We extend two theorems due to P. Flavell [6] to arbitrary fusion systems

Kessar, R.

Linckelmann, M.

English

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Kessar, R. & Linckelmann, M. (2009). On two theorems of Flavell. Archiv der Mathematik, 92(1), 1 - 6. doi: 10.1007/s00013-008-2911-6 <http://dx.doi.org/10.1007/s00013-008-2911-6>City Research OnlineOriginal citation: Kessar, R. & Linckelmann, M. (2009). On two theorems of Flavell. Archiv der Mathematik, 92(1), 1 - 6. doi: 10.1007/s00013-008-2911-6 <http://dx.doi.org/10.1007/s00013-008-2911-6>Permanent City Research Online URL: http://openaccess.city.ac.uk/1884/ Copyright & reuseCity  University  London has developed City  Research Online  so that  its  users  may access the research outputs of City University London's staff. Copyright © and Moral Rights for this paper are retained by the individual author(s) and/ or other copyright holders.  All material in City Research Online is checked for eligibility for copyright before being made available in the live archive. URLs from City Research Online may be freely distributed and linked to from other web pages. Versions of researchThe version in City Research Online may differ from the final published version. Users are advised to check the Permanent City Research Online URL above for the status of the paper.EnquiriesIf you have any enquiries about any aspect of City Research Online, or if you wish to make contact  with the author(s) of this paper, please email the team at publications@city.ac.uk.On two theorems of FlavellRadha Kessar, Markus LinckelmannAbstractWe extend two theorems due to P. Flavell [6] to arbitrary fusion systems.The fusion system of a finite group G at a prime p encodes the structure of a Sylow-p-subgroupS together with extra information on G-conjugacy within S in category theoretic terms. Fromwork of Alperin and Broue´ [1] it emerged that fusion systems of finite groups are particular casesof fusion systems of p-blocks of finite groups. In the early 1990’s, Puig introduced the notion ofan abstract fusion system on a finite p-group S as a category whose objects are the subgroupsof S and whose morphism sets satisfy a list of properties modelled around what one observes inthe case of fusion systems of finite groups and blocks. There are ‘exotic’ fusion systems which donot arise as fusion system of a block. Benson [2] suggested that nonetheless any fusion systemshould give rise to a p-complete topological space which should coincide with the p-completion ofthe classifying space BG in case the fusion system does arise as fusion system of a finite group G.Broto, Levi and Oliver laid in [4] the homotopy theoretic foundations of such spaces - called p-localfinite groups - and gave in particular a cohomological criterion for the existence and uniqueness ofa p-local finite group associated with a given fusion system. While the existence and uniqueness ofp-local finite groups for arbitrary fusion systems is still an open problem, there has been in recentyears a steadily growing body of work by many authors trying to add to the understanding offusion systems by extending classical concepts and results on the p-local structure of finite groups(some of which were relevant for the classification of finite simple groups) to all fusion systems.This is also the underlying philosophy of the present note. The following two theorems are Flavell’sTheorem A and Theorem B in [6], extended to arbitrary fusion systems. Our general terminologyon fusion systems follows [7]; in particular, by a fusion system we always mean a saturated fusionsystem.Theorem 1. Let p be an odd prime, S a finite p-group, F a fusion system on S and α anautomorphism of S acting freely on S−{1}, which stabilises F and whose order is a prime numberr which does not divide the orders of the automorphism groups AutF(R) for all F-centric radicalsubgroups R of S. Then F = NF(S).The hypothesis that α stablises F means that for any two subgroups Q, R of S and anymorphism ϕ : Q → R in F the morphism α ◦ ϕ ◦ α−1|α(Q) : α(Q) → α(R) is again a morphismin F . Moreover, a subgroup Q of S is called F -centric if CS(Q′) = Z(Q′) for all subgroups Q′ ofS such that Q′ ∼= Q in F ; a subgroup Q of S is called F -radical if Op(AutF (Q)) = AutQ(Q), thegroup of inner automorphisms of Q. By Alperin’s fusion theorem, F is completely determined bythe automorphism groups in F of F -centric radical subgroups of S.1Theorem 2. Let p be an odd prime, S a finite p-group and F a fusion system on S. Let T be asubgroup of Z(S) such that NF(J(S)) ≤ NF (T ). Then one of the following hold:(a) T is weakly F-closed.(b) There exists a non-trivial cyclic p′-subgroup X of AutF(T ) such that X acts transitively on[T,X ]− {1}.Here J(S) is the Thompson subgroup of S; that is, J(S) is generated by all abelian subgroupsof S of maximal order, [T,X ] is the subgroup of T generated by the set of elements of the formψ(t)t−1, where ψ ∈ X and t ∈ T , and T is weakly F-closed if ϕ(T ) = T for all ϕ ∈ HomF (T, S). Asubgroup Q of S is called fully F-normalised if |NS(Q)| ≥ |NS(Q′)| for any subgroup Q′ of S suchthat Q′ ∼= Q in F . In that case, by a result of Puig, there is a fusion system NF(Q) on NS(Q),which is called the normaliser of Q in F . More precisely, for any two subgroups U , V of NS(Q),the morphism set HomNF (Q)(U, V ) consists of all group homomorphisms ϕ : U → V for whichthere exists a morphism ψ : QU → QV in F satisfying ψ|U = ϕ and ψ(Q) = Q. Moreover, NF(Q)induces a fusion system NF(Q)/Q on NP (Q)/Q. Given two normal subgroups Q, Q′ of S suchthat F = NF (Q) = NF(Q′) one checks that then F = NF(QQ′); we denote by Op(F) the largestnormal subgroup R of S such that F = NF (R). Similarly, if Q is a fully F -centralised subgroup ofS, there is a fusion system CF (Q) on CS(Q) such that for any two subgroups U , V of CS(Q), themorphism set HomCF (Q)(U, V ) consists of all group homomorphisms ϕ : U → V for which thereexists a morphism ψ : QU → QV in F satisfying ψ|U = ϕ and ψ(Q) = IdQ. If Z is a subgroup ofZ(S) such that F = CF (Z) then F induces a fusion system, denoted by F/Z, on S/Z.The strategy to prove Theorem 2 is as in [7]: we show that a minimal counterexample is p-constrained, hence a fusion system of a finite group and thus [6, Theorem B] applies. Theorem1 follows from Theorem 2 exactly along the lines of the proof of [6, Theorem A]. We need thefollowing well-known statements:Lemma 3. Let F be a fusion system on S. Suppose that T is a weakly F-closed subgroup of Z(S).Then F = NF (T ).Proof. Let R be an F -centric subgroup of S and let ϕ ∈ AutF(R). Since R contains Z(S), henceT , the assumptions on T imply that ϕ(T ) = T . Alperin’s fusion theorem implies the lemma.Lemma 4. Let F be a fusion system on S. For any subgroup Q of S there is a morphismτ : NS(Q)→ S such that τ(Q) is fully F-normalised.Proof. See for instance [7, Lemma 2.2].Lemma 5. Let F be a fusion system on S, let Z be a subgroup of Z(S) such that F = CF (Z).Set S¯ = S/Z and F¯ = F/Z. We have F = NF(S) if and only if F¯ = NF¯(S¯).Proof. This is a special case of more general results; see e.g. [7, Corollary 3.3].Lemma 6. Let F be a fusion system on S, let Q be a subgroup of S and r a prime divisor of|AutF (Q)|. Then there is an F-centric radical subgroup R of S such that r divides |AutF (R)|.2Proof. Wemay assume thatQ is fully F -normalised. Then, by the Sylow axiom, the group AutS(Q)of automorphisms of Q induced by conjugation with elements in NS(Q) is a Sylow-p-subgroup ofAutF (Q); hence Op(AutF(Q)) = AutR(Q) for a unique subgroup R of NS(Q) containing QCS(Q).The extension axiom implies that any automorphism in AutF(Q) extends to an automorphismin AutF(R). Thus AutF(Q) is a quotient of a subgroup of AutF (R); in particular, r divides|AutF (R)|. Since Q < R unless R is F -centric radical, repeating this argument shows that rdivides |AutF(R)| for some F -centric radical subgroup R of S.Proof of Theorem 2. We use the notation and hypotheses of 2. We argue by induction over thenumber of morphisms in F . Let F be a minimal counterexample to 2. That is, T is not weaklyclosed, and no non trivial cyclic p′-group X of AutF(T ) acts transitively on the non identityelements in [T,X ]. Note that the latter of these two conditions passes down to any subsystem ofF on any subgroup of S containing T .The purpose of the first part of the proof is to show that T ≤ Op(F); in particular, Op(F) 6= 1.By Alperin’s fusion theorem, there exists a fully F -normalised subgroup Q of S such that T ≤ Qand such that T is not weakly NF(Q)-closed. We choose Q with these properties such that NS(Q)has maximal order (but we do not require Q to be centric). Note though that then NS(Q) iscentric, hence T ≤ Z(S) ≤ Z(NS(Q)); in particular, T ≤ J(NS(Q)).Consider first the case where NS(Q) = S. Then J(NS(Q)) = J(S). Since NF(J(S)) ≤ NF(T )by the hypotheses we get that NNF(Q)(J(S)) ≤ NNF (Q)(T ). This shows that NF(Q) is also acounterexample, hence F = NF(Q). This implies T ≤ Q ≤ Op(F) 6= 1 in the case S = NS(Q).Consider next the case where NS(Q) < S. By Lemma 4 applied to J(NS(Q)) there is amorphismτ : NS(J(NS(Q)))→ Ssuch that τ(J(NS(Q))) is fully F -normalised. Note thatQ < NS(Q) < NS(NS(Q)) ≤ NS(J(NS(Q)))Thus τ maps NS(Q) to NS(τ(Q)). Since Q is fully F -normalised, it follows that τ(NS(Q)) =NS(τ(Q)). Thus both τ(Q) and J(NS(τ(Q))) = τ(J(NS(Q))) are fully F -normalised. Next,observe that τ(T ) = T . Indeed, by Alperin’s fusion theorem, τ is a composition of restrictions ofautomorphisms of fully F -normalised subgroups of S whose order is at least that of NS(J(NS(Q))),which is bigger than the order of NS(Q). By the maximality assumptions on Q we therefore haveτ(T ) = T . Since T is not weakly NF(Q)-closed, there is ϕ ∈ AutF (Q) such that ϕ(T ) 6= T .Then, setting ψ = τ ◦ ϕ ◦ τ−1 and using τ(T ) = T we get that ψ(T ) 6= T , and hence T is notweakly NF (τ(Q))-closed. Therefore, after possibly replacing Q by τ(Q), we may assume thatboth Q and J(NS(Q)) are fully F -normalised. The point of making this assumption is that thenNF (J(NS(Q))) is a fusion system on NS(J(NS(Q))). We claim that NF(J(NS(Q))) ≤ NF(T ).Since Q was chosen with NS(Q) of maximal order subject to T not being weakly NF (Q)-closed, Tis weakly NF (J(NS(Q)))-closed. As T ≤ J(NS(Q)) this is equivalent to NF(J(NS(Q))) ≤ NF(T ).But then also NNF (Q)(J(NS(Q))) ≤ NNF (Q)(T ), hence NF(Q) is a counterexample to Theorem 2as well, a contradiction to the minimality of F . This shows that T ≤ Op(F) 6= 1.The second part of the proof consists of showing thatOp(F) is in fact F -centric. SetQ = Op(F).Since T is not weakly F -closed there is ϕ ∈ AutF(Q) such that ϕ(T ) 6= T . The morphism ϕ3extends to a morphism ψ ∈ AutF(QCP (Q)). Thus T is not weakly NF(QCP (Q))-closed, andhence NF (QCP (Q)) is also a counterexample to Theorem 2. By the minimality of F it followsthat F = NF(QCP (Q)), or equivalently, QCP (Q) ≤ Op(F) = Q, which implies that Q is F -centric.It follows now from [3, Proposition C] that F = FS(L) for some finite group L. But thenFlavell’s Theorem B in [6] applies, showing that F cannot be a counterexample. This contradictionconcludes the proof of Theorem 2.Proof of Theorem 1. We show that Theorem 2 implies Theorem 1, closely following the proof of[6, Theorem A]. As before we argue by induction over the number of morphisms in F . Let F be aminimal counterexample to 1; that is, with the notation and hypotheses of 1 we have NF (S) < F .We first show that Op(F) = 1. Suppose not; that is, F = NF (Q) for some non trivial normalsubgroup Q of S. Since α stablises F it stablises Op(F). Choose a minimal α-stable normalsubgroup V of S such that F = NF(V ). Since Z(V ) is then also α-stable satisfying F = NF(Z(V )),the minimality of V implies that V is abelian. The subgroup Ω1(V ) generated by all elements oforder p in V is then again α-stable and satisfies F = NF(Ω1(V )). The minimality of V impliesthat V is elementary abelian and that V is a simple module for the group AutF (V )⋊ 〈α〉 over Fp.By Lemma 6, the prime r does not divide the order of AutF (V ). Using that AutF(V ) and 〈α〉have coprime orders one checks easily that V is in fact a faithful simple AutF (V ) ⋊ 〈α〉-moduleover Fp. Since α acts freely on V −{1}, the hypotheses of [5, Theorem A] are satisfied, and hence[AutF (V ), α] is a 2-group. Since α acts freely on S − {1} we have S = [S, α]. Thus the image ofS = [S, α] in AutF (V ) is contained in the 2-group [AutF(V ), α], and hence must be trivial as p isodd. Therefore S centralises V , or equivalently, V ≤ Z(S). It follows that CF (V ) is an α-stablefusion system on S satisfying the hypotheses of Theorem 1. If CF (V ) = F then the fusion systemF¯ = F/V on S¯ = S/V inherits the hypotheses in 1, and hence F¯ = NF¯(S¯) by induction. Lemma5 implies that F = NF(S), a contradiction. Thus CF (V ) < F , and hence CF (V ) = NCF (V )(S)by induction. Since F = NF(V ) and fusion in V ≤ Z(S) is controlled by AutF (S) it follows againthat F = NF(S), so the assumption Op(F) 6= 1 leads to a contradiction.Set T = Ω1(Z(S)). Assume first that T is weakly F -closed. Then, by Lemma 3, we haveF = NF(T ), contradicting Op(F) = 1. Thus T is not weakly F -closed. Clearly NF(S) ≤NF (J(S)) < F , and hence NF(S) = NF(J(S)) by induction. As T = Ω1(Z(S)) ≤ J(S) itfollows that NF (J(S)) ≤ NF (T ). Since T is not weakly F -closed, Theorem 2 implies that there isa non trivial cyclic p′-subgroup X of AutF(T ) which acts transitively on the non identity elementsin [T,X ]. Now T can be viewed as module of AutF(T )⋊〈α〉 over Fp. Then [6, Theorem 3.2] impliesthat α has a non trivial fixpoint in T , a contradiction. This completes the proof of Theorem 1.References[1] J. L. Alperin, M. Broue´, Local methods in block theory, Ann. Math. 110 (1979), 143–157.[2] D. J. Benson, Cohomology of sporadic groups, finite loop spaces, and the Dickson in-variants, in: Geometry and Cohomology in Group Theory (eds. P. H. Kropholler, G. A.Niblo, R. Sto¨hr), London Math. Soc. Lecture Ser. 252 (1998), 10–23.4[3] C. Broto, N. Castellana, J. Grodal, R. Levi, B. Oliver, Subgroup families controllingp-local finite groups, Proc. London Math. Soc (3) 91 (2005), 325–354.[4] C. Broto, R. Levi, B. Oliver, The homotopy theory of fusion systems, J. Amer. Math.Soc. 16 (2003), 779–856.[5] P. Flavell, A Hall-Higman-Shult Type Theorem for Arbitrary Finite Groups, Invent. Math.164 (2006), 361–397.[6] P. Flavell, Automorphisms and fusion in finite groups, J. Algebra 300 (2006), 472–479.[7] R. Kessar, M. Linckelmann, ZJ-theorems for fusion systems, Trans. Amer. Math. Soc.360 (2008), 3093–3106.5

On two theorems of Flavell

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