A GENERALIZATION OF THE Z^{∗}-THEOREM

E. HENKE AND J. SEMERARO

Abstract. Glaubermann’s Z^{∗}-theorem and analogous statements for odd primes show that, for any prime p and any finite group G with Sylow psubgroup S, the centre of G is determined by the fusion system F_{S}(G). Building on these results we show a statement that can be considered as a generalization: For any normal subgroup N of G, the centralizer C_{S}(N) is expressed in terms of the fusion system F_{S}(G) and its normal subsystem induced by N.

Throughout p is a prime. Glauberman’s Z^{∗}-theorem [3] and its generalization to odd primes, which is shown using the classification of finite simple groups (see [4] and [7]), can be reformulated as follows:

Theorem A. Let G be a finite group with O_{p′}(G) = 1, and S ∈ Syl_{p}(G). Then Z(G) = Z(F_{S}(G)).

We refer the reader here to [2] for basic definitions and results regarding fusion systems; see in particular Definitions I.4.1 and I.4.3 for the definition of central subgroups and the centre Z(F).

Given a saturated fusion system F on a finite p-group S and a normal subsystem E of F on T ⩽ S, Aschbacher [1, (6.7)(1)] showed that the set of subgroups X of C_{S}(T) with E ⊆ C_{F}(X) has a largest member C_{S}(E). He furthermore constructed a normal subsystem C_{F}(E) on C_{S}(E), the centralizer of E in F; see [1, Chapter 6]. Note that C_{S}(E) depends not only on S and E but also on the fusion system F in which both S and E are contained.

The definition of C_{S}(E) generalizes the definition of Z(F) since C_{S}(F) = Z(F). Moreover, for every normal subgroup H of a finite group G with Sylow p-subgroup S, F_{S∩H}(H) is a normal subsystem of F_{S}(G) by [2, I.6.2]. Thus, the following theorem, which we prove in this short note, can be seen as a generalization of Theorem A.

Theorem B. Let G be a finite group and S be a Sylow p-subgroup of G. Let H ⊴G with O_{p′}(H) = 1. Then C_{S}(F_{S∩H}(H)) = C_{S}(H).

In the statement of Theorem B it is understood that C_{S}(F_{S∩H}(H)) is formed inside of F_{S}(G). The result says in other words that, under the hypothesis of Theorem B, for any X ⩽ S with F_{S∩H}(H) ⊆ C_{F_{S}(G)}(X), we have X ⩽ C_{S}(H). This is not true if one drops the assumption that H is normal in G as the following example shows: Let G := G_{1} × G_{2} with G_{1} ≅ G_{2} ≅ S_{3}. Set p = 3, S = O_{3}(G), S_{i} := O_{3}(G_{i}) and let R be a subgroup of G of order 2 which acts fixed point freely on S. Set H := S_{1} ⋊ R. Then S_{1} = S ∩ H ∈ Syl_{3}(H) and F_{S_{1}}(H) = F_{S_{1}}(G_{1}) ⊆ C_{F_{S}(G)}(S_{2}) as S_{2} = C_{S}(G_{1}). However, S_{2} ⩽̸ C_{S}(H) by the choice of R.

Theorem B was conjectured by the second author of this paper in [6]. Our proof of Theorem B builds on Theorem A and the reduction uses only elementary group theoretical results. Essential is the following lemma, whose proof is self-contained apart from using the conjugacy of Hall-subgroups in solvable groups.

Lemma 0.1. Let G be a finite group with Sylow p-subgroup S and a normal subgroup H. Let P ⩽ S such that P ∩ H is centric in F_{S∩H}(H). Then for every p′-element ϕ ∈ Aut_{G}(P) with [P, ϕ] ⩽ P ∩ H and ϕ|_{P ∩H} ∈ Aut_{H}(P ∩ H), we have ϕ ∈ Aut_{H}(P).

Proof. This is [5, Proposition 3.1]. □

Proof of Theorem B. We assume the hypothesis of Theorem B. Furthermore, we set F := F_{S}(G), T := S ∩ H and E := F_{T} (H). If a homomorphism ϕ between subgroups A and B of T is induced by conjugation with an element h ∈ H, then ϕ extends to c_{h} : AC_{S}(H) → BC_{S}(H) and c_{h} restricts to the identity on C_{S}(H). Thus E ⊆ C_{F}(C_{S}(H)), so by the definition of C_{S}(E), we have C_{S}(H) ⩽ C_{S}(E). To prove the converse inclusion, choose t ∈ C_{S}(E). Define:

G_{0} := H⟨t⟩ and S_{0} := T ⟨t⟩,

so that plainly S_{0} is a Sylow p-subgroup of G_{0} and F_{0} := F_{S_{0}}(G_{0}) is a saturated fusion system on S_{0}. Note also that O_{p′}(G_{0}) = 1 as O^{p}(G_{0}) = O^{p}(H) and O_{p′}(H) = 1 by assumption.

By Theorem A, Z(F_{0}) = Z(G_{0}) ⩽ C_{S}(H). It thus suffices to prove t ∈ Z(F_{0}). As t ∈ C_{S}(E) ⩽ C_{S}(T), t ∈ Z(S_{0}). Let P be a subgroup of S_{0} which is centric radical and fully normalized in F_{0}. Then t ∈ Z(S_{0}) ⩽ C_{S_{0}}(P) ⩽ P. It is sufficient to prove [t, Aut_{F_{0}}(P)] = 1. For as P is arbitrary, Alperin’s fusion theorem [2, Theorem 3.6] implies then t ∈ Z(F_{0}). As P is fully F_{0}-normalized, Aut_{S_{0}}(P) ∈ Syl_{p}(Aut_{F_{0}}(P)) and thus Aut_{F_{0}}(P) = Aut_{S_{0}}(P)O^{p}(Aut_{F_{0}}(P)). Note that [t, Aut_{S_{0}}(P)] = 1 as t ∈ Z(S_{0}). Hence, it is enough to prove

[t, O^{p}(Aut_{F_{0}}(P))] = 1.

Let ϕ ∈ Aut_{F_{0}}(P) be a p′-element. Since O^{p}(H) = O^{p}(G_{0}), we have O^{p}(Aut_{F_{0}}(P)) = O^{p}(Aut_{H}(P)). In particular, ϕ ∈ Aut_{H}(P) and thus ϕ|_{P ∩T} ∈ Aut_{H}(P ∩T) = Aut_{E}(P ∩T). As t ∈ P ⩽ S_{0} = T ⟨t⟩, we have P = (P ∩T)⟨t⟩. Moreover, t ∈ C_{S}(E) implies that E ⊆ C_{F}(⟨t⟩). Hence, φ|_{P ∩T} extends to ψ ∈ Aut_{F}(P) with the property that tψ = t. Note that o(ψ) = o(ϕ|_{P ∩T}) and thus ψ is a p′-element as ϕ has order prime to p. Moreover, plainly [P, ψ] ⩽ P ∩ T and ψ|_{P ∩T} = ϕ|_{P ∩T} ∈ Aut_{H}(P ∩ T). Since E ⊴ F_{0}, P ∩ T is E-centric by [1, 7.18]. Now it follows from Lemma 1 that ψ ∈ Aut_{H}(P). Thus, χ := φ ∘ ψ^{−1} ∈ Aut_{H}(P) ⩽ Aut_{F_{0}}(P). Clearly χ|_{P ∩T} = Id as ψ extends ϕ|_{P ∩T}. Moreover, using that H is normal in G, we obtain [P, χ] ⩽ [P, Aut_{H}(P)] = [P, N_{H}(P)] ⩽ P ∩ H = P ∩ T. Hence, by [2, Lemma A.2], χ ∈ C_{Aut_{F_{0}} (P)}(P/(P ∩ T)) ∩ C_{Aut_{F_{0}} (P)}(P ∩ T) = O_{p}(Aut_{F_{0}}(P)) = Inn(P) as P is radical in F_{0}. As Inn(P) ⩽ Aut_{S_{0}}(P) and [t, Aut_{S_{0}}(P)] = 1, it follows tχ = t. By the choice of ψ, also tψ = t and consequently tϕ = t. Since ϕ was chosen to be an arbitrary p′-element in Aut_{F_{0}}(P) and O^{p}(Aut_{F_{0}}(P)) is the subgroup generated by these elements, it follows that [t, O^{p}(Aut_{F_{0}}(P))] = 1. As argued above, this yields the assertion. □

References

[1] M. Aschbacher, The generalized fitting subsystem of a fusion system Mem. Amer. Math. Soc. 209 (2011), no. 986, vi+110.

[2] M. Aschbacher, R. Kessar, and B. Oliver, Fusion systems in algebra and topology London Math. Soc. Lecture Note Series, vol. 391, Cambridge University Press, 2011.

[3] G. Glauberman, Central elements in core-free groups J. Algebra 4 (1966), 403-420.

[4] R. M. Guralnick and G. R. Robinson, On extensions of the Baer–Suzuki theorem J. Algebra 82 (1993), no. 1–3, 281-297.

[5] E. Henke, Products in fusion systems J. Algebra 376 (2013), 300-319.

[6] J. Semeraro, Centralizers of subsystems of fusion systems J. Group Theory, to appear.

[7] W. J. Xiao, Glaubermans conjecture, Mazurovs problem and Pengs problem Sci. China Ser. A 34, no. 9.

Institute of Mathematics, University of Aberdeen, U.K.

  E-mail address: ellen.henke@abdn.ac.uk

Heilbronn Institute for Mathematical Research, Department of Mathematics, University of Bristol, U.K.

  E-mail address: js13525@bristol.ac.uk