The determination of Abelian hall subgroups by a conjugacy class structure

Publicacions Ma emá iques, Vol 36 (1992), 685-691 . A bs ac THE DETERMINATION OF ABELIAN HALL SUBGROUPS BY A CONJUGACY CLASS STRUCTURE WOLFGANG KIMMERLE AND ROBERT SANDLING Pe e Menal in memo iam The objec o he a icle is o show ha a Jo dan-H&lde class s uc u e o a ini e g oup de e mines abelian Hall subg oups o he g oup up o isomo phism . The p oo uses he classi ica ion o he ini e simple g oups . A conjugacy class s uc u e on a g oup cap u es in o ma ion abou i s no mal subse s, o example, ha which is deducible om i s cha ac e able o om i s in eg al g oup ing . Va ious such s uc u es we e in- oduced in [KS] and used he e o d aw conclusions abou he g oup p e iously in es iga ed only using he cha ac e able o he in eg al g oup ing . The mos basic class s uc u e conside ed was a Jo dan-Hdlde class s uc u e, one which cap u es he pose o no mal subse s o a g oup, eco ds hei sizes, indica es which no mal subse s a e no mal subg oups and which a e he p eimages o he conjugacy classes o i s quo ien g oups . I was shown in [KS] ha such a class s uc u e de e mines he chie ac o s o a g oup . Mo e echnically, g oups G and G* ha e he same chie ac o s i hey a e in class co espondence o ype JH, ha is, i he e is a bijec i e co espondence be ween hem p ese ing no mal subse s and hei assumed p ope ies . (Mo e o mal de ini ions a e a ailable in ou ea lie pape .) I was also shown ha he isomo phism ype o an abelian Sylow subg oup is de e mined by a class s uc u e o ype JH, and ha o an abelian Hall subg oup by a s onge class s uc u e (namely, one in which he se o p imes in ol ed in he o de s o he elemen s in conjugacy classes is also posi ed) . In his pape , his ex a hypo hesis is emo ed o gi e he ollowing . 68 6  W . KIMMERLE, R . SANDLING Main Theo em . A Jo dan-Hólde class s uc u e de e mines abelian Hall subg oups up o isomo phism . I should be no ad ha he esul s o ou ea lie pape a e dependen on he classi ica ion o he ini e simple g oups . Apa om building on ou ea lie esul s, his pape makes u he in oca ions o he classi ica- ion . Ou collabo a ion on he opic o abelian Sylow and Hall subg oups was a di ec consequence o he second au ho 's pa icipa ion [San] in he 1986 ing heo y con e ence in G anada in which Pe e Menal played a p ominen ole . We begin he p oo o ou heo em by ixing some no a ion . The g oups G and G* will be assumed o be in class co espondence o ype JH wi h * being used o deno e he bijec ion (on subse s and subg oups as well as on elemen s) . We assume ha , o a ixed se 7 o p imes, G has abelian Hall 7 -subg oups . We mus show ha G* also has abelian Hall 7 -subg oups and ha hey a e isomo phic o hose o G . By ou esul on abelian Sylow subg oups, i su ices o show ha G* has Hall 7 -subg oups and ha hey a e abelian (o me ely nilpo en ) . The p oo o he de e mina ion o abelian Hall 7 -subg oups when 17 i >_ 2 is mo e elabo a e han ha o abelian Sylow subg oups . One eason o his is he absence o a di ec analogue o g oups wi h abelian Hall 7 -subg oups o he c i e ion [KS, 2 .1] o de ec ing abelian Sy- low subg oups . This can be seen om he g oup ob ained by ex end- ing L2(7 5) by he cyclic g oup C5 ac ing as ield au omo phisms (he e 7 = {3, 5}) . (Con as he beha iou o Sylow subg oups as seen in he P oposi ion below wi h ha o Hall subg oups .) Unde a class co espondence o ype JH, he o de s o x and x* need no coincide ( iz ., he qua e nion and dihed al g oups o o de 8) . This ende s subg oups like 0"(G), gene a ed by all 7 -elemen s o G, less use ul han hey would be i o de s we e p ese ad as hey a e by s onge class co espondences (sea [KS]) . In hei place, we use he ollowing cha ac e is ic subg oup o a g oup . De ini ion . Le Z be a g oup and p a se o p imas . De ine C p (Z) as he subg oup gene a ed by all x in Z o which IZ : C Z (x) j is a p'-numbe , whe e CZ(x) deno es he cen aliza o x in Z . Unde a class co espondence o ype JH, Ci,(G*) = C, (G)* . No e, in addi ion, ha , i G has abelian Hall 7 -subg oups, hen hey a e all con ained in C, (G) ; i ollows om a heo em o Wieland [Suz, 5 .3 .2] ha 0-'(G) < C, (G) . In he p oo o Theo em 2 .1 o [KS], elemen s no malising each simple Lemma 1 . Suppose ha ABELIAN HALL SUBGROUPS  68 7 ac o o each pe ec minimal no mal subg oup o a g oup we e exam- ined . Fo he de e mina ion o abelian Hall subg oups, such elemen s play a mo e p ominen ole which is made mo e explici in he ollowing de ini ion o he cha ac e is ic subg oup K(G) which hey comp ise . I he g oup G has a unique minimal no mal subg oup and his subg oup is nonabelian (so ha G is embedded in a w ea h p oduc [Rose, p .223]), K(G) is he in e sec ion o G wi h he base g oup o he w ea h p oduc . De ini ion . The cha ac e is ic subg oup K(G) o G is de ined as he in e sec ion o all NG(S) whe e S is a nonabelian simple subg oup o G which is no mal in Soc G, he socle o G . Fo a nonabelian minimal no mal subg oup M o G, le K(GmodCG(M)) deno e he in e se image in G o K(G/CG(M)) . I is easy o see ha K(GmodCG(M)) is he in e sec ion o all NG(S), S a simple subg oup o G no mal in M . I ollows ha K(G) is he in e - sec ion o all such K(GmodCG(M)) . The de ini ion o K(G) is simila o ha o McB ide's k(G) [McB, p .217], and he wo subg oups coin- cide in he case, impo an he e in educ ion a gumen s, whe e G has a unique and nonabelian minimal no mal subg oup . One well-known consequence o he classi ica ion o he ini e simple g oups which was used in [KS] is needed again he e, and is s a ed o he eade 's con enience (c . [GL, 7 .10]) . P oposi ion . Le S be a simple g oup o o de di isible by a p ime p . I X is a subg oup o Au S in which S <_ X and p di ides he index o S in X, hen X has nonabelian Sylow p-subg oups . The de e mina ion o abelian Hall subg oups is accomplished h ough a se ies o educ ion s eps (c . he p oo o he main heo em) . They lead o he ollowing si ua ion which is exploi ed in he subsequen lemmas . (i) G has abelian Hall 7 -subg oups ; (ii) each minimal no mal subg oup M o G has o de di isible by some p ime in 7 ; (iii) each minimal no mal subg oup M o G is nonabelian . Then C (G) < K(G) and C (G*) < K(G*) . P oo . . Fo he i s conclusion, i su ices o show ha C  (G) _< K(GmodCG(M)) o each minimal no mal subg oup M o G . Le x be an elemen o G o which IG : CG(x)j is a 7 '-numbe . By (ii), he e is some pE n o which M con ains non i ial Sylow p-subg oups . Le P be a Sylow p-subg oup o G con ained in CG(x) . Then P n M is a 68 8  W . KIMMERLE, R . SANDLING non i ial Sylow p-subg oup o M . I ollows ha , i S is a simple sub- g oup o G no mal in M, hen x ixes a non i ial elemen o S . Thus, xE NG(S), which su ices . The second conclusion ollows in a simila way using he esul om [KS] ha each minimal no mal subg oup M* o G* is isomo phic o such a subg oup o G (namely M) so ha pa s (ii) and (iii) apply o M* . While Soc G* = (Soc G) * and is isomo phic o Soc G by [KS] in his case, we do no know whe he simila s a emen s hold o K(G) ( o example, unde pa (iii) abo e) . We e such s a emen s ue, he de e - mina ion o abelian Hall subg oups would be accomplished a his poin . In hei s ead, we u n o a close examina ion o wo ques ions : when does a g oup X, S <_ X <_ Au S o a simple g oup S, ha e an abelian Hall subg oup? How can his be ecognised using p ope ies de es able unde a class co espondence o ype JH? Suppose ha G has he unique minimal no mal subg oup M, M ~ Sa, o he nonabelian simple g oup S ; hen G may be iden i ied wi h a sub- g oup o Au S' which is a w ea h p oduc o Au S and he symme ic g oup o deg ee a . As ema ked, K(G) is he in e sec ion o G wi h he base g oup (Au S)a . In ac , he e is a subg oup X o Au S, namely he image o NG(S)/CG(S), o which K(G) is a subg oup o Xa whose p ojec ion o each componen is su jec i e . The nex wo lemmas enable us o de e mine he isomo phism ype o an abelian Hall subg oup o Xa on he basis o in o ma ion deducible unde a class co espondence o ype JH . I will be con enien o use wo i ems o s anda d no a ion in ol ing a se p o p mms and a g oup Z : ¡Zl p o he p -pa o he o de o Z and Z p o a Hall p-subg oup o Z . The i s o he lemmas makes ano he appeal o he classi ica ion o ini e simple g oups . Lemma 2 . Le S be a ini e simple g oup and le Y be a subg oup o Au S con aining S . Le k be a di iso o ¡Y¡ which is ela i ely p ime o ¡Si . Then Y has a unique conjugacy class o subg oups o o de k and hese subg oups a e cyclic . P oo .. Fo al e na ing and spo adic g oups, whose ou e au omo - phism g oups a e 2-g oups, he s a emen is acuous . Cyclic simple g oups ha e cyclic ou e au omo phism g oups so he lemma is s aigh - o wa d . Fo a simple g oup S o Lie ype, only a ield au omo phism has o de ela i ely p ime o 151 (see, o example, [MeB, Lemma 4 .1]) . Le o, be he se o p ime di iso s o k .  Now Ou S has a cyclic subg oup ABELIAN HALL SUBGROUPS  68 9 (o ield au omo phisms) which con ains a cyclic Hall Q-subg oup . As Ou S is soluble, Y/S con ains a cyclic Hall Q-subg oup, and so Y does as well . By he heo em o Wieland ci ed ea lie , all Hall -subg oups o Y a e conjuga e and e e y subg oup o o de k is con ained in a Hall u-subg oup so ha hese subg oups a e also conjuga e in Y . Lemma 3 . Le S be a ini e simple g oup . Suppose ha X and Y a e subg oups o Au S con aining S and ha IX I, = ¡Y¡, . Then X has abelian Hall 7 -subg oups i and only i Y does ; i X, and Y, a e abelian Hall 7 -subg oups o X and Y espec i ely, hen X, and Y, a e conjuga e in Au S . P oo . Le u = 7 - 7 (S) and le k = IXI o = ¡Y¡, . Suppose ha X, is an abelian Hall 7 -subg oup o X . By he P oposi ion, IX : Si = k . By Lemma 2, X, = S~C whe e S, = S nX, and whe e C is cyclic o o de k . Lemma 2 also shows ha Y has a subg oup D o o de k and ha he e is an au omo phism a E Au S such ha D =Ca . I ollows ha Xa~ is a Hall 7 -subg oup o Y . Tha all such Hall 7 -subg oups a e conjuga e in Y ollows om he p e iously ci ed heo em o Wieland . Thus, X, and Y, a e conjuga e in Au S . P oo o Main Theo em : Recall ha ou objec is o show ha G* has an abelian Hall 7 -subg oup . We may assume ha 0, , (G) = 1 = O, , (G*) . No e ha 0 7 "(G*) = 0"'(G)* ; as 0"'(G) < C, (G), 0" (G) * <_ C,,(G)* = C,, (G*) so ha C, (G*) con ains all 7 -elemen s o G* . By induc ion, we may assume ha any p ope quo ien o G* has abelian Hall 7 -subg oups . We may also assume ha G and G* a e no simple by [KS] . Le M be a minimal no mal subg oup o G so ha G*1M* has an abelian Hall 7 -subg oup . As M* ~:d M, M* also has an abelian Hall 7 -subg oup . By he heo em o Wieland ci ed abo e and a heo em o Hall [Suz, 5 .3 .12], G* has a Hall 7 -subg oup . By ou ea lie Sylow esul , each Sylow p-subg oup o pE 7 is abelian ; i emains o show ha a Hall 7 -subg oup is i sel abelian . As i s image in any p ope quo ien o G* is abelian, we may assume ha M* is he unique minimal no mal subg oup o G* and hus ha M is he same in G . I M and M* a e abelian, hen, o some pE 7 , hey a e (elemen a y abelian) p-g oups . Fo x* E G* such ha IG* : CG * (x*)j is 7 ', M* <_ CG * (x*) . As hese elemen s gene a e C, (G*), M* cen alises C,(G*) whence M* commu es wi h all 7 -elemen s o G* . Thus M* is a cen al subg oup o each Hall 7 -subg oup o G* . Le H* be a Hall 7 -subg oup o G* . By induc ion, H*/M* has an abelian Hall Q-subg oup L*/M* o u = 7 - {p} . As M* is o cop ime 69 0  W . KIMMERLE, R . SANDLING index in L*, i has a complemen K* which is an abelian Hall Q-subg oup o G* . Le P* be a Sylow p-subg oup o G* con ained in H* . As M* <_ P*, K* no malises P* . Mo eo e , K* s abilises he no mal se ies P* >M* >_ 1 so ha K* cen alises P* [Suz, 4 .1 .13] . Thus H* is abelian as equi ed . Suppose, hen, ha M and M* a e nonabelian . By Lemma 1, K(G) con ains all Hall 7 -subg oups o G . As desc ibed abo e, K(G) is iso- mo phic o a subg oup o X' whe e M .z ; S°, S< X <_ Au S, K(G) p ojec s on o each di ec ac o X and X has abelian Hall 7 -subg oups . In a simila manne K(G*) is isomo phic o a subg oup o Y° whe e M* i M z ~ S°, S _< Y <_ Au S and K(G*) p ojec s on o each di ec ac o Y . By Lemma 1, C, (G*) _< K(G*) . Bu C,(G*) = C,(G)* and IGI, = IC,(G)j, so ha K(G*), being no mal, con ains a Hall 7 -subg oup o G* . We show ha i is abelian by p o ing ha Y has abelian Hall 7 -subg oups . Fo his, i su ices by Lemma 3 o show ha IXI71 = IYL11 . Fo each pEnn 7 (S), X and Y ha e abelian Sylow p-subg oups, in he la e case because Y is he image o K(G*) which has abelian Sylow p-subg oups . By he P oposi ion, X p and Y p a e con ained in S so ha Ap = I SI p = Mp . I emains o show ha IXI, = jY1, o u = 7 - 7 (S) . By Lemma 2, X Q is cyclic so ha IXw = expX Q . As K(G) _< X' and as K(G) p ojec s on o X, exp X Q = exp K(G) Q . Also, exp K(G) o = exp G Q = exp(G/M) asince u n 7 (M) is emp y . As in he p e ious case G/M and G*/M* ha e isomo phic abelian Hall -subg oups so ha exp(G/M) Q = exp(G*/M*)Q . As be o e, exp(G*/M*) o = exp G* a = exp K(G*) Q . Finally, Y has cyclic Hall o , -subg oups by Lemma 2 and again exp K(G*)Q = expY, = ¡Y j, . T ac- ing h ough he equali ies, we conclude ha I X I Q = ¡Y¡, as equi ed . I may be possible o ecognise whe he a g oup has nilpo en Hall subg oups unde a class co espondence o ype JH . The p oo abo e in he M abelian case, o an easy a gumen using he Fi ing subg oup, p o ides he ollowing esul as e iden e . P oposi ion . A class co espondence o ype JH de e mines whe he o no Hall subg oups o a soluble g oup a e nilpo en . ABELIAN HALL SUBGROUPS  69 1 Re e en es [GL] GORENSTEIN, D . AND LYONS, R ., The local s uc u e o ini e g oups o cha ac e is ic 2 ype, Mem . Ame . Ma h . Soc . 42, no . 276 (1983) . [KS] KIMMERLE, W . AND SANDLING, R ., G oup heo e ic and g oup ing heo e ic de e mina ion o ce ain Sylow and Hall subg oups and he esolu ion o a ques ion o R . B aue , J . Algeb a ( o appea ) . [McB] McBRIDE, P ., Nonsol able signalize unc o s on ini e g oups, J . Algeb a 78 (1982), 215-238 . [Rose] RoSE, J . S ., "A cou se on g oup heo y," Camb idge Uni e si y P ess, Camb idge, 1978 . [San] SANDLING, R ., "A p oo o, he class sum co espondence us- ing he eal g oup algeb a," Ring heo y (G anada 1986), 237-244, Lec u e No es in Ma h . 1328, Sp inge , Be lin, 1988 . [SUZ] SUZUKI, M ., "G oup heo y II," Sp inge , New Yo k, 1986 . Wol gang Kimme le :  Robe Sandling : Ma h . Ins i u B  Ma hema ics Depa men Uni e si i S u ga  The Uni e si y 7 S u ga 80  Manches e GERMANY  M13 9PL ENGLAND Rebu el 9 de Gene de 1992