NORMAL SUBSYSTEMS AND CONTROL OF FUSION

Markus Linckelmann

April, 2003

Abstract. We prove a “control of fusion” type Theorem which says that if a normal subsystem F′ of a fusion system F is controlled by a non-trivial subgroup, then so is F itself. This is an analogue for arbitrary fusion systems of [2, Theorem 1.5] which proves this fact for fusion system of p-blocks of finite groups.

Theorem 1. Let p be a prime and let F be a fusion system on a finite p-group P. Let P′ be a strongly F-closed subgroup of P and let F′ be a fusion system on P′ such that F′ ⊴ F. Suppose that there is a subgroup Q′ of P′ such that F′ = N_{F′}(Q′). Let Q be the subgroup of P′ generated by all subgroups of P′ isomorphic to Q′ in F. Then F = N_{F}(Q).

We refer to the appendix for the terminology regarding fusion systems. The proof of Theorem 1 is based on one hand on ideas of G. R. Robinson, which have led to [2, 1.5], and on the other hand on the notion of normal subsystems developed in [3, §3]. It seems to be an open question as to whether a simple fusion system in the sense of [3, 4.1] can have non trivial proper strongly closed subgroups. The following result, due to Stancu [6], can be obtained as a consequence of Theorem 1 and provides a partial answer to this problem:

Corollary 2. ([6, Proposition 6.2]) Let p be a prime and let F be a fusion system on a finite p-group P. Let Q be a subgroup of P. We have F_{Q}(Q) ⊴ F if and only if F = N_{F}(Q).

Corollary 2 applies in the following situation: if Q is an abelian strongly F-closed subgroup of P, then by [3, 3.4], F_{Q}(Q) is normal in F, and hence F = N_{F}(Q).

We divide the proofs in a series of statements (the first of which is well-known but included for the convenience of the reader). Throughout the rest of this note, p is a prime.

Typeset by AMS-TEX

Proposition 3. Let F be a fusions system on a finite p-group P. Let P′ be a strongly F-closed subgroup of P and let F′ be a category on P′ such that F′ ⊆ F. Let Q be a subgroup of P′.

(i) If Q is fully F-centralised then Q is fully F′-centralised. (ii) If Q is fully F-normalised then Q is fully F′-normalised.

Proof. (i) Let τ : Q → R be an isomorphism in F′ such that R is fully F′-centralised. If Q is fully F-centralised, then τ^{−1} : R → Q extends to a morphism σ : RC_{P′}(R) → QC_{P} (Q) in F. Since P′ is strongly F-closed, this restricts to a morphism C_{P′}(R) → C_{P′}(Q) in F, in particular |C_{P′}(R) ≤ |C_{P′}(Q)|. Since Q, R are isomorphic in F′ and since R is fully F′-centralised, this inequality is in fact an equality and so is Q is fully F′-centralised, too.

(ii) Let now τ : Q → R be an isomorphism in F′ such that R is fully F′-normalised. Then there is an automorphism α of Q in F such that α ∘ τ^{−1} : R → Q extends to a morphism ψ : N_{P} (R) → N_{P} (Q). Since P′ is strongly F-closed, ψ induces a morphism N_{P′}(R) → N_{P′}(Q); in particular, |N_{P′}(R)| ≤ |N_{P′}(Q)|. Since Q, R are isomorphic in F′ and R is fully F′-normalised, this inequality is in fact an equality. □

Corollary 4. Let F be a fusion system on a finite p-group P, let F′ be a fusion system on a subgroup P′ of P such that F′ ⊴ F and let Q be a subgroup of P′. If Q is fully F-normalised then Aut_{F′}(Q) ∩ Aut_{P} (Q) = Aut_{P′}(Q).

Proof. If Q is fully F-normalised then Aut_{P} (Q) is a Sylow-p-subgroup of Aut_{F}(Q). Since Aut_{F′}(Q) ⊴ Aut_{F}(Q) the intersection Aut_{F′}(Q) ∩ Aut_{P} (Q) is then also a Sylowp-subgroup of Aut_{F′}(Q). By Proposition 3, Q is also fully F′-normalised and hence Aut_{P′}(Q) is a Sylow-p-subgroup of Aut_{F′}(Q) as well. Clearly Aut_{P′}(Q) ⊆ Aut_{F′}(Q) ∩ Aut_{P} (Q), which implies the equality as stated. □

Proposition 5. Let F be a fusion system on a finite p-group P. Let F′ be a fusion system on a subgroup P′ of P such that F′ ⊴ F. Let φ : Q → Q′ be an isomorphism in F. Suppose that Q, Q′ are contained in P′ and that Q′ is fully F′-normalised. Then there is an automorphism τ ∈ Aut_{F′}(Q′) such that the isomorphism τ ∘ φ : Q → Q′ can be extended to a morphism ψ : N_{P′}(Q) → P in F.

Proof. Since F′ ⊴ F, the morphism φ induces an isomorphism Aut_{F′}(Q) ≅ Aut_{F}(Q′) given by “conjugation” with φ. Since Q′ is fully F′-normalised, the group Aut_{P′}(Q′) is a Sylow-p-subgroup of Aut_{F′}(Q′). Thus the image φ ∘ Aut_{P′}(Q) ∘ φ^{−1} of Aut_{P′}(Q) in Aut_{F′}(Q′) is conjugate to a subgroups of Aut_{P′}(Q′). In other words, there is τ ∈ Aut_{F′}(Q′) such that (τ ∘ φ) ∘ Aut_{P′}(Q) ∘ (τ ∘ φ)^{−1} ⊆ Aut_{P′}(Q′), or equivalently, the isomorphism Aut_{F′}(Q) ≅ Aut_{F′}(Q′) induced by “conjugation” with τ ∘φ maps Aut_{P′}(Q) to Aut_{P′}(Q′).

Let now β : Q′ → R be an isomorphism in F such that R is fully F-normalised. By Proposition 3, R is then also fully F′-normalised and by Corollary 4, Aut_{F′}(R) ∩ Aut_{P} (R) = Aut_{P′}(R) is a Sylow-p-subgroup of Aut_{F′}(R). Thus some Aut_{F′}(R)- conjugate of β ∘ Aut_{P′}(Q′) ∘ β^{−1} is contained in Aut_{P′}(R). Thus, up to replacing β, we may assume that β ∘ Aut_{P′}(Q′) ∘ β^{−1} = Aut_{P′}(R). Note that then N_{P′}(Q′) ≅ N_{P′}(R) because both Q′, R are fully F′-normalised. Since R is in fact fully F-normalised, β extends to an isomorphism σ : N_{P′}(Q′) ≅ N_{P′}(R) in F. Also β ∘ τ ∘ φ maps Aut_{P′}(Q) into Aut_{P′}(R), hence extends to a morphism ρ : N_{P′}(Q) → N_{P′}(R) in F. Then σ^{−1}ρ : N_{P′}(Q) → N_{P′}(Q′) is a morphism in F which extends τ ∘ φ. □

Corollary 6. Let F be a fusion system on a finite p-group P. Let F′ be a fusion system on a subgroup P′ of P such that F′ ⊴ F. Let Q be a normal subgroup of P′. For any morphism φ : Q → P in F there is a morphism ψ : φ(Q) → P′ in F′ and an automorphism α ∈ Aut_{F}(P′) such that ψ ∘ φ = α|_{Q}.

Proof. Clearly there is a morphism ψ : φ(Q) → P′ in F′ such that ψ(φ(Q)) is fully F′-normalised. It follows from Proposition 3 that ψ can be chosen in such a way that ψ∘φ extends to a morphism α from N_{Q}(P′) = P′ to P, and since P′ is strongly F-closed it follows that α is in fact an automorphism of P′. □

Corollary 7. Let F be a fusion system on a finite p-group P. Let F′ be a fusion system on a subgroup P′ of P such that F′ ⊴ F. Let Q be a strongly F′-closed subgroup of P′. For any morphism φ : Q → P in F the subgroup φ(Q) is a strongly F′-closed subgroup of P′ and there are automorphisms ψ ∈ Aut_{F′}(φ(Q)) and α ∈ Aut_{F}(P′) such that ψ ∘ φ = α|_{Q}; in particular, α(Q) = φ(Q).

Proof. By Corollary 6 there is a morphism ψ : φ(Q) → P′ and an automorphism α ∈ Aut_{F}(P′) such that ψ ∘ φ = α|_{Q}. Since F′ ⊴ F clearly α(Q) is strongly F′-closed. Since ψ is a morphism in the category F′, this forces in particular φ(Q) = α(Q). □

We mention two well-known consequences of Proposition 5 applied to the case P = P′ and F = F′ (which could also be proved easily directly): if φ : Q → Q′ is an isomorphism in F such that Q′ is fully F-normalised then there is an automorphism τ ∈ Aut_{F}(Q′) such that N_{τ∘φ} = N_{P} (Q). In other words, for any subgroup Q of P there is a morphism φ : Q → P such that φ(Q) is fully F-normalised and such that φ extends to a morphism ψ : N_{P} (Q) → P in F.

Proposition 8. Let F be a fusion system on a finite p-group P, let F′ be a fusion system on a subgroup P′ of P such that F′ ⊴ F, and let Q be a subgroup of P′. If Q is strongly F′-closed and weakly F-closed then Q is strongly F-closed.

Proof. Suppose that Q is not strongly F-closed. Then there is a subgroup R of Q and a morphism φ : R → P such that φ(R) is not contained in Q. Choose R maximal with this property. Certainly φ(R) ⊆ P′ because P′ is strongly F-closed. Hence there is a morphism ψ : φ(R) → P′ in F′ such that ψ(φ(R)) is fully F′-normalised. Also ψ(φ(R)) is again not contained in Q because Q is strongly F′-closed and ψ is a morphism in F′. Thus, up to replacing φ by ψ ∘ φ we may assume that φ(R) is fully F′-normalised. Hence, by Proposition 5, up to replacing φ by φ composed with a suitable automorphism of φ(R) in F′ we may assume that φ extends to a morphism σ : N_{P′}(R) → N_{P′}(φ(R)) in F. Restricting σ to N_{Q}(R) yields a morphism (abusively still denoted by the same letter) σ : N_{Q}(R) → P′. Since σ extends φ we still have that σ(N_{Q}(R)) is not contained in Q. By the maximality of R with this property this forces R = Q. But then Q cannot be weakly F-closed. □

The next Proposition is an analogue of [2, 7.1] for arbitrary fusion systems.

Proposition 9. Let F be a fusion system on a finite p-group P and let P′ be a strongly F-closed subgroup of P. For any subgroup Q of P there is an isomorphism Q ≅ R in F such that both R and R ∩ P′ are fully F-normalised.

Proof. Since every subgroup of P is isomorphic to a fully F-normalised subgroup of P we may assume that Q is fully F-normalised. Then there is an isomorphism ψ : Q∩P′ → T in F such that T is fully F-normalised, and by Proposition 3, we may choose ψ such that N_{ψ} = N_{P} (Q ∩ P′). Then ψ extends to a morphism φ : N_{P} (Q ∩ P′) → P. Clearly Q ⊆ N_{P} (Q) ⊆ N_{P} (Q ∩ P′). Set R = φ(Q). Since φ maps N_{P} (Q) to N_{P} (R) and Q is fully F-normalised, R is fully F-normalised, too, and since P′ is strongly F-closed we have R∩P′ = φ(Q)∩P′ = φ(Q∩P′) = ψ(Q∩P′) = T, which is also fully F-normalised. □

The following result generalises [2, 7.2] to arbitrary fusion systems and is the main tool in the proof of Theorem 1.

Proposition 10. Let F be a fusion system on a finite p-group P, let P′ be a strongly F-closed subgroup of P and let F′ be a fusion system on P′ contained in F. Let Q be a subgroup of P such that both Q, Q ∩ P′ are fully F-normalised. Suppose that Q is F-radical centric. Then Q ∩ P′ is F′-centric. If in addition F′ is normal in F, then Q ∩ P′ is F′-radical centric and fully F′-normalised.

Proof. Any isomorphism ψ : S → Q ∩ P′ in F extends to a morphism C_{P} (S)S → C_{P} (Q ∩ P′)(Q ∩ P′) because Q ∩ P′ is fully F-centralised. Any such morphism sends C_{P′}(S) to C_{P′}(Q ∩ P′) because P′ is strongly F-closed. Thus we have |C_{P′}(S)| ≤ |C_{P′}(Q ∩ P′)|, and therefore, in order to show that Q ∩ P′ is F′-centric, it suffices to show that C_{P′}(Q ∩ P′) ⊆ Q ∩ P′.

Let Y be the subset of all φ ∈ Aut_{F}(Q) satisfying [Q, φ] ⊆ Q ∩ P′, where [Q, φ] is as usual the subgroup generated by the set of elements of the form u^{−1}φ(u), with u running over the set of elements of Q.

One easily checks that Y is in fact a subgroup of Aut_{F}(Q). Since P′ is strongly F-closed, every automorphism of Q in F restricts to an automorphism of Q ∩ P′ in F. Denote by ρ : Aut_{F}(Q) → Aut_{F}(Q∩P′) the group homomorphism sending φ ∈ Aut_{F}(Q) to its restriction to Q ∩ P′. Set X = Y ∩ ker(ρ); that is, X consists of all φ ∈ Aut_{F}(Q) such that [Q, φ] ⊆ Q ∩ P′ and such that φ|_{Q∩P′} = Id_{Q∩P′}. Both X, Y are normal subgroups of Aut_{F}(Q). Moreover, since any φ ∈ X restricts to the identity on Q ∩ P′, we have [[Q, φ], φ] = 1. Thus, if we write φ(u) = uτ(u) for u ∈ Q and some τ(u) ∈ Q∩P′, we get φ^{n}(u) = uτ(u)^{n} for any positive integer n, and hence the order of φ is a power of p. It follows that X is a p-group.

Clearly X contains all automorphisms of Q induced by conjugation with elements in C_{P′}(Q ∩ P′) ∩ N_{P} (Q). As Q is F-radical centric, we have X ⊆ Aut_{Q}(Q), and hence C_{P′}(Q ∩ P′) ∩ N_{P} (Q) ⊆ Q. This forces C_{P′}(Q ∩ P′) ⊆ Q, or equivalently, C_{P′}(Q ∩ P′) = Z(Q ∩ P′). This proves that Q ∩ P′ is F′-centric.

Assume now that F′ is normal in F. Then Q∩P′ is fully F′-normalised by Proposition 3. It remains to show that Q ∩ P′ is F′-radical. Let S be the subgroup of N_{P′}(Q ∩ P′) containing Q ∩ P′ such that O_{p}(Aut_{F′}(Q ∩ P′)) = Aut_{S}(Q ∩ P′) (this makes sense as Q ∩ P′ is fully F-normalised and hence fully F′-normalised by Proposition 1). This is a characteristic p-subgroup of Aut_{F′}(Q∩P′), hence a normal p-subgroup of Aut_{F}(Q∩P′). Note that in particular Q normalises S; thus QS is a subgroup of P. If Q∩P′ is a proper subgroup of S then in particular S is not contained in Q and hence Q is a proper subgroup of QS, thus a proper subgroup of N_{QS}(Q). But the image of N_{QS}(Q) in Aut_{F}(Q) is then a non-trivial normal p-subgroup of Aut_{F}(Q) because it is the inverse image of the normal p-subgroup Aut_{S}(Q∩P′) under the canonical restriction map ρ. This contradicts however the hypothesis that Q is F-radical. □

Proof of Theorem 1. Let α ∈ Aut_{F}(P′). Since F′ is normal in F we have F′ = N_{F′}(Q′) = N_{F′}(α(Q′)). Thus in fact F′ = N_{F′}(Q), and in particular, given an F′-centric radical subgroup S of P′, we have Q ⊆ S by A.9. Futhermore, given φ ∈ Aut_{F}(S), by Corollary 6, there are α ∈ Aut_{F}(P′) and ψ ∈ Aut_{F′}(S) such that ψ ∘ φ = α|_{S}. Note that by the construction of Q we have α(Q) = Q. Also, since F′ = N_{F′}(Q) we have ψ(Q) = Q and hence φ(Q) = Q. Thus φ belongs to N_{F}(Q). Let R be an F-radical centric subgroup of P. After possibly replacing R by a subgroup isomorphic to R in F we may assume that R, R ∩ P′ are fully F-normalised by Proposition 9. Then R ∩ P′ is F′-centric radical by Proposition 10. In particular, Q ⊆ R by Lemma A.9. Thus any automorphism of R in F restricts to an automorphism of the F′-centric radical subgroup S = R ∩ P′ of P′, and hence belongs to N_{F}(Q) by the first part of the proof. Alperin’s fusion theorem implies that F = N_{F}(Q). □

Proof of Corollary 2. Suppose that F_{Q}(Q) ⊴ F. In particular Q is strongly F-closed in P. Then F = N_{F}(Q) by Theorem 1. The converse is trivial. □

Appendix: Normal and simple fusion systems

Let p be a prime and let P be a finite p-group.

Definition A.1. A category on P is a category F whose objects are the subgroups of P and whose morphism sets Hom_{F}(Q, R) are sets of injective group homomorphisms from Q to R, where Q, R are subgroups of P, with the following properties:

• the composition of morphisms in F is given by the usual composition of group homomorphisms;

• if Q ⊆ R then the inclusion map from Q to R is a morphism in F;

• if φ : Q → R is a morphism in F then so is the induced isomorphism Q ≅ φ(Q) and its inverse.

If Q, R are subgroups of P we denote by Hom_{P} (Q, R) the set of group homomorphisms from Q to R induced by conjugation with elements in P, and we write Aut_{P} (Q) = Hom_{P} (Q, Q).

Definition A.2. Let F be a category on P and let Q be a subgroup of P. We say that

- Q is fully F-normalised if |N_{P} (Q)| ≥ |N_{P} (Q′)| for any subgroup Q′ of P such that there is an isomorphism Q′ ≅ Q in F;

- Q is fully F-centralised if |C_{P} (Q)| ≥ |C_{P} (Q′)| for any subgroup Q′ of P such that there is an isomorphism Q′ ≅ Q in F;

- Q is F-centric if C_{P} (Q′) = Z(Q′) for any subgroup Q′ of P such that there is an isomorphism Q′ ≅ Q in F;

- Q is F-radical if O_{p}(Aut_{F}(Q)) ⊆ Aut_{Q}(Q);

- Q is weakly F-closed if for every morphism φ : Q → P we have φ(Q) = Q;

- Q is strongly F-closed if for every morphism φ : R → P we have φ(R∩Q) = φ(R)∩Q;

Definition A.3. (cf. [1, 1.2]) Let F be a category on P, let Q be a subgroup of P and let φ : Q → P be a morphism in F. We denote by N_{φ} the subgroup of all elements y in N_{Q}(P) for which there exists an element z ∈ N_{P} (φ(Q)) such that φ(yuy^{−1}) = zφ(u)z^{−1} for all u ∈ Q. Note that QC_{P} (Q) ⊆ N_{φ} ⊆ N_{P} (Q).

Definition A.4. ([3, 3.1]) Let F be a category on P, let P′ be a subgroup of P and let F′ be a category on P′. We say that F normalises F′ or that F′ is normalised by F if P′ is strongly F-closed in P and if for any isomorphism φ : Q → Q′ in F and any two subgroups R, R′ of P′ ∩Q we have φ∘Hom_{F′}(R, R′)∘φ^{−1}|_{φ}(R) = Hom_{F′}(φ(R), φ(R′)). We say that F′ is normal in F and write F′ ⊴ F if F′ is a subcategory of F and if F normalises F′.

Definition A.5. A category F on a finite p-group P is called a fusion system if Hom_{P} (Q, R) ⊆ Hom_{F}(Q, R) for all subgroups Q, R of P and if the following properties hold:

(I-S) Aut_{P} (P) is a Sylow-p-subgroup of Aut_{F}(P);

(II-S) every morphism φ : Q → P such that φ(Q) is fully F-normalised extends to a morphism ψ : N_{φ} → P in F (that is, ψ|_{Q} = φ).

The concept of a fusion system is due to L. Puig [4]; the above definition appears in Stancu [5] and is, by [5] or [3] equivalent to the definition given in [1, 1.2] where the two axioms (I-S) and (II-S) are replaced by

(I-BLO) if Q is a fully F-normalised subgroup of P then Q is fully F-centralised and Aut_{P} (Q) is a Sylow-p-subgroup of Aut_{F}(Q); (II-BLO) every morphism φ : Q → P such that φ(Q) is fully F-centralised extends to a morphism ψ : N_{φ} → P in F (that is, ψ|_{Q} = φ).

If G is a finite group and P a Sylow-p-subgroup of G we denote by F_{P} (G) the category on P whose morphism sets are the group homomorphisms induced by conjugation with elements in G; that is, Hom_{F_{P} (G)}(Q, R) = Hom_{G}(Q, R) for any two subgroups Q, R of P. It is well-known and easy to verify that F_{P} (G) is a fusion system on P. In particular, F_{P} (P) is the fusion system on P consisting precisely of all morphisms given by conjugation with elements in P. For any fusion system F on P we have F_{P} (P) ⊆ F and F_{1}(1) ⊴ F.

Definition A.6. ([3, 4.1]) A fusion system F on a finite p-group P is called simple if P ≠ 1 and if F has no proper non trivial normal fusion subsystem.

Definition A.7. Let F be a category on P and let Q be a subgroup of Q. We denote by N_{F}(Q) the category on N_{P} (Q) whose objects are the subgroups of N_{P} (Q) and whose morphisms from a subgroup R of N_{P} (Q) to a subgroup Q of N_{P} (Q) are all morphisms φ : R → S in F for which there exists a morphism ψ : RQ → SQ in F satisfying ψ|_{R} = φ and ψ(Q) = Q.

The following result is due to Puig; see [1, A6] for a proof.

Proposition A.8. Let F be a fusion system on P and let Q be a subgroup of P. If Q is fully F-normalised then N_{F}(Q) is a fusion system on N_{P} (Q).

We need the following well-known fact:

Lemma A.9. Let F be a fusion system on P and let Q be a subgroup of G such that F = N_{F}(Q). Then Q is contained in any F-centric radical subgroup R of P.

Proof. Since F = N_{F}(Q), the image Aut_{Q}(R) of Q in Aut_{F}(R) is normal in Aut_{F}(R). Since R is F-radical this implies that Aut_{Q}(R) ⊆ Aut_{R}(R) = O_{p}(Aut_{F}(R). Thus Q ⊆ RC_{P} (R) = R, where the last equation uses the fact that R is F-centric. □

References

1. C. Broto, R. Levi, R. Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), 779–856.

2. R. Kessar, M. Linckelmann, G. R. Robinson, Local control in fusion systems of p-blocks of finite groups, J. Algebra 257 (2002), 393–413.

3. M. Linckelmann, Simple fusion systems and the Solomon 2-local groups, J. Algebra 296 (2006), 385–401.

4. L. Puig, Unpublished notes.

5. R. Stancu, Equivalent definitions of fusion systems, preprint (2003).

6. R. Stancu, Control of fusion in fusion systems, preprint (2004).

Markus Linckelmann 130 Math Tower 231 West 18th Avenue Columbus, Ohio 43210

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