On two theorems of Flavell

Radha Kessar, Markus Linckelmann

Abstract

We extend two theorems due to P. Flavell [3] to arbitrary fusion systems.

The following two theorems are Flavell’s Theorem A and Theorem B in [3], extended to arbitrary fusion systems. Our general terminology on fusion system follows [4]; in particular, by a fusion system we always mean a saturated fusion system.

Theorem 1. Let p be an odd prime, S a finite p-group, F a fusion system on S and α an automorphism of S acting freely on S −{1}, which stabilises F and whose order is a prime number r which does not divide the orders of the automorphism groups Aut_{F}(R) for all F-centric radical subgroups R of S. Then F = N_{F}(S).

The hypothesis that α stablises F means that for any two subgroups Q, R of S and any morphism φ : Q → R in F the morphism α ∘ φ ∘ α^{−1}|_{α(Q)} : α(Q) → α(R) is again a morphism in F. Moreover, a subgroup Q of S is called F-centric if C_{S}(Q′) = Z(Q′) for all subgroups Q′ of S such that Q′ ≅ Q in F; a subgroup Q of S is called F-radical if O_{p}(Aut_{F}(Q)) = Aut_{Q}(Q), the group of inner automorphisms of Q. By Alperin’s fusion theorem, F is completely determined by the automorphism groups in F of F-centric radical subgroups of S.

Theorem 2. Let p be an odd prime, S a finite p-group and F a fusion system on S. Let T be a subgroup of Z(S) such that N_{F}(J(S)) ≤ N_{F}(T). Then one of the following hold:

(a) T is weakly F-closed.

(b) There exists a non-trivial cyclic p′-subgroup X of Aut_{F}(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 subgroups of S of maximal order, [T, X] is the subgroup of T generated by the set of elements of the form ψ(t)t^{−1}, where ψ ∈ X and t ∈ T, and T is weakly F-closed if φ(T) = T for all φ ∈ Hom_{F}(T, S). A subgroup Q of S is called fully F-normalised if |N_{S}(Q)| ≥ |N_{S}(Q′)| for any subgroup Q′ of S such that Q′ ≅ Q in F. In that case, by a result of Puig, there is a fusion system N_{F}(Q) on N_{S}(Q), which is called the normaliser of Q in F. More precisely, for any two subgroups U, V of N_{S}(Q), the morphism set Hom_{N_{F} (Q)}(U, V) consists of all group homomorphisms φ : U → V for which there exists a morphism ψ : QU → QV in F satisfying ψ|_{U} = φ and ψ(Q) = Q. Moreover, N_{F}(Q) induces a fusion system N_{F}(Q)/Q on N_{P} (Q)/Q. Given two normal subgroups Q, Q′ of S such that F = N_{F}(Q) = N_{F}(Q′) one checks that then F = N_{F}(QQ′); we denote by O_{p}(F) the largest normal subgroup R of S such that F = N_{F}(R). Similarly, if Q is a fully F-centralised subgroup of S, there is a fusion system C_{F}(Q) on C_{S}(Q) such that for any two subgroups U, V of C_{S}(Q), the morphism set Hom_{C_{F} (Q)}(U, V) consists of all group homomorphisms φ : U → V for which there exists a morphism ψ : QU → QV in F satisfying ψ|_{U} = φ and ψ(Q) = Id_{Q}. If Z is a subgroup of Z(S) such that F = C_{F}(Z) then F induces a fusion system, denoted by F/Z, on S/Z.

The strategy to prove Theorem 2 is as in [4]: we show that a minimal counterexample is pconstrained, hence a fusion system of a finite group and thus [3, Theorem B] applies. Theorem 1 follows from Theorem 2 exactly along the lines of the proof of [3, Theorem A]. We need the following 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 = N_{F}(T).

Proof. Let R be an F-centric subgroup of S and let φ ∈ Aut_{F}(R). Since R contains Z(S), hence T, 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 morphism τ : N_{S}(Q) → S such that τ(Q) is fully F-normalised.

Proof. See for instance [4, Lemma 2.2].

Lemma 5. Let F be a fusion system on S, let Z be a subgroup of Z(S) such that F = C_{F}(Z). Set S̄ = S/Z and F̄ = F/Z. We have F = N_{F}(S) if and only if F̄ = NF̄(S̄).

Proof. This is a special case of more general results; see e.g. [4, 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 |Aut_{F}(Q)|. Then there is an F-centric radical subgroup R of S such that r divides |Aut_{F}(R)|.

Proof. We may assume that Q is fully F-normalised. Then, by the Sylow axiom, the group Aut_{S}(Q) of automorphisms of Q induced by conjugation with elements in N_{S}(Q) is a Sylow-p-subgroup of Aut_{F}(Q); hence O_{p}(Aut_{F}(Q)) = Aut_{R}(Q) for a unique subgroup R of N_{S}(Q) containing QC_{S}(Q). The extension axiom implies that any automorphism in Aut_{F}(Q) extends to an automorphism in Aut_{F}(R). Thus Aut_{F}(Q) is a quotient of a subgroup of Aut_{F}(R); in particular, r divides |Aut_{F}(R)|. Since Q < R unless R is F-centric radical, repeating this argument shows that r divides |Aut_{F}(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 the number of morphisms in F. Let F be a minimal counterexample to 2. That is, T is not weakly closed, and no non trivial cyclic p′-group X of Aut_{F}(T) acts transitively on the non identity elements in [T, X]. Note that the latter of these two conditions passes down to any subsystem of F on any subgroup of S containing T.

The purpose of the first part of the proof is to show that T ≤ O_{p}(F); in particular, O_{p}(F) ≠ 1. By Alperin’s fusion theorem, there exists a fully F-normalised subgroup Q of S such that T ≤ Q and such that T is not weakly N_{F}(Q)-closed. We choose Q with these properties such that N_{S}(Q) has maximal order (but we do not require Q to be centric). Note though that then N_{S}(Q) is centric, hence T ≤ Z(S) ≤ Z(N_{S}(Q)); in particular, T ≤ J(N_{S}(Q)).

Consider first the case where N_{S}(Q) = S. Then J(N_{S}(Q)) = J(S). Since N_{F}(J(S)) ≤ N_{F}(T) by the hypotheses we get that N_{N_{F} (Q)}(J(S)) ≤ N_{N_{F} (Q)}(T). This shows that N_{F}(Q) is also a counterexample, hence F = N_{F}(Q). This implies T ≤ Q ≤ O_{p}(F) ≠ 1 in the case S = N_{S}(Q).

Consider next the case where N_{S}(Q) < S. By Lemma 4 applied to J(N_{S}(Q)) there is a morphism

τ : N_{S}(J(N_{S}(Q))) → S

such that τ(J(N_{S}(Q))) is fully F-normalised. Note that

Q < N_{S}(Q) < N_{S}(N_{S}(Q)) ≤ N_{S}(J(N_{S}(Q)))

Thus τ maps N_{S}(Q) to N_{S}(τ(Q)). Since Q is fully F-normalised, it follows that τ(N_{S}(Q)) = N_{S}(τ(Q)). Thus both τ(Q) and J(N_{S}(τ(Q))) = τ(J(N_{S}(Q))) are fully F-normalised. Next, observe that τ(T) = T. Indeed, by Alperin’s fusion theorem, τ is a composition of restrictions of automorphisms of fully F-normalised subgroups of S whose order is at least that of N_{S}(J(N_{S}(Q))), which is bigger than the order of N_{S}(Q). By the maximality assumptions on Q we therefore have τ(T) = T. Since T is not weakly N_{F}(Q)-closed, there is φ ∈ Aut_{F}(Q) such that φ(T) ≠ T. Then, setting ψ = τ ∘ φ ∘ τ^{−1} and using τ(T) = T we get that ψ(T) ≠ T, and hence T is not weakly N_{F}(τ(Q))-closed. Therefore, after possibly replacing Q by τ(Q), we may assume that both Q and J(N_{S}(Q)) are fully F-normalised. The point of making this assumption is that then N_{F}(J(N_{S}(Q))) is a fusion system on N_{S}(J(N_{S}(Q))). We claim that N_{F}(J(N_{S}(Q))) ≤ N_{F}(T). Since Q was chosen with N_{S}(Q) of maximal order subject to T not being weakly N_{F}(Q)-closed, T is weakly N_{F}(J(N_{S}(Q)))-closed. As T ≤ J(N_{S}(Q)) this is equivalent to N_{F}(J(N_{S}(Q))) ≤ N_{F}(T). But then also N_{N_{F} (Q)}(J(N_{S}(Q))) ≤ N_{N_{F} (Q)}(T), hence N_{F}(Q) is a counterexample to Theorem 2 as well, a contradiction to the minimality of F. This shows that T ≤ O_{p}(F) ≠ 1.

The second part of the proof consists of showing that O_{p}(F) is in fact F-centric. Set Q = O_{p}(F). Since T is not weakly F-closed there is φ ∈ Aut_{F}(Q) such that φ(T) ≠ T. The morphism φ extends to a morphism ψ ∈ Aut_{F}(QC_{P} (Q)). Thus T is not weakly N_{F}(QC_{P} (Q))-closed, and hence N_{F}(QC_{P} (Q)) is also a counterexample to Theorem 2. By the minimality of F it follows that F = N_{F}(QC_{P} (Q)), or equivalently, QC_{P} (Q) ≤ O_{p}(F) = Q, which implies that Q is F-centric.

It follows now from [1, Proposition C] that F = F_{S}(L) for some finite group L. But then Flavell’s Theorem B in [3] applies, showing that F cannot be a counterexample. This contradiction concludes the proof of Theorem 2.

Proof of Theorem 1. We show that Theorem 2 implies Theorem 1, closely following the proof of [3, Theorem A]. As before we argue by induction over the number of morphisms in F. Let F be a minimal counterexample to 1; that is, with the notation and hypotheses of 1 we have N_{F}(S) < F.

We first show that O_{p}(F) = 1. Suppose not; that is, F = N_{F}(Q) for some non trivial normal subgroup Q of S. Since α stablises F it stablises O_{p}(F). Choose a minimal α-stable normal subgroup V of S such that F = N_{F}(V). Since Z(V) is then also α-stable satisfying F = N_{F}(Z(V)), the minimality of V implies that V is abelian. The subgroup Ω_{1}(V) generated by all elements of order p in V is then again α-stable and satisfies F = N_{F}(Ω_{1}(V)). The minimality of V implies that V is elementary abelian and that V is a simple module for the group Aut_{F}(V) ⋊ ⟨α⟩ over 𝔽_{p}. By Lemma 6, the prime r does not divide the order of Aut_{F}(V). Using that Aut_{F}(V) and ⟨α⟩ have coprime orders one checks easily that V is in fact a faithful simple Aut_{F}(V) ⋊ ⟨α⟩-module over 𝔽_{p}. Since α acts freely on V − {1}, the hypotheses of [2, Theorem A] are satisfied, and hence [Aut_{F}(V), α] is a 2-group. Since α acts freely on S − {1} we have S = [S, α]. Thus the image of S = [S, α] in Aut_{F}(V) is contained in the 2-group [Aut_{F}(V), α], and hence must be trivial as p is odd. Therefore S centralises V, or equivalently, V ≤ Z(S). It follows that C_{F}(V) is an α-stable fusion system on S satisfying the hypotheses of Theorem 1. If C_{F}(V) = F then the fusion system F̄ = F/V on S̄ = S/V inherits the hypotheses in 1, and hence F̄ = NF̄(S̄) by induction. Lemma 5 implies that F = N_{F}(S), a contradiction. Thus C_{F}(V) < F, and hence C_{F}(V) = N_{C_{F} (V)}(S) by induction. Since F = N_{F}(V) and fusion in V ≤ Z(S) is controlled by Aut_{F}(S) it follows again that F = N_{F}(S), so the assumption O_{p}(F) ≠ 1 leads to a contradiction.

Set T = Ω_{1}(Z(S)). Assume first that T is weakly F-closed. Then, by Lemma 3, we have F = N_{F}(T), contradicting O_{p}(F) = 1. Thus T is not weakly F-closed. Clearly N_{F}(S) ≤ N_{F}(J(S)) < F, and hence N_{F}(S) = N_{F}(J(S)) by induction. As T = Ω_{1}(Z(S)) ≤ J(S) it follows that N_{F}(J(S)) ≤ N_{F}(T). Since T is not weakly F-closed, Theorem 2 implies that there is a non trivial cyclic p′-subgroup X of Aut_{F}(T) which acts transitively on the non identity elements in [T, X]. Now T can be viewed as module of Aut_{F}(T)⋊⟨α⟩ over 𝔽_{p}. Then [3, Theorem 3.2] implies that α has a non trivial fixpoint in T, a contradiction. This completes the proof Theorem 1.

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