}'eadthofGUUistheintegerb(G)equaltothemaximumofb(x)asxrangesoverG.GtLet7c(G)denote7thenilpGotenceclassofG.gTheclass-br}'eadthxconjecture7(alsoknownastheBreadthConjecture)UUstatesthattheinequalityp?Qc(G)b(G)8+1always1holds.[AlthoughcounterexampleshavebGeenfoundforp ܲ=2,none1isknownforpoGdd.[F*or background8andrecentresults7abGouttheclass-breadthconjecture,>werefer7thereaderto[LGNW69)]andT[ENO06 [].DInparticular,UseveralcasesareUknowntobGetrue,Uandmoreover,TtheUbGoundisoptimal,inp7thep8sensethattherearegroupsforwhichtheequalityc(G)=b(G)J+J1holds.nThe niteabGelianp-groupsnPandnOthe nitep-groupsofmaximalnilpGotenceclassaresuchinstances,and[LGNW69)]presentsUUfurthercases.color push Black color popExerciseT4.25.qDzFindUUthebreadthofa nitep-groupGwhen:8 6color push Blackq(i) color pop;GUUisabGelian;6color push Black(ii) color pop;GUUhasmaximalnilpGotenceclass.Gtcolor push BlackTheoremT4.19. color pop[LGNW69),UUTheoremsA,and2.3]F;oranyprimepandany nitep-gr}'oupG,theinequalityp?c(G)2b(G) 8holds.Mor}'eprecisely,theinequalityt>c(G)<<$ pKwfe@ߟ (֍p81n]b(G)8+1 8holds. Obviously*,UUthesecondpartofthestatementisrelevqantonlyforoGddprimes!1PropGositionUUCin[GM10]statesthefollowing.color push BlackPropQosition@4.20.7 color popL}'et鳵pbeanoddprimeandGa nitep-gr}'oup.Assumethatthep}'osetE2E(G) hasmor}'ethanonecomponent.Thentheclass-breadthconjectureholdsforG. color push Black color popanL͍color push BlacktFINITEp-GROUPSINREPRESENTZATIONTHEORYl31 color pop37L͍color push BlackPr}'oof. color pop4CW*ritec=c(G)forthenilpGotenceclassofG.* LetEk=hx;zpibGeamaximalelementary abGelian subgroupof G,;withzj2Z(G).hBy[Bla85w,Theorem],weobtainthe equalitiesCGڲ(E)=hxi8Z(NGڲ(E)),UUwithZ(NGڲ(E))cyclic,and܍霸jG:CGڲ(E)j=jG:CG(x)j=pc1OU:|HenceUUc=b(x)8+1.qSinceb(G)b(x),theclass-breadthconjecturecb(G)8+1holdsforG.jսcolor push BlackRemark޻4.21. color popObserve]thata]similarproGofshowsthattheclass-breadth]conjectureholdsforany niteop-groupGohavingsomesoftsubgroupAsuchothat,foreverypropGersubgroupH?ofGcontainingA,]the)nilpGotenceclassof(NGڲ(H)isonemorethanthenilpGotenceclassofH.BInturn,]thisraisestheUUquestion:O&color push Black color popT&Do}'estheequality'εc(G)=b(G)8+1holdforGhavingasoftsubgroup?F*romh;HGethelyi'sarticle[HGet90Y],lonemayhopGethatindeedvc(G)=旵b(G)Ey+Ez1inthiscase.yStill,lthisremainsUUtobGeproved.卑4.6.2.Oliver'sp-gr}'oupconjecture.Oliver'sYp-groupYconjecturedealswith nitep-groups,ڤwherepdenotesanoGddprimenumbGer(seeV[Oli04UZ]).v,ItVreducesthequestionofexistenceanduniquenessofalinkingsystemassoGciatedtohanarbitrarysaturatedfusionisystemintermsof nitep-groups. TheintentofhthissectionistogiveUUabriefoverviewoftheconjecture.卑color push BlackDe nitionT4.22. color popW*riteUU6%n eufm10X(S)forthelargestsubgroupQtګofSforwhichthereisasequence=q1=Q0CQ18Qt WwithUUQidESanda[ 1|s(CS(Qi1 ));QiTL;p81]=1 8i:W*eUUcallX(S)theOliversub}'groupofS.color push Blackc!c color pop;If+p=2,3thenX(S)=CS( 1|s(S)).PdIn+particular,4X(S)=Z(S)*if+Sisgeneratedbyelements;ofUUorder2.color push Blackc!c color pop;X(S)UUiswell-de ned,inthesensethatifpm1=Q0CQ18Qt Wand'1=Q0፱0Q0፱18Q0፴u;areUUsequencesasinthede nition,thensois1=Q0CQ18QtLnQtVQ0፱18QtVQ0፴uz;;i.e.,UUanyp-groupShasauniquelargestsubgroupX(S2})tasinthede nition.qInparticular,oX(S)UUchar S:InUUordertostatetheconjecture,weneedtorecallthefollowingnotion.䍑color push BlackDe nitionT4.23. color popLetUUGbGeap-group.qTheThompsonsub}'groupUUisthesubgroupw:J9(G)=hEZjE2E(G)m:rankE=rank:Gi ofUUG.卑Recallqthatthereqare(atleast)twoqversionsofqtheThompsonsubgroupofa nitep-groupandourchoice ismotivqatedbythesettingofOliver'sconjecture.`SIntheliterature,+qthisThompsonsubgroupissometimescalledµJeKK(G)insteadofJ9(G),bGecausethen,J(G)denotestheThompsonsubgroupthatisgeneratedbyalltheabGeliansubgroupsofGofmaximalorder(see[Gor80Ò,ͼp.271]).Ingeneral,J9(G)6=JeKK(G),UUbuttheyhaveUUsimilarpropGerties.ConjectureUU[Oli04UZ,Conjecture3.9]F*orUUanyoGddpandanyS,wehaveJ9(S)X(S).color push Black color popExerciseT4.26.qDzFindUUa nite2-groupSforwhichJ9(S)X(S).卑color push BlackRemark4.24.F color popThevconjectureuisnotane}'cessaryconditionCزtouprovethevexistenceanduniqueness ofoalinkingsytemassoGciatedtoanarbitrarysaturatedofusionsystem.Indeed,vwhatisneededistoprovedthatdX(S)alwayscontainsdauniversally.we}'akly/closedsubgroupdڲofd۵Sgforany nitep-groupS.ByG[Oli04UZ,p.340],auniversallyHweaklyclosedsubgroupHofS;ԲisasubgroupQofS;ԲsuchthatwheneverF+isafusionsystemonap-groupS^06containingShasastronglyF9-closedsubgroup,thenQisweaklyF9-closedinS^0aƲ.SHence,astheThompsonsubgroupofap-groupisuniversallyweaklyclosed,andthisUUsortofsubgroupisveryrare,J9(S)appGearstobethe\obvious"choiceforanyS. color push Black color pop RnL͍color push Black328N.MAZZA color pop37L͍color push Black color popExerciseT4.27.qDzFindUUap-groupSandacharacteristicsubgroupQofSwhichisnotuniversally weaklyUUclosed.q(Hence\char!6)universallyweaklyclosed".)^InUUOliver'sarticle,itisshownthat:ፑcolor push BlackPropQositionT4.25. color popThefollowinghold._6color push BlackqŲ(i) color pop;X(S)xc}'ontainsallthenormalabelianxsubgroupsofS.OInparticular,}X(S)iscentric,}thatis;CS(X(S))X(S).6color push Black򪨲(ii) color pop;X(S)c}'ontainsallthenormalsubgroupsQofS'tsuchthat[ 1|s(Z(X(S)));Q;p81]=1.6color push Black㋲(iii) color pop;IfX(S)GhastobGeunderstoodas9thequotientST=X(S).hEitherSZ=X(S), orUUwehavethatJ9(S)X(S)UUifandonlyifST=X(S)hasnoF-moGdulesatisfying(PS).T*ounderstandtherecastofOliver'sconjecture,werecallfrom[GLS96!,˸x26]thenotionofF-mo}'dule.⍑color push BlackDe nition;4.29.5] color popLetFGEbGea nitegroupandV*afaithfulFpRG-module.|F*oraFsubgroupH~CofG,setAf>jH(V8)=<$KjHjjCVɲ(H)jKwfe3; (֍JjVj;/:-A@nontrivial@ elementary@ abGeliansubgroupEӘofGisano enderQofGRonVxifjEm(V8)1.jIf@ thereisnoUUconfusion,simplysaythatEisano ender.IfUUV9hasano ender,thenViscalledanFc-mo}'dule[.AnUUelement16=g"2GUUisquadr}'atic(onV8)if[V9;g[;g]=0.Hence,UUano enderEisquadr}'aticif[V9;E;E]=0.De neUUthesetofb}'esto enders0bKP:=fEZGjE2E(G)m:jEm(V8)jF(V) ,81F*Eg:ፑcolor push BlackRemarkT4.30. color pop^6color push Black color pop;NoteUUthateveryEZ2P'Ҳisano ender,andeveryminimalo enderliesinP}. 6color push Black color pop;V9isUUanF-moGduleifandonlyifP'Ҳisnonempty*.6color push Black color pop;Timmesfeld_$r}'eplacement_%theorem.If2>EԸ27FP},iythereis2?aquadratic2>o enderFո2Pwhich;satis esX˵jF(V8)w=jEm(V)XandFwE.|)Hence,PJ6=;X˲ifXandonlyifthereisaquadratic;o ender.qMoreover,UUasFpR-vectorspaces,mCVɲ(Fc)=[V9;E]8+CV(E).6color push Black color popE([V9;g[ٲ;p]E=[V;g[ٟ^p+],_yfor g2G) &(poGdd)&(V/isfaithful))any quadratic;elementUUhasorderp. color push Black color pop!nL͍color push BlacktFINITEp-GROUPSINREPRESENTZATIONTHEORYl33 color pop37L͍TheUUmainresult[GHL08]istheasfollows.ԍcolor push BlackTheoremw4.31. color pop[GHL08]Oliver'sc}'onjectureholdswheneverϵSL]isap-gr}'oupsuchthatST=X(S)has nilp}'otenceclass2.ՍInUUthesubsequentarticle[GHMar!n=],itwasshownabitmore.color push BlackTheoremv4.32.T# color pop[GHMar!n=,Theorem1.1]Supp}'osethatpisano}'ddprimeandSwisap-gr}'oupsuchthatST=X(S)satis esanyofthefollowingc}'onditions 6color push BlackqŲ(i) color pop;ST=X(S)hasnilp}'otenceclassatmostfour;6color push Black򪨲(ii) color pop;ST=X(S)ismetab}'elian; 89j cmti9(i.e.@;cmmi6S=De{eufm6DX(SQl) hasNjG>khasnilpGotenceclassatmost4Q$:fQby(v)and(iii),CEisnotano ender()4.32(i)).color push Black(vii)!c color pop;IfUUGismetabGelian$:qbyTheorem4.34,Eisnotano ender()4.32(ii)).color push Black (viii)!c color pop;IfUUa;b2GarequadraticandCVɲ(a)=CV(b),UUthen[a;b]=1.color push Black:(ix)!c color pop;IfUUthep-rankofGisatmostp$()4.32(iii)). color push Black color pop"2enL͍color push Black348N.MAZZA color pop37L͍Բ5.AglanceaUTtrepresentationtheory5.1.In9troQduction.In=this=sectionweapplytheresultsobtainedintheprevioussectionstothestudyofendotrivial moGdules=VAsinmanytopicsinmodularrepresentationtheoryof nitegroups,wVusefulinformationis6oobtainedbyconsideringtherestrictionsof6pagivenrepresentationtotheelementaryabGelianp-subgroupsoftheconsidered nitegroup.&F*orinstance, sChouinar}'dtheorem(Theorem5.3)detectspro8jectivityUUofmoGdulesinthisway*.Hence,UUthroughoutthissection:Xcolor push Blackc!c color pop;weUU xa eldkofprimecharacteristicp,color push Blackc!c color pop;weUUassumethatkisalgebraicallyclosed,andcolor push Blackc!c color pop;byUUmo}'dule[,wealwaysmeana nitelygeneratedleftmoGdule.Our7interest7isinthestudyofkPG-moGdulesfora nitegroupGwhoseorderisdivisiblebyp.gThen,wereferto[Alp86Z,Ben98a",CR62(,CR87,CR90,ThGe95z]forthebackgroundmaterialonthestructureof theŸgroupalgebraandthe\wild"situationarising.@Theshortest(condensed)accountofthistopicispresentedin[Ben98a [,Chapters1and3].VThemainpGointtokeepinmindisthatundertheabGoveassumptions,5Maschke's-theoremfails,i.e.dmoGdulesare-notcompletelyreducible.As-asubstituteofMaschke'sUUtheorem,thereisKrull-Schmidtthe}'orem.HPcolor push BlackTheoremT5.1. color pop[Ben98a [,UUTheorem1.4.6]Every nitelygener}'atedkPG-modulehastheuniquedecompositionproperty.HQAUUkPG-moGduleMlphastheuniquede}'compositionpropertyUUifMlpsatis esthefollowingtwoconditions:color push BlackX(i)!c color pop;MlpisUUadirectsumofindecompGosablekPG-modules;andcolor push Black;(ii)!c color pop;if^n;Zi=1 tOUiOand^m;Zi=1ViaretwodecompGositionsofMϲintoadirectsumofindecompGosable;kPG-moGdules,:[then ms=tnand,:Zafterapossiblereorderingofthesummands,UiT͍L+3L= QVi`ٲfor;allUUi.A?spGecial@class?ofrepresentationsisformedbythepro8jectivemoGdules.jRecallthatakPG-moGduleMisC(fr}'eeC'ifMZChasabasis.;?Thatis,~thereexistsasubsetXWSvMZBsuchthateveryC'mSu2SvMZBcanbGeuniquely_expressed`asakPG-linearcombinationofelements`inX.InthecaseofmoGdulesover_ nitegroup)algebras,^thefreemoGdulesareprecisely)thoseisomorphictodirectsumsofkPG(whichhasbasisUUf1g).HQcolor push BlackDe nition5.2.. color popA!CkPG-moGdule!OM8kispro8jective!PifandonlyifM8jsatis esanyofthefollowingequiv-alentUUconditions:Xߍ6color push Blackq(i) color pop;GivenanykPG-moGdulesAandBq,andhomomorphismsofkG-moGdulesf:rM;!Bq,and G;gxF:mA!Bq,UUwithg.surjective,thenthereexists\q|v~fj:mM3!Asuchthatg\q~[ٵf ײ=f.[pInUUotherwords,givenadiagram=ꪵM"yfyꪺF`ꪄfd=%A dgnv"UT//>"fdZ8nv%B4(withUUg.surjective,8A;0#thereUUexists\q|v~f S2Homqk+BG"۲(M;A)suchthatthediagramW`GꪵM_Ŝf\Ŝꪺ\jꪄfd VCw~㍒Af<`H~~<`H͹}?Of}BV}E]}Hd-1}L k :}O,r Hc}RNyU}1G%AGngU"UT//<`H"fdZ8X%Bqtcommutes.3%6color push Black(ii) color pop;AnyUUshortexactsequenceofkPG-moGdulesandhomomorphismsofkG-moGdulesjR0r//Z2fduꪵA!//!2fd!ꪵB80//812fd80ꪵMڟ//ڟ2fdڟ0splits. (SeeUUde nitionbGelow.)i6color push Black(iii) color pop;MlpisUUadirectsummandofafreemoGdule.6color push Black*(iv) color pop;MlpisUUinjective.HQNote2that2ifGisap-group,j8thenallthreeconceptsoffree,j8pro8jectiveandinjectivemoGdulesareequivqalenttbGecausethetgroupalgebrakPGisthenindecomposable.ЄInthecaseoftanarbitrary nitegroup:G,(rthen9apro8jectivemoGduleisinjectivebutnotfreeingeneral,(sasitisisomorphictoadirectsumUUofdirectsummandsofkPG. color push Black color pop#J9nL͍color push BlacktFINITEp-GROUPSINREPRESENTZATIONTHEORYl35 color pop37L͍ChouinardUUtheoremisadete}'ctiontheoremforpro8jectivemoGdules.color push BlackTheorem5.3. j color pop[Ben98b ,|TheoremF5.2.4F(Chouinard)]A nitelygener}'atedkPG-moduleMisprojective ifandonlyMispr}'ojectiveonr}'estrictiontoeveryelementaryabelianp-subgroupofG.AnotherUUusefulresultthatwerecordhereisSchanuel'slemmac.color push BlackTheoremT5.4. color popL}'etn0//2fdꪵMy//lß2fd [yꪵPK#f T//H;2fd ZTꪵN4t//t2fd䔟TIda㍄fdc㍄fd7t0&0Q#N7//#ifd7àQ&Llß#N7//n#ifd8lß%Qƴg T#N7//T#ifdT&N4t#N7//t#ifd7t&07찍b}'eshortexactsequencesofkPG-moduleswithPvandQprojectives.Then,bfMO8QT͍+3= UNLPv:color push BlackPr}'oof. color pop4CSince>PCͲispro8jective,thereexistsamapP*!QliftingtheidentityonN,sothatwecancompletethediagramintotwocommutativesquares.^RearrangingthemoGdulesintothediagrambGelow,!weconsiderthepullbackXofthearrowsf Pandg[ٲ,!whichwecompleteusingthestandardmethoGds:/iꪵL893fdJ891fdJQj5D xybsql10_㍺㍄fdꪵLij_'fꪺ̟ꪄfd}&MůBĄfdïBĄfd; 9H9#N7//H9#ifdH9&X㍒~㍒f #N7// #N7//Y#ifd7ō~g?ĺBĺBĄfd%Q7g?ĺBĺ̟BĄfd}LM;Fp鎟In//H9IOfdU鎟LPDf YIn// YIn//󸫟IOfdTYLN\SinceKbGothLPyڲandQarepro8jectivethemiddlerowLandthemiddlecolumnsplit.Thisyieldstheisomorphisms/SwMO8QT͍+3= UNXT͍+3=0LP casUUrequired.I%F*arKawayLfromthepro8jectivemoGdules,Ievery niteLgrouphasatrivialwkPG-mo}'dule.aThisistheonedimensionalvectorspacekNequippGedwiththetrivialgroupaction.kThatis,(-g[a8=7aforalla27kandallgp25G.Anotherusefulconceptisthatofthedualx(orykP-dual),ofakG-moGduleM.ItisthekPG-moGduleM^d=eHomkN(M;k). RInotherwords,OweconsiderthesetofkP-linearmapsM3!ekP. RThestructure!(of!)kPG-moGduleonM^'isgivenasaparticularcaseofthefollowingconstruction.`cLetM;NbGePtwoPkPG-modules.dT*akenPaskP-vectorPspaces,wesetPHom&k(M;N)PforthesetofkP-linearmapsM3!N.qThen,UUHom*k>(M;N)UUisakPG-moGdulefortheactionofGgivenbyT(g[')(m)=g'(g1 Mm) forUUallm2M;g"2GUUand'2Homqk(M;N).bHence,UUforM^,wetakeN3=kP,andwegetbl~J(g[')(m)='(g1 Mm) forUUallm2M;g"2GUUand'2M^.bAnxessentialwtoGolwhenstudyingkPG-moGdulesistheanalysisofkPG-homomorphisms,#andhenceofexactse}'quences!.qAnUUexactsequenceofkPG-moGdulesisasequence(possiblyin nite)::://2fdMn+1˶ifn+1ꘟ//ꘟ2fdꘞ*Mn7#fn//2fdMn1Gq$///q%2fdJq$:::ofѹkPG-moGdulesandkG-homomorphismssuchthatѸker.+(fnq~)i=him(fn+1)foralln.W*ecallanexactsequenceUUshortifitisanexactsequenceoftheform 0 // 2fd ꪵL//2fdꪵM //񹐟2fd ꪵN0َ//َ2fd3َ0?.:An½exact¾sequencesplitsifforeachnthereisakPG-homomorphismhn :@-Mn1+a!}pMn 4(M;N)asabGoveUUisakPG-module. 6color push Black(ii) color pop;ProveUUthatforanysubgroupH%SofG,the xe}'dpointsUUofHom*k>(M;N)undertheaction;ofUUH%SisthesetHom*k+BH$+(M;N)ofkPH-homomorphismsM3!N.6color push Black(iii) color pop;ProveUUthatHom*k>(M;N)T͍+3= UNM^߸ 8NlpasUUkPG-moGdules,foranykG-moGduleM.J5.2.EndotrivialTmoQdules.TwospGecialclassesofrepresentationsof nitep-groupshavebGeenidenti edbyE.Dade([Dad78a!1,Dad78b6|]):Zendo-p}'ermutation^andendotrivialmoGdules.>ThestudyofendotrivialmoGdulesfor nitep-groups/has0sincethenbGeenextendedtoarbitrary nitegroups,,fandthisisournexttopicofdiscussion.[color push BlackDe nition5.5. color popLet68GbGea nitegroup.ghA60kPG-moduleMMSisendotrivialifEnd kMT͍3+33=likK>(pro8jX)askPG-moGdules,forsomepro8jectivemodule(pro8jX).8Equivqalently*,byExercise5.29(iii),MisendotrivialifUUandonlyifM^߸ 8MT͍3+33=likw(pro8jX).\HereUUis\THE"example,splitintothreesteps:qtrivial,constructive,and... .color push BlackExampleT5.6. color popLetUUGbGea nitegroup.,J6color push Blackq(i) color pop;kisUUendotrivial. j6color push Black(ii) color pop;Let0///2fd2ꪵLT//<2fdWꪵPz_//b_2fd}_ꪵN//2fd0bGeashortexactsequenceofkPG-moduleswith;Ppro8jective.qThen,UULisendotrivialifandonlyifNlpisendotrivial.6color push Black(iii) color pop;Let5(P;@)4bGeaminimalpro8jectiveresolutionofthetrivialmoGdulekP.eF*oralln61let; ^nq~(kP)=ker݈(@n1),andsetalso ^0|s(k)=k,and ^nq~(k)= ^n (k)^]Ҳforallintegersn<0.;Then,h ^nq~(kP)isanindecompGosableendotrivialkG-moGdule.E)W*ecall ^nq~(k)then-thԺsyzygy;ofkP.,Icolor push Black color popExerciseT5.30.qDzProveUUthethreeassertionsinExample5.6.(Hint:thethirdassertionfollowsfromtheUU rsttwo.qF*orUUthesecond,Schanuel'sUUlemmamaybGeuseful.)ByKrull-Schmidttheorem(Theorem5.1), everykPG-moGdulecanbewrittenasadirectsumofinde-compGosable}Jmodules}Kwhichareuniqueuptoisomorphism(andorderofthesummands).ApplyingKrull-SchmidtUUtheoremtoendotrivialmoGdulesgivesthefollowing.[color push BlackPropQositiont5.7. color popEveryfendotrivialgmo}'duleMsplitsasadir}'ectgsumM⺲=˟M0":(pro8jX),wheregM0isaninde}'composableendotrivialmo}'duleand(pro8jX)issomeprojectivemodule.5ecolor push BlackPr}'oof. color pop4CW*riteM3=iTLUinasadirectsumofindecompGosablekPG-modulesUiTL.^6HenceM^T͍w+3w=MiU^;ZiʢandweUUhaveby\distributivity"of over,^WM߸ 8MT͍3+33=liM t(i;j3U፴i Uj:wSinceeM1isendotrivial,&/M^s MT͍3+33=lik(pro8jX),andeKrull-SchmidttheoremfsaysthatthereisexactlyonesummandU^;Zi8 UjisomorphictokI(pro8jX)andalltheothersarepro8jective.Thus,wemust color push Black color pop%{nL͍color push BlacktFINITEp-GROUPSINREPRESENTZATIONTHEORYl37 color pop37L͍havethati=j[oandthereisauniqueindecompGosabledirectsummandofMwhichisendotrivial, allUUtheothersbGeingpro8jective.#׋ color push BlackDe nition=5.8. color popTheindecompGosableendotrivialsummandM0 (oftheendotrivialmoduleMѲinPropGositionUU5.7isthec}'apofM.@UniquenessfinKrull-SchmidttheoremimpliesthatthecapofanendotrivialmoGduleMˁcharac-teriseseM,iandeitenablesustoworkuniquelywithindecompGosableendotrivialmoGdulesifwewish.This^fact^hadalreadybGeennotedbyDade,aWandmoregenerally*,aWthisisalways^thestandpGoint^takenwhenGworkingGinthestablemoGdulecategory*.mWethusGde neanequivqalencerelationontheclassofendotrivialUUmoGdulesbysetting:;gM3N(UX)M0T͍C+3C=N0|tforUUanyendotrivialmoGdulesM;N.Equivqalently*,RweRcanSsaythatStheequivqalencerelationistheisomorphismofthemoGduleswhen consideredUUinthestablemo}'dulecategoryUU(see[Ben98a [,CT03]).Qcolor push Black color popExerciseT5.31.qDzSuppGoseUUthatM;NlpareendotrivialkPG-modulesandletHG.qProveUUthat Res${GR${H-SMlpisUUanendotrivialkPH-moGdule,andthatM^ TandMO 8NareendotrivialkPG-moGdules.@color push BlackDe nition5.9.i color popThegsetofequivqalenceclassesTc(G)offendotrivialmoGdulesformagroup,1kcalledtheUUgr}'oupofendotrivialmodules!.qTheUUgroupstructureisinducedbythetensorproGduct.Thatis,;`[M]8+[N]=[MO 8N] forUUendotrivialkPG-moGdulesMlpandN.In/particular,7theidentity/ofTc(G)istheclass0=[kP]=fk=ݸF(pro8jX)g/ofalltrivialsourceendotrivialmoGdulese(seeparagrapheonthetheoryofverticesandsources,ionpage39).8Theadditiveinverseof[M]2Tc(G)UUis[M]=[M^]UUtheclassofthedualofM.Qcolor push Black color popExerciseT5.32.qDzLetUUGbGea nitegroupandn2Z.ProveUUthat;ko[ nq~(kP)]=n[ (k)] inUUTc(G).;F*romExercise5.31,weobservethatgivenasubgroupHXofG,therestrictionmapalongtheinclusionH,UX!GUUinducesawell-de nedgrouphomomorphismjMPResz PGRz PHn:mTc(G)!T(H) with)tRes8tGR8tH@r%([M])=[ResGRHU(M)]UU:WhatcanwesayabGoutthedimensionofanendotrivialkPG-moGdule?wjThe rstthingtoremembGeristhatthedimensionofapro8jectivemoGduleisdivisiblebytheorderofaSylowp-subgroupofG.Thus,UUfrom;-M߸ 8MT͍3+33=likw(pro8jX) weUUgetthatGcdimX@(M)2C1 (moGdjPcj) whereUUP*2Sylcqƴp߲(G).InUUotherwords,wehave:R (dim(M)81)(dim(M)+1)0 (moGdjPcj)whichUUgives׍RdimP(M)^dG1 (moGdjPcj)oifUUpisoGdd; G1 (moGdjPcj=2)oifUUp=2.%\Motivqationsߓforߔstudyingandseekingaclassi cationofallindecompGosableendotrivialmoGdulesarise from twomainstreamsinrepresentationtheory*.] Ontheonehand,#inthecasewhenGisap-group,the!endotrivialmoGdulesarethebuildingbricksoftheendo-permutationkPG-modules,,8whichinturnappGearinthedescriptionofthesourcealgebraofbloGcksofgroupalgebrashavingGasdefectgroup.On&theother'hand,"forany nitegroupG,"theendotrivialmoGdulesformpartofthePic}'ardYgroupofthe3+stable3*moGdulecategory*.fdWewill3+notdiscussanyofthesetopics,:butmentioningthemmayhelpto]vkeepin]wmindthat\endotrivialmoGdulesareuseful"... whatever]vthismeans.+Anotherfactwhichcomesm#straightoutofthede nitionofm$Tc(G)isthatclassifyingendotrivialmoGdulesisequivqalenttoydeterminingyTc(G).!Hence,wemayyalsoaimyatthemoremoGdestgoalto\only"loGokfortheisomorphismUUtypGeofthegroupTc(G)ratherthanitspresentationbygeneratorsandrelations.AnPessentialresultOabGoutthedetectionofTc(G)isduetoLuisPuig,andanother,provenPbyCarlsonandThGevenazisananalogueofChouinard'stheorem.ɈRecallthatE(G)denotesthesetofallthenontrivialUUelementaryabGelianp-subgroupsofG.@color push BlackTheoremT5.10. color popL}'etGbea nitep-group. color push Black color pop&nL͍color push Black388N.MAZZA color pop37L͍6color push BlackqŲ(i) color pop;(Carlson-ThGevenaz) VA2kPG-mo}'dule2MIisendotrivial2ifandonlyifResGRE^MIisanendotrivial ;kPE-mo}'duleforeachEZ2E(G).ֱ6color push Black򪨲(ii) color pop;(Puig#)jTher}'estrictionmapse Res% :E6(G)a:ZTc(G)!zyY jEb}2E6(G)% T(E) 8has nitekernel.YR;Inp}'articular,Tc(G)isa nitelygeneratedabeliangroup.FActually*,RtheR resultalsoRholdsforarbitrary nitegroups.pButbGeforeswitchingtonon-p-groups,RletusUUgivea rstclassi cationforabGelianp-groups.Fcolor push BlackTheoremT5.11. color pop[Dad78b!,UUTheorem10.1]Supp}'osethat汵Gisanabelian汵p-group.ThenTc(G)]=]h[ (kP)]iiscyclic,dspannedbytheclassofthesyzygy (kP)ofk~(se}'eExercise5.32).Hence,fFTc(G) 8isD88 D8< D8:d+'DztrivialZifjGj2,T͍+'Ǹ+3+'Dz=5Z=2ZifGiscyclicofor}'deratleast3,T͍+'Ǹ+3+'Dz=5ZZotherwise.$FAnotherlqkeylpfacttoinferfromPuig'sresultisthat,r7sinceTc(G)isabGelianand nitelygenerated,r8itcanUUbGewrittenasadirectsum {s_Tc(G)=TT(G)8TF(G) withTcT(G)thetorsionasub}'group,3andTcF(G)atorsion-free nitelygeneratedabGeliansubgroupofTc(G)whichisalsoadirectsumcomplementtoTcT(G).XByextension,wewillcallanendotrivialmoGduletorsionifitsclassisinTcT(G),@i.e.has niteorderinTc(G).Notethat,whileTcT(G)isunique,TcF(G)isnot,unique!@ThereareingeneralseveraldirectsumcomplementstoTcT(G).@Thelack=wof=v\non-canonical"complementis=vanissuewhich=vstillneedstobGeovercome=winthegeneralcase(workginhprogress...).mNevertheless,langimpGortantconsequencehofTheorems5.10and5.11isthat,fora nitep-groupG,^thetorsion-freerankofTc(G)(i.e.BKtheZ-rankofTF(G))equalsthenumbGerofconjugacyMclassesofMconnectedcompGonentsofthepGosetE2QL(G)fromNotation4.2.o?Inotherwords,WRcolor push Blackc!c color pop;ifsGhastrankatmost2,{therankofTcF(G)sequalsthenumbGersofG-conjugacyclassesof;maximalUUelementaryabGelianp-subgroupsofrank2;andcolor push Blackc!c color pop;ifGhasrankatleast3,therankofTcF(G)equalsthenumbGerofG-conjugacyclassesof;maximalUUelementaryabGelianp-subgroupsofrank2plusTone.The?achievement>oftheclassi cationtheoremforendotrivialmoGdulesoverp-groupshas>bGeenob-tained4FbyCarlson4GandThGevenaz,labGout30years4GafterDade'sresultabGove.F*orconvenience,lwerecapUUthewholeclassi cationinTheorem5.12.Fcolor push BlackTheoremT5.12. color pop[CT05q,UUCT04]L}'etGbeanontrivial nitep-group.WS6color push BlackqŲ(i) color pop;IfGiscyclicofor}'deratle}'ast3,)thenTc(G)T͍+3= UNZ=2.mQIfGiscyclicofor}'der2,)thenTc(G)=0.6color push Black򪨲(ii) color pop;If.Gis.gener}'alised.quaternion,UthenTc(G)T͍+3=Z=201Z=4(assumingthatk}isalgebr}'aically;close}'difGhasorder8).6color push Black㋲(iii) color pop;IfGissemi-dihe}'dral,thenTc(G)T͍+3= UNZ=28Z.6color push Black*(iv) color pop;IfGisnotcyclic,gener}'alisedquaternion,orsemi-dihe}'dral,thenTc(G)istorsion-fr}'ee.6color push BlackIJ(v) color pop;If4#every4"maximalelementaryab}'eliansubgroup4#ofGhasr}'ankatle}'ast3,1thenTc(G) =;h[ (kP)]iT͍+3= UNZ.6color push Black*(vi) color pop;Ifsomemaximalelementaryab}'eliansubgroupofGhasrank2andifGisnotsemi-dihedral,;thenTc(G)isfr}'eeabelianonasetofexplicitgenerators(see[CT04q,UUTheorem3.1]).FE.ZDade'scontributionZtotheclassi cationofendotrivialandendo-pGermutationmodulesgoesbeyondhisCseminal1978Dtwo-partCarticle[Dad78a!1,Dad78b$I].RIndeed, inanunpublishedmanuscript[Dad801],healsoprovesaresult(Theorem7.1inthatpapGer)whichhasbGeenkeyinextendingtheclassi cationof2endotrivialmoGdulestoarbitrary nite2groupswithanormalSylowp-subgroup.fGF*orconvenience,werecastDade'stheoreminaslightlydi erentway*,andonlyforendotrivialmoGdules.xRecallthatif/H.isa0normalsubgroupofG, thenGactsonakPH-moGduleMKbyconjugation: (foraneasierreading,UUwepGointoutwherethemultiplicationsoGccur)g"WM6isUUthekPH-moGdulegivenbyh:g_m.=gw( hg:m) forUUallg"2G,allh2H,andallm2M. color push Black color pop'nL͍color push BlacktFINITEp-GROUPSINREPRESENTZATIONTHEORYl39 color pop37L͍Hence,[theZZG- xe}'dpointsinmoQd((kPH)Z[areZZthekH-moGdulesZZMqvsuchthatZ[MT͍摸+3摲=}&^gyMaskH-moGdules. IfUUwerestrictourattentiontoendotrivialmoGdules,Dade'stheoremreadsasfollows.color push BlackTheoremT5.13. color pop[Maz07v,UUCorollary3.2(Dade)]L}'et P2lYSylΟqƴp (G)andsupposethatPElYG.@ThenTc(G)isgeneratedbytheclassesoftheindecom- 2p}'osableendotrivialkPG-modulesMsuchthatResSGRSPiMisinde}'composable.Inparticular,themapǍS(RescN(GRcN(Ppc:ZTc(G)!T(P) 8induc}'esanisomorphismofabeliangroups퍑TcF(G)T͍D+3D= eTF(P)^G=P,wher}'ejTF(P)^G=PisithesubsetoftheG=Pc- xe}'dpointsjofTcF(P).Moreover,wehavethatifTcT(P)=0,thenTcT(G)isgener}'atedbythe1-dimensionalkPG-mo}'dules.InTheorem5.13,TcF(P)^G=PmisthesubgroupofTF(P)generatedbytheclassesofthe\G-stable"endotrivialյkPPc-moGdules,twhichallextendintokPG-moGdulesandgenerateԵTc(G). GSinceTF(P)isnotycanonical,BTheorem5.13shouldbGeunderstoodasfollows:(foranychoiceofTcF(P),Btheysetof a xedpGointsεTcF(P)^G=P|isasetofequivqalenceclassesofendotrivialkPPc-moGduleswhichextendtoGײ(i.e.hMtheyareinfactkPG-moGdules),&andtheirclassesformasetofgeneratorsforatorsion-freesubgroup\gTcF(G)of\fT(G),^,directsum\gcomplementtoTcT(G).In\gpractice,^+onceTc(P)\gisknown,^+thecomputationsRgivingQTc(P)^G=PcomedownRtoamatterofcalculatingG-conjugacyclassesofelementsofUUPc.qF*ormoredetails,includingexamples,see[Maz07v].At}this}stage,ourob8jective}fortheclassi cationofendotrivialmoGdulesgetstrickier.Indeed, thereisYnowayZweYcansimply\extend"amoGdulefromthenormaliserofaSylowp-subgrouptothewholegroup.^BThemaintoGolavqailableistheGreencorrespondence(see[Ben98a [,&|x3]),butstillitdoGesnotwork@thatwell.jLetusbrie yreviewthesituation.SuppGosethatGisa nitegroupandPiaSylowp-subgroup ofG.4SetN3=NGڲ(Pc).Then,µanykPG-moGdulecanbeconsideredasakPN-module,¶viathe usualurestriction,}sothatweuobtainawell-de nedugrouphomomorphism=ResNN=GRNN=NWfڧ=Id ğb#imb#(Res {Gύ {N)3{ :hThe-\obvious"-candidateforbGeingthewantedsplitting-istheusualinductionofmoGdules,5(orrathertheVGr}'eencorrespondence.v;Namely*,0givenVanyindecompGosablekPG-moduleM,/weassociatetoitavertexandasour}'ce,РwhicharerespGectivelyap-subgroupQofGòandanindecompGosablekPQ-moGduleV8.jLThesubgroupQisuniqueuptoconjugacyandV6fuptoisomorphismandconjugate.The*subgroupQisa)smallestp-subgroup(fortheorder)suchthatMEcanbGewrittenasadirectsummand/ofamoGduleinducedfromQ,fLi.e.suchthatMFϲispr}'ojective\relativeto/Q.Hence,fLthe 35source~SV7isanindecompGosable~RkPQ-module~SsuchthatMnj(IndGōQXĵV8.Itisnotatrivialresult,andso UQwe4referto[Ben98a [,;3Section3]foraproGofofthisandthefollowingresultsused4withoutproGof.fTheGreenxcorrespGondencexreliesonthemarvellouspropGertiesofverticesandsources.Namely*,~suppGosethatUUQisap-subgroupofGandthatH%SisasubgroupofGcontainingNGڲ(Q).qThen,ffff4ffdthereUUisa181UUcorrespGondencebetweenUUtheindecomposablekPG-moduleswithvertexQandUUtheindecompGosablekPH-moduleswithvertexQ.hffffff4-TheZrelationshipbGetweenZtwoGreencorrespGondents, sayanindecompGosablekPG-moduleMqϲwithvertex*ٵQ,3Xand*anindecompGosablekPH-module*صUAalsowith*vertexQ,3Xare*summarizedbythe*2facts:˸color push Blackc!c color pop;ResJߍGRJߍHTMT͍3+33=liUL,wherealltheindecompGosablesummandsinLhavevertexsmallerthanQ. Ncolor push Blackc!c color pop;IndIsGōIsHS̵UT͍)+3)=AM:#L^09, where~alltheindecompGosablesummandsinL^0have~vertexsmaller;thanUUQ.Ing'suchg(acon guration,wesaythatg(M~BisthekPG-Gr}'eencorrespondentZofg'U,andthatU~CisthekPH-Gr}'eencorrespondentUUofM.ThePessentialassumptionintheGreencorrespGondenceisthatthetwogroupsGandH containthenormaliserofavertex.#SincethedimensionofendotrivialmoGdulesiscoprimetothecharacteristicofthe eld,indecompGosableendotrivialkPG-moduleshavevertexaSylowp-subgroupofG.KHence,asUUcorollaryoftheGreencorrespGondence,wegetthefollowinginjectivityresult.color push BlackPropQosition5.14.е color popL}'etcGbea nitegroupcwithSylowp-subgroupPandHNGڲ(Pc).yTherestric-tionmap M|ResGRHɲ:ZTc(G)!T(H) 8isinje}'ctive. color push Black color pop(;nL͍color push Black408N.MAZZA color pop37L͍color push BlackRemarkt5.15. color popW*ealreadyknewthattherestrictionmapbGetweengroupsofendotrivialmoGdulesis well-de ned,bGecauseҪtheҩrestrictionofapro8jectivemoGduleisapro8jectivemoGdule.F8Thefactthatthe GreenΔcorrespGondenceisΕa11correspondenceimpliestheΕinjectivityofResGRH$E.݅However,forthereversesopGeration,i.e..thepassagefromrTc(H)toT(G)rusingtheinductionandthentakingtheGreencorrespGondent,0doesnotgiveanendotrivialkPG-moGduleingeneral.V;Indeed,1ifM isthekPG-GreencorrespGondentOofU,withU kanindecomposableendotrivialkPH-module,thenM jisPendotrivialifandonlySifRResRGRRHMT͍.+3.=_U޲(pro8jX).`ThisisRfalseingeneral.Indeed,#notRalltrivialsourcemoGdulesareendotrivialUU(see[CHM10!]foradiscussionofsomeinterestingcases).TheUUsituationwheretheGreencorrespGondenceissucientisnext.color push BlackPropQosition=5.16.޽ color popL}'etnGbea nitegroupandoHlastronglyp-embeddedsubgroupofoG.+Ther}'estrictionmap IResКGRКHJ:ZTc(G)!T(H) 8isanisomorphism.RecallthatasubgroupHZofGisstr}'onglyop-embeddedifpdividesjHjandifpdoGesnotdividejHs\^gH jforeveryg"2GT@H.KThisisthecasewhenH=NGڲ(S)isthenormaliserofaSylowp-subgroupSvofG%whenever$SrisaTIsubgroupofG,i.e.Jbtrivial'+interse}'ction.Asubgroup%(orsubset)$SrofagroupGisUUtrivialintersectionifandonlyifSm\8^g5?S=1(or;)foreveryg"2G8NGڲ(S).2color push Black color popExerciseT5.33.26color push BlackqŲ(i) color pop;ProveUUthatanendotrivialmoGdulehasvertexaSylowp-subgroup.6color push Black(ii) color pop;ProveUUthatifaSylowp-subgroupPofagroupGhasorderp,thenNGڲ(Pc)isTI.;HenceUUNGڲ(Pc)isstronglyp-embGedded.6color push Black(iii) color pop;ProveUUPropGosition5.16Leavingaparttheseobstacles,LletusturntowardsothermatterswhichwillhelpuslinkbackwiththeUUresultsintheprevioussection.color push BlackTheoremT5.17. color pop[CMN06!]L}'etR{GbeaRz nitegroup.Thetorsion-freeRzranknG UofTc(G)equalsthenumberRzofconjugacyclassesofthej}p}'osetj~E2n(G).Thatis,rifGj~hasacyclicSylowp-sub}'group,rthenj}nG =0,ifGj~hasp-r}'ank2,thennG isVtheVnumb}'erofG-c}'onjugacyclassesofelementaryab}'elianp-subgroupsVofrank2VofG,c,andifGhasr}'ankatleastŲ3,thennG isoneplusthenumb}'erofG-c}'onjugacyclassesofmaximalelementaryab}'elianp-subgroupsofrank2.CarryingBonAwiththeinformationonTc(G)thatcanbGeobtainedbythestructureofE2Ӳ(G),}andtheUUresultsin[Maz08v,GM10X],wealsoobtainthefollowing.color push BlackPropQositionT5.18. color popL}'etGbea nitegroup.WriteTcT(G)forthetorsionsubgroupofTc(G).6color push BlackqŲ(i) color pop;Ifthep-r}'ankofGisgreaterthanpifpisodd,orgreaterthan4ifp=2,thene6Tc(G)=TT(G)8h[ (kP)]iT͍+3= UNTT(G)Z:6color push Black򪨲(ii) color pop;Thetorsion-fr}'eerankofTc(G)isatmostp8+1ifpiso}'dd,andatmost5ifp=2.color push Black color popExerciseT5.34.qDzUseUUtheresultsinSection4toproveUUPropGosition5.18.16.şOOddbitsandendsBelowaresomeusefulresultsregardingthestudyof nite(p-)groupswhicharenottreatedinthe lectures,UUalthough,theremayalsobGeafewredundancieswithnotionsfromSection1.e6.1.OnTnilpQoten9tandsolv\rablegroups.F*romDe nition1.18, itiseasytoseethatnilpGotentgroupsaresolvqable.RIndeed, wehavethatifthederived-seriesconverges-to1,5thensodoGesanycentralseriesbGecause-ٵidGi$foralli.dHowever,5theconverseUUdoGesnotholdingeneral.color push Black color popExerciseT6.35.qDzProveUUthatthesymmetricgroupS4ȲissolvqablebutnotnilpGotent.NipGotentgroupsarenotabelian,Gbuttheyarestillaclassofgroupseasiertostudy*.BsThenextpropGositionUUmotivqatesthisclaim. color push Black color pop)nL͍color push BlacktFINITEp-GROUPSINREPRESENTZATIONTHEORYl41 color pop37L͍color push BlackPropQositionT6.1. color popL}'etGbeagroup.J6color push BlackqŲ(i) color pop;Gisnilp}'otentifandonlyifGisthedirectproductofitsSylowp-subgroups. 6color push Black򪨲(ii) color pop;IfGisnilp}'otent,rthenanypropersubgroupofGisproperlycontainedinitsnormaliserin;G.6color push Black㋲(iii) color pop;IfGisap-gr}'oup,thenGisnilpotent.color push Black color popExerciseT6.36.qDzProveUUPropGosition6.1.܍color push BlackLemmaT6.2. color pop[Hal34G!,UU2]AssumethatGisap-gr}'oupofclassc.6color push BlackqŲ(i) color pop;ZciF<GidZci+1G1,foralli. 6color push Black򪨲(ii) color pop;L}'etKKbeasubgroupofG.;color push BlackUV(a)u color popP<IfK~4EGandKZap,thenK\8(Za+1/SZa)6=;andjKj>p^a.;color push Black(b)u color popP<IfK~4EGandKGap,thenhK(;Ga+1siGajWandjG:Kjp^ap.;color push Black㐲(c)u color popP<IfK~4ap,thenhK(;a+1siajWandjG:Kj>p^ap.6color push Black㋲(iii) color pop;[Zj6;GiTLjZjgi J,foralli;j.Henc}'e,ZjĸCGڲ(Gj6),forallj.6color push Black*(iv) color pop;[GiTL;Gj6]Gi+j ..Thus,if2ic,thenGi3isab}'elian.6color push BlackIJ(v) color pop;If/ H is/ asub}'group/ ofGgener}'ated/ byc}'ommutatorsofweightwD,UthenH!$Gwy1.kThus,if;w c,thenanyc}'ommutatorofweightwistrivial.6color push Black*(vi) color pop;idGM2i ;RandiCGڲ(ZM2ik).Thus,ifajWisnontrivial,thenc2^iTL.6color push Blackc(vii) color pop;(GiTL)jIJ=Gij .Thus,ifijY>c,thentheclassofGjʓissmallerthani.6color push Blackm(viii) color pop;If-KIisnormalinG.andKM7GiTL,"thenthefactorgr}'oupsKsF:=Ks+1ljandZ(K)s=qZ(K)s1;have@or}'der@atleastp^iTL,kexcept@maybeforthe@lasttermsine}'achseries..Thus,kifKisnot;ab}'elian,thenjZ(K)jp^i3andjK(=K2|sjp^iTL,forac}'ertainintegeri,atleast2.6color push Black*(ix) color pop;IfYb+1?6=;1,thenjbD=b+1$Gj>p^2b andthefactorsXoftheLCSandoftheUCShaveor}'derat r;le}'astp^iTL.ۍInUUthislastitem,thesetM(G)isde nedasfollows:color push Blackc!c color pop;E(G)UUisthesetofallelementaryabGeliansubgroupsofG.color push Blackc!c color pop;A(G)UUisthesetofallEZ2E(G),suchthatrankصE=rank:G.color push Blackc!c color pop;Ny=(G)UUisthesetofallnormalelementaryabGeliansubgroupsofG.color push Blackc!c color pop;M(G)isthesetofmaximalelementsinNy=(G).XDThatis,M(G)isformedbyallEZ2Ny=(G),;suchUUthatEZFc,withF*2Ny=(G)impliesEZ=Fc.color push BlackLemmar6.3(N.8Blackburn). color popL}'etfGebeafp-group.1IfG2Bisnoteabelian,then2="(G2|s)2Bhaseindexatle}'astp^3ZinG2andZ(2|s)isnotcyclic.܍F*or8 completeness,pletusrecallthata nite8 groupGismetab}'elianifitscommutatorsubgroupisabGelian.6.2.Comm9utators.LetYusrecapsomewell-knownYcommutatoridentities.LetGbGeagroup,HA;K(;LGUUandx;y[;z72G.color push BlackX(i)!c color pop;[xy[;zp]=[x;zp]^y·[y[;z]=[x;zp][x;z;y[ٲ][y;zp];color push Black;(ii)!c color pop;[x;y[zp]=[x;zp][x;y[ٲ]^zb=[x;zp][x;y[ٲ][x;y;zp];color push BlackG(iii)!c color pop;[x;y[ٲ]^1 =[y;x];color push Black:(iv)!c color pop;[HA;K]EhH;K8߸i;color push BlackUW(v)!c color pop;HNGڲ(K)UUifandonlyif[HA;K]K;color push Black:(vi)!c color pop;[HA;K]=[K(;H];color push Black(vii)!c color pop;IfUUHA;K~4EGwithKH,thenHA=KZ(G=K)UUifandonlyif[HA;K]K;color push Black (viii)!c color pop;SuppGoseUUthatK~4EG.qThenG=K qisabelianifandonlyif[G;G]K;color push Black:(ix)!c color pop;IfUUHA;K(;LEG,then[HK(;L]=[HA;L][K(;L];color push BlackUW(x)!c color pop;'([HA;K])=['(H);'(K)],forany'2Homq(G;G).[Inparticular,[HA;K]EGifandonlyif;bGothUUH%SandK qarenormalinG;color push Black:(xi)!c color pop;LetUUz7=[x;y[ٲ]andsuppGosethat[zp;x]=[zp;y[ٲ]=1.qThen,;color push BlackUV(a)u color popP<[x^iTL;y[ٟ^i%]=zp^ij forUUalli;j;;color push Black(b)u color popP<(xy[ٲ)^id=y^i%x^iTLzp^dI,UUwithd= K1K&fes2 )i(i81); ;color push Black(c)u color popP<[x;y[ٲ]^1 =[x^1 t;y]=[x;y^1 M]. color push Black color pop*rnL͍color push Black428N.MAZZA color pop37L͍AusefulresultisThompson'sreplacementtheorem.L`Therefore,weintroGducethefollowingnotation. LetKA(G)bGethecollectionofKallabeliansubgroupsofap-groupG,MandKletma"bethemaximumoftheݞordersoftheelementsofݝA(G). SetA0|s(G)forthesubsetofA(G)formedbytheelementsoforderUUmap.4>color push BlackTheoremT6.4(Thompson'sUUreplacementtheorem). color pop[Gor80Ò,Theorem8.2.5]L}'etGbeap-group, +Av׸2A0|s(G),andBH2vA(G).Supp}'osethatAv׵NGڲ(Bq)andthatBGNGڲ(A).Then,ther}'eexistsA^_2A0|s(G)withtheproperties:GP6color push BlackqŲ(i) color pop;A8\BG(A^ĸ\Bq,and6color push Black򪨲(ii) color pop;A^_NGڲ(A).4?AsUUaconsequenceofThompson'sreplacementtheoremweobtainthefollowing.color push BlackTheorems/6.5.2 color pop[Gor80Ò,Theorem8.2.6]EdL}'etEcGbeEcap-groupandEcAanabelianEcnormalsubgroupEcofG.Then,ther}'eexistssomeBG2A0|s(G)withANGڲ(Bq).color push BlackPr}'oof. color pop4CObserve$Othat$NA NGڲ(Bq)forallA 2A0|s(G).޵Hence,X letuschoGoseA2A0|s(G)$NsuchthatjA8\BqjUUismaximal.qByTheorem6.4,wemusthavethatBGNGڲ(A).x/vF*urtherYconsequencesandimprovementsYofThompson'sYreplacementtheoremarediscussedinGoren-stein'sUUbGook([Gor80Ò,8]).ۍ6.3.Constructions.When~studyingan~abstractgroup,onestrategyistoreducetheproblembycuttingitintosmallerpieceswhicharegroupsinthemselvesandwhichareassembled\methoGdically".NFThepurposeofthissection"is!todescribGethree\assembling"constructions:thesemi-directproGduct,thewreathproGduct,and*Lthecentral*KproGductofgroups.Theseprovideus*Kwithusefulmeansofconstructingp-groupswithUUprescribGedproperties(seealso[Maz03b"*]).AsUUfordirectproGducts,therearetwoUUsortsofsemi-directproducts.4?color push BlackDe nitionT6.6. color popLetUUGbGeagroup,andN;H%SsubgroupsofGsuchthat:GP6color push Blackq(i) color pop;N3EG;6color push Black(ii) color pop;NO\8H=1;6color push Black(iii) color pop;G=NH.GOThen,UUGisthe(internal)semi-dir}'ectproductAn2internalsemi-directproGductcanalsobeseenasexternal,jviatheidenti cationHT͍+3=[G=qNO;Out%(N),Mwith.OutU(N)T͍+3=Aut (N)=AutrN2#(N).thegroupofouterautomorphismsof-N.RTherefore,weͿdo;notdistinguishbGetweenͿbothsemi-direct;products.Clearly*,directproducts;aresemi-directproGducts,@and;theconverse;is;trueifandonlyifthegroupisaninternaldirectproGductG=NoŵHwithUUHCGڲ(N)(whichimpliesN3CGڲ(H)).color push BlackDe nition(6.8.A color popLetcWƧbGecasubgroupofsomesymmetricgroupSnԗofdegreen,fandletH3beanygroup.qTheUUwr}'eathproductH-ofH%SbyWisthesemi-directproGductJH޸o8W*=(HgH)1|31{z31}1nȱtimesB=oWwhereW%actsonthencopiesofHbypGermutationonthefactors.@Explicitly*,|everyelementofHAoDWcanwXbGewritenwYas(h1|s;:::;hnq~;wD)wXwithhiT2ȵHGVforalliandwD2ȵWc,andthemultiplicationruleisasfollows: [mg=f(h1|s;:::;hnq~;wD)(k1;:::;knq~;v[ٲ)=(h1k㐴w081 (1)u;:::;hnq~k㐴w081 (n);wDv[ٲ)forUUall(h1|s;:::;hnq~;wD);(k1;:::;knq~;v[ٲ)2H޸o8Wc. color push Black color pop+ nL͍color push BlacktFINITEp-GROUPSINREPRESENTZATIONTHEORYl43 color pop37L͍color push Black color popExerciseT6.37.*E6color push BlackqŲ(i) color pop;ProveUU(ifnotalreadydone)thataSylowp-subgroupPofthesymmetricgroupSp2 Dzof Gf;degreeUUp^2ȲisisomorphictoawreathproGductCp2o8CpR. 6color push Black(ii) color pop;ProveUUthatC2So8C2T͍C+3C=D8|s.6color push Black(iii) color pop;FindUUthenilpGotenceclassofCp2o8CpR.qWhataboutLCSandUCSofCp2o8CpR?color push BlackDe nition.6.9.`r color popLetG;HbGegroupsandZ/anabeliangroupsuchthatthereareinjectivegrouphomomorphisms !Y :mZ~4!Z(G) and$wxF:Z!Z(H)UU::W*e>|de nethec}'entral~productֱG .Z{H%ofGandHNwithrespecttoZas>|follows.j)ConsiderthedirectproGductUUG8H%SandtheequivqalencerelationTbg(g[;h)(g0*;h09)(UX)9UUz72Z with(((g0*;h09)=(g 8(zp);(z1 - )h)T(or#simply(g[;h)(g^0*;h^09)(UX)9#z2Z with(((g[ٟ^0*;h^09)=(g[zp;z^1 - h)#,Vvia$identi cationofZ?asacommoncentralsubgroupofGandofH).ZGThen,thegroupGZ=HӲisthesetofequivqalenceclasses.Inthecasethatthereisanobviousidenti cationZ.=3Z(H)3=Z(G),QthenwesimplywriteGdHinsteadUUofG8Z;H.WiththeabGovenotions,wwecannowde nea\veryspGecial"classofp-groups.TheycanbGefoundinseveralcriticalturningpGointsinthestudyof nitegroups,offusionsystems,andofcoursealsoinrepresentationtheory*._Moreover,#theircohomologyringisknown(seee.g.[BC92N=])._Theirclassi cationispresentedinmanyplaces,forinstancein[Hup67w,IGII.13],[Gor80Ò,5.5]..Sincethereareslightհdi erencesձinthede nitions(depGendingonauthors'schoices),8weհpresenthereհthatof[Hup67w,IGII.13],UUcomplementedwith[BM04@]andwith[Sta02q].color push BlackDe nitionT6.10. color popLetUUGbGeap-group.*E6color push Blackq(i) color pop;GUUissp}'ecialUUifeither;color push BlackUV(a)u color popP<GUUiselementaryabGelian,or;color push Black(b)u color popP<(G)=G^0Q=Z(G)UUiselementaryabGelian.;InUUparticular,aspGecialp-grouphasnilpotenceclassatmost2.6color push Black(ii) color pop;GUUisextr}'aspecialUUifGisanonabGelianspecialp-groupwithj(G)j=p.6color push Black(iii) color pop;Gisalmost,extr}'aspecialifGisthecentralproGductofacyclicgroupoforderp^2x withan;extraspGecialUUp-groupofexponentpifpisodd,oroftypeD8S8g8D8Ȳifp=2.6color push Black*(iv) color pop;GUUisgener}'alisedextraspecialUUifj(G)j=pUU(andisthuscentralinG).TheA(classi cationtheoremsA)explicitlydescribGethestructureoftheseclassesofp-groups.kItmustbGenoted!that theyallhave!thesamepropGertythattheirF*rattinifactorgroupisanelementaryabGelianp-group.MHence,appGealingtolinearalgebra,wegetvectorspaceswhichhavea\nice"geometricalbGehaviourUU(see[BM04@]forinstance).ūcReferencescolor push Black[AÎG98]E color popAG*J.\qAlp cmmi10 0ercmmi7O \cmmi5K`y cmr10ٓRcmr7Zcmr5u cmex10"