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Journal of Algebra
www.elsevier.com/locate/jalgebra
A generalization of Hall-Wielandt theorem
M. Yasir Kızmaz
DepartmentofMathematics,BilkentUniversity,06800Bilkent,Ankara,Turkey
a r t i c l e i n f o a bs t r a c t
Article history:
Received15August2018 Availableonline17October2019 CommunicatedbyE.I.Khukhro
MSC:
20D10 20D20
Keywords:
Controllingp-transfer p-Nilpotency
Let G be a finite group and P ∈ Sylp(G). We denote the k’thtermoftheuppercentralseriesofG byZk(G) andthe normofG byZ∗(G).Inthisarticle,weprovethatifforevery tameintersectionP∩ Q suchthatZp−1(P )< P∩ Q< P ,the groupNG(P∩Q) isp-nilpotentthenNG(P ) controlsp-transfer in G.Forp= 2,wesharpenourresultsbyprovingifforevery tame intersection P ∩ Q such that Z∗(P ) < P ∩ Q < P , the group NG(P ∩ Q) is p-nilpotent then NG(P ) controls p-transferinG.Wealsoobtainseveralcorollarieswhichgive sufficient conditions for NG(P ) to control p-transfer in G as a generalization of somewellknown theorems,including Hall-Wielandt theorem and Frobenius normal complement theorem.
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1. Introduction
Throughoutthearticle,weassumethatallgroupsarefinite.Notationandterminology arestandardasin[1].LetG beagroupandP ∈ Sylp(G).WesaythatG isp-nilpotent if ithasanormalHallp-subgroup.LetN beasubgroupofG suchthat|G: N| iscoprime top.ThenN issaidtocontrol p-transfer inG ifN/Ap(N )∼= G/Ap(G).Afamousresult
E-mailaddress:yasirkizmaz@bilkent.edu.tr.
https://doi.org/10.1016/j.jalgebra.2019.10.018 0021-8693/©2019ElsevierInc. Allrightsreserved.
of Tate in [2] shows that N/Ap(N ) ∼= G/Ap(G) if and only if N/Op(N ) ∼= G/Op(G).
Thus,N controls p-transferinG ifandonlyifN/Op(N )∼= G/Op(G).Inthiscase,one canalsodeducethatN isp-nilpotentifandonlyifG isp-nilpotent.
Bya result due to Burnside, NG(P ) controls p-transfer in G if P is abelian. Later worksofHallandWielandtshowedthatNG(P ) controlsp-transferinG iftheclassofP isnot“toolarge”.Namely,theyprovedthefollowinggeneralizationofBurnside’sresult.
Theorem 1.1 (Hall-Wielandt).If the class of P is less than p, then NG(P ) controls p-transferinG.
In 1978, Yoshida introduced the concept of charactertheoretic transfer and by the meansofit,heobtainedthefollowinggeneralizationofHall-Wielandttheorem.
Theorem1.2. [7,Theorem 4.2] If P hasnoquotientisomorphic to Zp Zp thenNG(P ) controls p-transferinG.
Theoriginalproofofthisstrongtheoremdependsoncharactertheory.However,Isaacs providedacharacterfreeproofto Yoshida’stheorem inhis book(seesection10 in[1]).
Taking advantages of his method, we obtain another generalization of Hall-Wielandt theorem.
Before presenting ourmain theorem,it is convenient here to give someconventions thatweadoptthroughoutthepaper.LetP,Q∈ Sylp(G) (possiblyP = Q).Wesaythat P∩ Q isa tameintersection ifboth NP(P∩ Q) andNQ(P ∩ Q) areSylowp-subgroups of NG(P ∩ Q).For simplicity, we use directly“X∩ Y is a tame intersection” without specifyingwhatX and Y are. Inthis case,it shouldbe understoodthatX and Y are Sylowp-subgroupsofG foraprimep dividingorderofG andX∩Y isatameintersection accordingto theformaldefinition.
Thefollowingisthemaintheorem ofourarticle.
Theorem1.3. Assumethat foreachtame intersectionZp−1(P )< P ∩ Q< P , thegroup NG(P ∩ Q) isp-nilpotent. ThenNG(P ) controls p-transferinG.
ThenextremarkshowsthatourtheoremextendstheresultofHall-Wielandttheorem inadifferentdirectionthanwhatYoshida’stheoremdoes.
Remark 1.4.Let G be a group having a Sylow p-subgroup P isomorphic to Zp Zp. Clearly,Yoshida’s theoremis notapplicablehere.If NG(P ) doesnotcontrolp-transfer in G then there exists a Sylow p-subgroup Q of G such that |P : P ∩ Q| = p and NG(P∩ Q) isnotp-nilpotentbyTheorem1.3.Noticethatthisisexactlythecasewhere G= S4 andp= 2. Wecansay inanother way thatNG(P ) controls p-transfer in G if
|P : P ∩ Px|> p for eachx∈ G\ NG(P ).
SomeoftheimmediatecorollariesofTheorem1.3 areasfollows.
Corollary 1.5.Assume that for any two distinct Sylow p-subgroups P and Q of G, the inequality|P ∩ Q|≤ |Zp−1(P )| is satisfied.ThenNG(P ) controlsp-transferinG.
The next corollary is a generalization of the well-known Frobenius normal comple- menttheorem,whichguaranteesthep-nilpotencyofG ifNG(X) is p-nilpotentforeach nontrivial p-subgroupX ofP .
Corollary 1.6. Assume that for each tame intersection Zp−1(P ) < P ∩ Q, the group NG(P ∩ Q) isp-nilpotent. ThenG isp-nilpotent.
Remark1.7. Themainingredientinprovingmostof thep-nilpotencytheoremsinclud- ingThompson-Glaubermanp-nilpotencytheoremsistheFrobeniusnormalcomplement theorem,andhenceitsabovegeneralizationcanbeusedinprovingstrongerp-nilpotency theorems.
Whenp= 2,Theorem 1.3guaranteesthatifNG(P ∩ Q) isp-nilpotentforeachtame intersectionZ(P )< P∩ Q< P , thenNG(P ) controlsp-transfer inG.Infact,weshall extendthisresultfurther.
LetZ∗(P ) denotethenormofP ,whichisdefinedas Z∗(P ) :=
H≤P
NP(H).
We haveclearlyZ(P )≤ Z∗(P ).Onecanrecursively defineZi∗(P ) fori≥ 1 as thefull inverseimageofZ∗(P/Zi−1∗ (P )) inP andsetZ0∗(P )= 1.WealsosaythatP isofnorm lengthat mosti ifZi∗(P )= P .Weshouldalsonote thatitiswell knownthatZ∗(P ) is containedinthesecondcenterofP .
Theorem 1.8.Assume that for each tame intersection Z∗(P ) < P ∩ Q< P , the group NG(P ∩ Q) isp-nilpotent. ThenNG(P ) controlsp-transferin G.
The followingcorollary isstrongerthanCorollary1.6 whenp= 2 although itisalso trueforoddprimes(asTheorem 1.8isalsotrueforoddprimes).
Corollary1.9.AssumethatforeachtameintersectionZ∗(P )< P∩Q,thegroupNG(P∩ Q) is p-nilpotent.ThenG is p-nilpotent.
The following theorem is a generalization of a theorem due to Grün (see Theo- rem 14.4.4 in [4]), which states that the normalizer of a p-normal subgroup controls p-transfer inG.Wealsouseournexttheorem intheproofofTheorem 1.8.
Theorem 1.10.LetK≤ Z∗(P ) be aweaklyclosed subgroupof P . ThenNG(K) controls p-transferin G.
Thenextcorollary canalsobe easilydeducedbythemeansofTheorem1.8.
Corollary1.11. Assumethat forany twodistinct Sylow p-subgroups P and Q ofG, the inequality|P ∩ Q|≤ |Z∗(P )| is satisfied.ThenNG(P ) controlsp-transferinG.
Remark1.12.InTheorems1.3and1.8,theassumption“NG(P∩Q) isp-nilpotent”could be replacedwithaweakerassumption “NG(P ∩ Q)/CG(P ∩ Q) isap-group”. Thiscan beobservedwiththeproofsofTheorems 1.3and1.8.
2. Preliminaries
LetH ≤ G and T ={ti | i= 1,2. . . ,n} be aright transversalfor H inG.Themap V : G→ H definedby
V (g) =
n i=1
tig(ti.g)−1
iscalled apretransfermap fromG to H.Whentheorder oftheproduct isnotneeded to specify, we simply write V (g) =
t∈Ttg(t.g)−1. Notice that the kernel of “dot ac- tion” is CoreG(H), and so t.g = t for all g ∈ CoreG(H). In the case that G is a p-group, Z(G/CoreG(H)) = 1 whenever H is a proper subgroup of G. If x∈ G such thatxCoreG(H)∈ Z(G/CoreG(H)) isoforderp,theneachx -orbithaslengthp when weconsidertheactionofx onT .
Let t1,t2. . . ,tk be representatives of all distinct orbits of x on T . As t.x and tx represent the same right coset of H in G for each t ∈ T , the set T∗ = {tixj | i ∈ {1,2,...,k}andj ∈ {0,1,...,p− 1}} is alsoarighttransversal forH in G.LetV∗ be a pretransfermap constructedbyusing T∗.Since V (u)≡ V∗(u)modH,wemayreplace T withT∗ withoutloss ofgeneralitywheneversuchasituationoccurs.
Wedenoteallpretransfermapswithuppercaselettersandeachcorrespondinglower caselettershowsthecorrespondingtransfermap.
Theorem 2.1.[1,Theorem 10.8] Let G be a group, and suppose that H ≤ K ≤ G. Let U : G→ K,W : K → H andV : G→ H be pretransfermaps. Thenfor allg∈ G, we haveV (g)≡ W (U(g))mod H,thatis, v(g)= w(U (g)).
Theorem2.2. [1,Theorem 10.10] LetX beasetofrepresentativesforthe(H,K) double cosetsinagroupG,whereH andK aresubgroupsofG.LetV : G→ H beapretransfer map, and foreach element x∈ X, let Wx : K → K ∩ Hx be a pretransfer map. Then fork∈ K,wehave
V (k)≡
x∈X
xWx(k)x−1 mod H.
Now wegiveatechnicallemma,whichis essentiallythemethodused intheproof of Yoshida’stheorem(seeproofofTheorem 10.1in[1]).Forthesakecompleteness,wegive theproof ofthislemmahere.
Lemma 2.3. Let G be agroup and, letP ∈ Sylp(G) and NG(P ) ≤ N. Supposethat N does not control p-transfer in G and let X be a set of representatives for the (N,P ) double cosetsin G,which containstheidentity e.Thenthefollowinghold:
(a) There existsa normalsubgroup M of N of indexp such that V (G)⊆ M for every pretransfermapV fromG to N .
(b) For each u∈ P \ M, there exists a nonidentity x∈ X such that W (u)∈ P ∩ M/ x where W isapretransfer mapfromP toP∩ Nx.
(c) Forx in part(b),wehave P∩ Nx< P and|P ∩ Nx: P∩ Mx|= p.
Proof. (a) It followsby([1], Lemma 10.11).
(b) Let u∈ P \ M.Let Wx be apretransfermap from P toP ∩ Nx for eachx∈ X.
Thenwehave
V (u)≡
x∈X
xWx(u)x−1 mod N
byTheorem2.2.SinceN≤ M andV (u)∈ M,weget
x∈X
xWx(u)x−1∈ M.
Notice that for x = e, We : P → P and We(u) = u = eWe(u)e−1 ∈ M./ Thus, there alsoexists e = x∈ X suchthatxWx(u)x−1∈ M./ SetWx= W .Thenweget W (u)∈ P ∩ Nx\ P ∩ Mx.
(c) Set R = P ∩ Nx and Q = P ∩ Mx. If R = P then Px−1 ≤ N, and hence there exists y ∈ N suchthatPx−1y = P . Sincex−1y∈ NG(P )≤ N, weget x∈ N.This isnotpossible asN xP = N eP andx = e.It followsthat R < P . NotethatR = Q by part (b). Moreover, the inequality 1 < |R : Q| ≤ |Nx : Mx| = p forces that
|R : Q|= p. 2
3. Mainresults
The following lemma serves as the key tool in proving our main theorems since it enables us to useinduction intheproofs of “controlp-transfer theorems”. Throughout the section, G isagroupand P is aSylow p-subgroupofG for aprimep dividing the order ofG.
Lemma 3.1. Let NG(P ) ≤ N ≤ G, Z ≤ P and Z G. Assume that N/Z controls p-transferin G/Z andthat oneof thefollowingholds:
(a) [Z,g,. . . ,g]p−1≤ Φ(Z) forallg∈ P . (b) Z≤ Φ(P ).
ThenN controls p-transferinG.
Weneedthefollowing lemmaintheproof ofLemma3.1.
Lemma3.2. LetNG(P )≤ N ≤ G,Z ≤ P andZ G.Assume that N does notcontrol p-transferinG andN/Z controlsp-transferinG/Z. ThenZ M andwehaveW (u)∈ P∩ Nx\ P ∩ Mx foreach u∈ Z \ M whereW,M andx are asinLemma2.3.
Proof. Set G/Z = G. Let V be a pretransfer map from G to N . Let T be a right transversalsetusedforconstructingV .Itfollowsthatthereexist anormalsubgroupM ofN withindexp such thatV (G)⊆ M byLemma2.3(a).
NowweclaimthatZ M.Assumetothecontrary.NoticethatthesetT ={t | t∈ T } is a right transversal set for N in G. Thus if we construct a pretransfer map V by using T , then V (g) = V (g). It follows that V (G) = V (G) ⊆ M N. Let F be a pretransfer map from N to P and f be the corresponding transfer map. Note that ker(f ) = Ap(N ) ≤ M as |N : M| = p, and hence f (M ) < f (N ). It then follows thatf (V (G)) < f (N ). Since f ◦ V is thetransfer map from G to P byTheorem 2.1, weget|G : Ap(G)| = |N : Ap(N )|,which contradictsthehypothesis.Thus, thereexists u∈ Z\M.ThenwehaveW (u)∈ P ∩Nx\P ∩Mxforeachu∈ Z\M byLemma2.3(b). 2 Proof of Lemma3.1. SupposethatN does notcontrolp-transfer inG.We derivecon- tradictionforbothparts.WeassumethenotationofLemma2.3.
Firstassume that(b) holds, thatis, Z ≤ Φ(P ). Note that|P : P ∩ M| = p,and so Z ≤ Φ(P ) ≤ M ∩ P . However,this is not possible by Lemma 3.2. This contradiction showsthatN controls p-transferinG when(b) holds.
Nowassume that(a) holds.LetX be asetof representatives for the(N,P ) double cosets inG,which containstheidentity e. ByLemma3.2, wesee thatZ M andwe haveapretransfermapW : P → P ∩NxsuchthatW (u)∈ P ∩M/ xforsomenonidentity x∈ X whereu∈ Z \ M.SetR = P ∩ Nxand Q= P ∩ Mx.
NowletS bearight transversalset forR inP ,whichisused forconstructingW so thatwehaveW (u)=
s∈Ssu(s.u)−1. Sinceu∈ Z ≤ CoreP(R), wehave(s.u)= s for alls∈ S.Thus,we getW (u)=
s∈Ssus−1.
Set C = CoreP(R). Since R < P by Lemma 2.3(c), C is also proper in P . So we see thatZ(P/C) = 1. Now choosen∈ P such thatnC ∈ Z(P/C) isof order p. Then each orbitof n on S has length p. Let s1,s2. . . ,sk be representatives of all distinct orbits of n on S. Without loss of generality, we can suppose that S = {sinj | i ∈ {1,2,...,k} and j ∈ {0,1,...,p− 1}}. Now we compute the contribution of a single n -orbitto W (u).Fixs∈ S.
(snun−1s−1)(sn2un−2s−1) . . . (snp−1un−p+1s−1)(sus−1) = s(nu)p−1n−p+1us−1.
Wehaves(nu)p−1n−p+1us−1= (s(nu)ps−1)(su−1n−pus−1).SetH =n,u .Duetothe factthat|n C : C|= p,wehaveH≤ C.Notethatu∈ Z ≤ C, andso
[H, u , H, u ] ≡ 1 mod Φ(C).
Wecanexpandthepoweroftheproductasinthefollowingform
(nu)p≡ (npup)[u, n]p2[u, n, n]p3...[u, n, ..., n]pp−2[u, n, ..., n]p−1 mod Φ(C) due tothepreviouscongruence.
As C P , we observe that s[u,n,...,n]is−1 ∈ C for i = 1,...,p − 1, and so (s[u,n,...,n]is−1)p ∈ Φ(C) for i = 1,...,p− 1. By using the fact that p
i+1
is divisi- ble byp fori= 1,. . . ,p− 2,weseethat
(s[u, n, ..., n]is−1)i+1p ∈ Φ(C) for i = 1, . . . , p − 2.
Note also that [u,n,...,n]p−1 ∈ Φ(Z) ≤ Φ(C) by hypothesis, and so we get that s[u,n,...,n]p−1s−1 ∈ Φ(C) sinceΦ(C) P .As aconsequence,weobtainthat
s(nu)ps−1 ≡ (snps−1)(sups−1)≡ snps−1 mod Φ(C).
It thenfollowsthat
(s(nu)ps−1)(su−1n−pus−1)≡ (snps−1)(su−1n−pus−1)≡ s[n−p, u]s−1≡ 1 mod Φ(C).
We only need to explain why the last congruence holds: Since both n−p and u are elements of C,we see that[n−p,u]∈ Φ(C). It follows thats[n−p,u]s−1 ∈ Φ(C) dueto the normality of Φ(C) in P . Then W (u) ∈ Φ(C) as the chosen n -orbit is arbitrary.
Since |R : Q| = p by Lemma 2.3(c), the containment Φ(C) ≤ Φ(R) ≤ Q holds. As a consequence,W (u)∈ Q.Thiscontradiction completestheproof. 2
Remark3.3. Intheproofsofmanyp-nilpotencytheorems,theminimalcounterexample G isap-solublegroupsuchthatOp(G)= 1 andG/Op(G) isp-nilpotent.Lemma3.1(a) guarantees the p-nilpotency of G if [Op(G),g,. . . ,g]p−1 ≤ Φ(Op(G)) for all g ∈ P . In particular ifOp(G)≤ Zp−1(P ) thenthep-nilpotencyofG follows.Thisboundseemsto be bestpossible sinceinthesymmetricgroupS4,O2(S4)≤ Z2(P ) andO2(G) Z(P ).
EvenifS4/O2(S4) is2-nilpotent,S4 failsto be2-nilpotent.
It is well known that if G/Z is p-nilpotent and Z ≤ Φ(P ) then G is p-nilpotent.
Lemma 3.1(b)generalizes this particularcase bystating thatifG/Op(G)∼= N /Op(N ) then G/Op(G)∼= N/Op(N ) whereG = G/Z andZ ≤ Φ(P ).
WealsoshouldnotethatinLemma3.1,weprovelittlemorethanwhatweneedhere as weseethatitmayhaveotherapplications too.
Proposition 3.4.Assume that for every characteristic subgroup of P that contains Zp−1(P ) is weaklyclosed in P .ThenNG(P ) controls p-transfer.
Proof. WeproceedbyinductionontheorderG.LetZ = Zp−1(P ).ThenNG(Z) controls p-transferinG by([4], Theorem 14.4.2).IfNG(Z)< G thenNG(P ) controlsp-transfer withrespect to groupNG(Z) by inductionappliedtoNG(Z). It followsthatP ∩ G = P∩ NG(Z) = P ∩ NG(P ),thatis, NG(P ) controlsp-transfer inG.
ThereforewemayassumeZ G.ItiseasytoseethatG/Z satisfiesthehypothesisof theproposition, and hence weget NG/Z(P/Z)= NG(P )/Z controlsp-transfer inG/Z byinductionappliedtoG/Z.ThentheresultfollowsbyLemma3.1(a). 2
Remark3.5. Intheaboveproposition,theassumptionthateverycharacteristicsubgroup containingZp−1(P ) isweaklyclosedcanbeweakenedtoZk(p−1)(P ) isweaklyclosedfor eachk = 1,...,n where Zn(p−1)(P )= P .Yetweshallnotneed thisfact.
AfterProposition3.4,itisnaturaltoaskthefollowing question.
Question 3.6. Does a Sylow p-subgroup P of a group G have a single characteristic subgroupwhose beingweaklyclosed inP issufficientto concludethatNG(P ) controls p-transferinG?
Proof of Theorem1.3. Let Zp−1 ≤ C beacharacteristicsubgroup ofP .Weclaimthat C is normal ineach Sylow subgroup of G that contains C. Assume the contrary and letQ∈ Sylp(G) suchthatC≤ Q and NQ(C)< Q.There exists x∈ NG(C) suchthat NQ(C)x= NQx(C)≤ P ,and henceNQx(C)≤ P ∩ Qx.
Set Qx = R. By Alperin Fusion theorem,we have R ∼P P . Thus there are Sylow subgroupsQifori= 1,2,. . . ,n suchthatP∩R ≤ P ∩Q1and(P∩R)x1x2...xi≤ P ∩Qi+1
wherexi∈ NG(P ∩ Qi),P∩ Qi isatame intersectionandRx1x2...xn= P .
Note that NP(P ∩ Q1) is a Sylow p-subgroupof NG(P ∩ Q1) as P∩ Q1 is atame intersection. Moreover, NG(P ∩ Q1) is p-nilpotent by the hypothesis as Zp−1 ≤ C <
NQ(C)x≤ P ∩ R ≤ P ∩ Q1.Thenwehave
NG(P ∩ Q1) = NP(P ∩ Q1)CG(P ∩ Q1).
Thus,we canwritex1= s1t1 where t1 ∈ CG(P ∩ Q1) and s1∈ NP(P ∩ Q1). Notice that t1 also centralizes C as C ≤ P ∩ Q1 and s1 normalizes C as C P . It follows that Cx1 = Cs1t1 = C < (P ∩ R)x1 ≤ P ∩ Q2. Then we get that NG(P ∩ Q2) is p-nilpotentbythehypothesisand wemaywrite x2 = s2t2 where t2 ∈ CG(P ∩ Q2) and s2 ∈ NP(P ∩ Q2) in a similar way. Notice also that Cx1x2 = Cx2 = C. Proceeding inductively, we obtain that NG(P ∩ Qi) is p-nilpotent for all i and Cx1x2...xn = C.
Since Cx1x2...xn = C P = Rx1x2...xn, we getC R = Qx. Weobtain thatC Q as x ∈ NG(C). This contradiction shows that C is weakly closed inP and the theorem followsbyProposition3.4. 2
Proof of Theorem1.10. WriteN = NG(K),andletX beasetofrepresentativesforthe (N,P ) doublecosets inG, which containsthe identity e.Note that NG(P )≤ N as K is aweaklyclosed subgroupof P .AssumethatN doesnotcontrolp-transfer inG.By Lemma2.3(b),we haveapretransfermap W : P → P ∩ Nx suchthatW (u)∈ P ∩ M/ x for eachu∈ P \ M where e = x ∈ X andM isas in Lemma2.3(a). SetR = P ∩ Nx and Q= P ∩ Mx.
Now choose u ∈ P \ M and u∗ ∈ N \ M such that both u and u∗ are of minimal possible order. Wefirstarguethat|u|=|u∗|. Clearlywe have|u∗|≤ |u| asu∈ N \ M.
Note that (u∗)q ∈ M if q is a prime dividing the order u∗ by the choice of u∗. The previous argumentshows thatp= q as|N : M|= p,and so u∗ isap-element.Thus, a conjugateof u∗ viaanelementofN liesinP\ M. Itfollowsthat|u|≤ |u∗|, whichgive us thedesiredequality.
Let S bearight transversalset forR in P usedforconstructing W sothatwe have W (u) =
s∈Ssu(s.u)−1. Let S0 be a set of orbit representatives of the action of u on S. Then we have W (u)=
s∈S0sunss−1 by transfer evaluation lemma. Note that sunss−1 ∈ R ≤ Nx, andhence xsunss−1x−1 ∈ N. Ifns> 1 then |xsunss−1x−1|<|u|, and so xsunss−1x−1 ∈ M by theprevious paragraph.Thus we get sunss−1 ∈ Q. As a consequence,weobserve that
W (u)≡
s∈S∗
sus−1 mod Q
where S∗={s∈ S | s.u= s}.
WeclaimthatK isnotcontainedinR.Sinceotherwise:bothK andKxarecontained in Nx, and so Kx−1 and K arecontained in N . Since K is aweakly closed subgroup of P , there exists y ∈ N such that Kx−1 = Ky (see problem 5C.6(c) in [1]). As a result yx ∈ N, and so x ∈ N. Thus, we get N xP = N eP which is a contradiction as x = e. Since R < P by Lemma 2.3(c), CoreP(R) is also proper in P . So we see that Z(P/CoreP(R)) = 1. Since K is notcontained inCoreP(R) and K isnormal in P , we can pick k ∈ K such that kCoreP(R) ∈ Z(P/CoreP(R)) is of order p. Now consider the actionof k on S. Then each k -orbithas length p andlet s1,s2. . . ,sn be representatives of all distinct orbits of k on S. Note that we mayreplace S with {sikj | i∈ {1,2,...,n}andj∈ {0,1,...,p− 1}}.Wealsonotethat
s.(uk) = (s.(ku)).[u, k] = s.(ku) for all s∈ S.
Thelastequalityholdsas[u,k]∈ CoreP(R).ItfollowsthatS∗isk -invariant.Note that k normalizes u as k ∈ Z∗(P ), and so uk−1 = un where n is a natural number whichiscoprimetop.Clearlyn isoddwhenp= 2.Ontheotherhand,ifp is oddthen it iswellknownthatn= 1+ pr forsomer∈ N ask−1 induces ap-automorphismona cyclic p-group.Thus,weobtainn≡ 1mod p inbothcase.
Nowwecomputethecontributionofasinglek -orbitto W (u).Fixs∈ S∗.
(sus−1)(skuk−1s−1)(sk2uk−2s−1)...(skp−1uk−p+1s−1) = suunun2...unp−1s−1= suzs−1 wherez = 1+ n+ n2+ ...+ np−1.Notethatz≡ 0modp,sus−1∈ R and |R : Q|= p by Lemma2.3(c),andhencesuzs−1= (sus−1)z∈ Q.Sincethechosenk -orbitisarbitrary, weobtainW (u)∈ Q.This contradictioncompletestheproof. 2
Nowwearereadyto givetheproof ofTheorem1.8.
Proof of Theorem1.8. Firstnoticethatifp isoddthentheresultfollowsbyTheorem1.3 duetothefactthatZ∗(P )≤ Z2(P )≤ Zp−1(P ).Thus,itissufficienttoprovethetheorem forp= 2.LetG beaminimalcounterexampletothetheorem.Wederiveacontradiction overaseries ofsteps.WriteZ = Z∗(P ) andN = NG(P ).
(1) EachcharacteristicsubgroupC ofP thatcontainsZ isweaklyclosedinP .More- over,Z isanormalsubgroupofG.
Byusing thesamestrategyused intheproof ofTheorem 1.3,we canshow thatany characteristic subgroupC of P thatcontains Z isweaklyclosed inP .In particular,Z isweaklyclosed inP .
SupposethatNG(Z) < G.Clearly NG(Z) satisfiesthe hypothesisand N ≤ NG(Z).
Thus, N controlsp-transfer with respect to the groupNG(Z) by theminimality of G.
Ontheotherhand,NG(Z) controlsp-transferinG byTheorem1.10.Asaconsequence, G∩ P = (NG(Z))∩ P = N∩ P .Thiscontradiction showsthatZ G.
(2) N/Z controls p-transferinG/Z.
Write G = G/Z. Clearly N = NG(P ). If Y is acharacteristic subgroup of P then Y isacharacteristic subgroup of P thatcontains Z. Then Y isweaklyclosed inP by (1).It followsthatY isweaklyclosed inP . ThenwegetN controls p-transferinG by Proposition3.4.
(3)|P : R|= 2.
ByLemma3.2, there exists u∈ Z \ M such thatW (u)∈ P ∩ Nx\ P ∩ Mx where W,M andx areas inLemma2.3.SetR = P ∩ Nx andQ= P ∩ Mx. LetS bearight transversal set for R in P usedfor constructing W . Since u ∈ Z ≤ CoreP(R), we get W (u)=
s∈Ssu(s.u)−1 =
s∈Ssus−1.
Since R < P by Lemma 2.3(c), CoreP(R) is also proper in P . So we see that Z(P/CoreP(R)) = 1. Now choose n ∈ P such that nCoreP(R) ∈ Z(P/CoreP(R)) isoforderp(= 2) andconsidertheactionofn onS.Withoutlossofgenerality,wemay takeS ={sinj | i ∈ {1,2,...,k}andj ∈ {0,1}} wheres1,s2. . . ,sk arerepresentatives ofalldistinctorbitsofn onS.Fixs∈ S. Wehave
(sus−1)(snun−1s−1) = su2[u, n−1]s−1= su2s−1[u, n−1].
Thelastequalityholdsasu∈ Z = Z∗(P )≤ Z2(P ).Weseethatsu2s−1 ∈ Q assus−1∈ Z ≤ R and |R : Q|= 2.Thus thecontributionof asingleorbitiscongruentto [u,n−1] modQ byLemma2.3(c).Asaconsequence,weobtainthatW (u)≡ [u,n−1]|S|/2modQ.
Supposethat|S|/2 isanevennumber.Weget[u,n−1]|S|/2∈ Q as[u,n−1]∈ Z ≤ R.This contradictsthefactthatW (u)∈ Q,/ andso|S|/2 isodd.Itfollowsthat|P : R|=|S|= 2 as required.
(4) R = Z.
Suppose thatZ < R. Note thatR = P ∩ Nx = P ∩ NG(P )x, and so R = P ∩ Px. Since |P : R| = 2 by (3), |Px : R| is also equalto 2. As aresult, R is normal inboth P and Px,thatis,R isatame intersection.Thus, weseethatNG(R) is p-nilpotentby our hypothesis.Pickx0 ∈ NG(R) suchthatPx= Px0.Then x0x−1∈ N whichimplies x0= tx forsomet∈ N.WeobservethatN x0P = N txP = N xP ,andsowemayreplace thedouble cosetrepresentative x withx0.
We observe that NG(R) is p-nilpotent and x ∈ NG(R) in the previous paragraph.
Thus,wecanwritex= c1c2forsomec1∈ P andc2∈ CG(R).AsW (u)∈ Q/ = P∩Mx,we see thatxW (u)x−1 = c1c2W (u)c−12 c−11 = c1W (u)c−11 ∈ M./ Hence,W (u)∈ M/ c1 = M . Recall that|P : M ∩ P |= p= 2,andsoP≤ M.Hence,weobtainthat
W (u) =
s∈S
sus−1=
s∈S
[s−1, u−1]u≡
s∈S
u = u2≡ 1 mod M ∩ P.
It followsW (u)∈ M,whichisnotthecase.ThiscontradictionshowsthatZ = R.
(5) Final contradiction.
We observe that |P : Z| = |P : Z∗(P )| = 2 by (4). If P is ahomomorphic image of P ,wecanconcludethat|P : Z∗(P )|≤ 2.SinceN doesnotcontrolp-transferinG,P has ahomomorphic image whichis isomorphicto Z2 Z2∼= D8 byYoshida’s theorem.
However,|D8: Z∗(D8)|=|D8: Z(D8)|= 4.Thiscontradictioncompletestheproof. 2 4. Applications
Theorem 4.1. Assumethat for any twodistinct Sylow p-subgroups P and Q of G, |P ∩ Q|≤ pp−1.ThenNG(P ) controlsp-transferinG.
Proof. We maysuppose that cl(P )≥ p. Notice that the inequality |Zp−1(P )| ≥ pp−1 holds inthiscase.Then theresultfollowsbyCorollary 1.5. 2
Themain theorem of[3] states thatifNG(P ) isp-nilpotentandforany twodistinct Sylow p-subgroups P and Q of G, |P ∩ Q| ≤ pp−1 then G is p-nilpotent. The above theorem isageneralizationofthis fact.
Theorem4.2. LetP ∈ Sylp(G).Supposethat P isofclassesp andNG(P ) isp-nilpotent.
If NG(P ) is amaximal subgroupof G then G isap-solvablegroupof p-length1.
Proof. We maysuppose thatG is notp-nilpotent. Then there exists U ≤ G such that Zp−1 < U < P andNG(U ) isnotp-nilpotentbyCorollary1.6.SinceZp−1 < U ,U P . It follows thatU NG(P ) asNG(P ) is p-nilpotent. Note thatNG(P ) = NG(U ) since NG(U ) is not p-nilpotent.Thus we get thatNG(P ) < NG(U ), and henceU G since NG(P ) isamaximal subgroup ofG. Onthe other hand,G/U is p-nilpotentas P/U is anabelianSylowsubgroupofG/U whereNG(P )/U = NG/U(P/U ) isp-nilpotent.Then theresultfollows. 2
Thompson provedthatifG possesanilpotent maximal subgroupof oddorder then G issolvable.LaterJankoextendedthisresultin[5] asfollows;
Theorem 4.3 (Janko). Let G be a group having a nilpotent maximal subgroup M . If a Sylow2-subgroup ofM is of classatmost2 thenG is solvable.
The above theorem can be deduced by the means of Theorem 4.2. We extend the resultofJankobyusingCorollary 1.9withthefollowingtheorem.
Theorem 4.4. Let G be a group with a nilpotent maximal subgroup M . If a Sylow 2-subgroupof M is ofnormlengthatmost 2 thenG issolvable.
Proof. We proceed by induction on the order of G. Suppose Op(G) = 1 for a prime p dividing the order of M . If Op(G) ≤ M then G/Op(G) satisfies thehypothesis and henceG/Op(G) issolvable byinduction.If Op(G) M then G= M Op(G) due to the maximalityofM . Thus,G/Op(G) issolvable as M isnilpotent.Then wesee thatG is solvableinboth cases.Thus,wemaysuppose thatOp(G)= 1 foranyprimep dividing theorderofM .
NowletP ∈ Sylp(M ).SinceM isnilpotent,wegetM≤ NG(P ).Ontheotherhand, NG(P )< G as Op(G)= 1. Then wehaveNG(P ) = M by themaximalityof M . Thus P is also a Sylow p-subgroup of G, that is, M is a Hall subgroup of G. Let X be a characteristic subgroup of P . Then NG(X) = M with a similar argument, and hence NG(X) isp-nilpotent.ItfollowsthatG isp-nilpotentbyThompsonp-nilpotencytheorem whenp isodd.
Nowassumethatp= 2. LetZ∗(P )≤ U ≤ P = Z2∗(P ).SinceP/Z∗(P ) isaDedekind group, U/Z∗(P ) P/Z∗(P ). It follows that U P , and hence U M. Then we get NG(U )= M whichisp-nilpotent.Thus,weobtainthatG isp-nilpotentbyCorollary1.9.
AsaresultG isp-nilpotentforeachprimep dividingtheorderofM .ThenM hasa normalcomplementN inG.NoticethatM actsonN coprimely,andsowemaychoose an M -invariant Sylow q-subgroup Q of N fora prime q dividing the order of N . The maximality of M forces thatM Q = G, thatis, N = Q. Since N is aq-group, we see thatG issolvable. 2
Remark4.5.Weshouldnotethattherearegroupsofclass3,whichhavenormlength2.
Forexample,onecanconsider thequaterniongroupQ16.Wealso notethatthebound
intermsof normlengthis thebestpossible.Forexample, D16 isof normlength3 and it is isomorphicto a Sylow 2-subgroupP of P SL(2,17) andP is amaximal subgroup of G.
Definition 4.6.A groupG iscalled pi-centralof heightk if everyelement oforder pi of G iscontainedinZk(G).
Theorem 4.7. Let G be a group and P be a Sylow p-subgroup of G where p is an odd prime. Assumethat eitherP isp-centralofheightp− 2 or p2-centralof heightofp− 1.
Then NG(P ) controlsp-transferin G.
Remark4.8. LetG beagroupand P ∈ Sylp(G).AssumethatP is p-centralof height p− 2 for an odd prime p. By ([6], Theorem E), NG(P ) controls G-fusion in P if G is ap-solvable group. In this case, NG(P ) also controls p-transfer in G. On the other hand,Theorem4.7guaranteesthatNG(P ) controlsp-transferinG foranarbitraryfinite group G.
WeneedthefollowingresultintheproofofTheorem4.7.
Theorem 4.9.[6, Theorem B] Let G be a group. If G is p-central of height p− 2 or p2-centralof heightofp− 1, thensoisG/Ω(G).
Proof of Theorem4.7. WeproceedbyinductionontheorderG.SetZ = Ω(P ).Clearly, Z isweaklyclosedinP .SinceΩ(P )≤ Zp−1(P ),NG(Z) controlsp-transferinG by ([4], Theorem 14.4.2).
IfNG(Z)< G thenNG(Z) clearlysatisfiesthehypothesis,andhenceNG(P ) controls p-transfer inNG(Z) bytheinductiveargument.ItfollowsthatP∩ G= P ∩ NG(Z) = P ∩ NG(P ),andhenceNG(P ) controlsp-transferinG.
Now assume thatZ G. Clearly, P/Z is a Sylow p-subgroup of G/Z. We see that P/Z is p-central ofheight p− 2 or p2-centralof height ofp− 1 by Theorem4.9.Thus, NG(P )/Z = NG/Z(P/Z) controlsp-transfer inG/Z by induction.Since Z ≤ Zp−1(P ), theresultfollowsbyLemma3.1(a). 2
4.1. Conclusion
“Control p-transfer theorems” supplymany nonsimplicitytheorems bytheir nature.
Let N be a subgroup of a group G such that |G : N| is coprime to p. If N controls p-transfer inG andOp(N )< N thenG isnotsimpleofcourse.
It is an easy exerciseto observe that ifK is a normal p-subgroup of G, and write G = G/K, then N controls p-transfer in G if and only if N controls p-transfer in G.
However,thisneedsnotbetrueifK isap-group.Thus,Lemma3.1suppliesanimportant criterion for that purpose and it enables the usage of the induction in the proofs of
“controlp-transfertheorems”.ItalsoseemsthatLemma3.1canbe improvedfurtherby bettercommutatortricksormorecarefulanalysisofthetransfermap.
Proposition 3.4 shows that when some certain characteristic subgroups of a Sylow subgroup P areweakly closedinP , NG(P ) controls p-transfer inG.Onecanask that whethertheconverseofthisstatementistrue.Anothernaturalquestionisthatwhether
“controlfusion”analogues ofLemma3.1andProposition3.4arepossible.Wealsocon- siderthatQuestion 3.6isveryinteresting.
WhenwecombineProposition3.4withAlperin Fusiontheorem,weobtainourmain theorems,whichsimplysaythatNG(P ) tendsto controlp-transferinG iftheintersec- tion of Sylow subgroups is not “too big”. We also sharpen our result when p = 2 via Theorem 1.8 and deduce two new versions of Frobenius normal complement theorem;
namely,Corollary1.6andCorollary1.9.SincewecannotdirectlyappealtoThompson- Glaubermanp-nilpotencytheoremswhenp= 2 (andG isnotS4 free),thecontribution ofCorollary1.9isimportant.Itiswellknownthata“controlfusion”analogueofGrün’s theoremis correct,andso weask whethera“controlfusion”analogueofTheorem1.10 ispossibleor not.
Theorem4.7showsthatNG(P ) controlsp-transferforgroupswhichhaveSylowsub- groupisomorphic to oneofthe two importantclasses ofp-groups, namely, p-central of height p− 2 or p2-central of height of p− 1. We lastly note thatTheorem 4.4 gives a solvability criteriawhich extendsaresultof Janko[5]. Evenifwe supply somelimited applications here, we think that the above theorems have a nice potential of proving nonsimplicitytheorems infinite grouptheory.
Acknowledgments
Iwouldliketo thankProf.GeorgeGlaubermanforhis helpfulcomments.
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