ANALOGUES OF GOLDSCHMIDT’S THESIS FOR FUSION SYSTEMS

JUSTIN LYND AND SEJONG PARK

Abstract. We extend the results of David Goldschmidt’s thesis concerning fusion in finite groups to saturated fusion systems and to all primes.

1. Introduction

Just recently, David Goldschmidt published his doctoral thesis [6] which had gone unpublished since 1968. In it he shows that if G is a finite simple group and T is a Sylow 2-subgroup of G, then the exponent of Z(T) (and hence of T) is bounded by a function of the nilpotence class of T. He also includes in the write-up a fusion factorization result for an arbitrary finite group involving ℧^{1}Z and the Thompson subgroup. In this paper, we generalize these results to arbitrary saturated fusion systems.

Unless otherwise indicated, throughout this paper p denotes an arbitrary prime number, n a nonnegative integer, and P a nontrivial finite p-group.

Theorem 1. Suppose P is of nilpotence class at most n(p − 1) + 1 and F is a saturated fusion system on P with O_{p}(F) = 1. Then Z(P) has exponent at most p^{n}.

This bound is sharp for all n and p; see Example 1 in Section 3. This also gives a bound on the exponent of P itself, which we certainly do not expect to be sharp.

Corollary 1. Suppose P is of nilpotence class at most n(p − 1) + 1 and F is a saturated fusion system on P with O_{p}(F) = 1. Then P has exponent at most p^{n^{2}(p−1)+n}.

Proof. By Theorem 1, Z(P) has exponent at most p^{n}. We claim that then every upper central quotient also has exponent at most p^{n}, and the proof is by induction. Let k ⩾ 1, and let x ∈ Z^{k+1}(P). If x^{p^{n}} does not lie in Z^{k}(P), then there exists t ∈ P such that [x^{p^{n}}, t] does not lie in Z^{k−1}(P). But by a standard commutator identity, [x^{p^{n}}, t] ≡ [x, t]^{p^{n}} ≡ 1 modulo Z^{k−1}(P), since by induction Z^{k}(P)/Z^{k−1}(P) has exponent at most p^{n}. This contradiction establishes the claim. The nilpotence class of P is at most n(p − 1) + 1 by hypothesis, so the exponent of P is at most p^{n(n(p−1)+1)}. □

Theorem 1 follows from the following, which we prove as Theorem 5 below.

Theorem 2. Suppose P has nilpotence class at most n(p − 1) + 1 and F is a saturated fusion system on P. Then ℧^{n}(Z(P)) is normal in F.

In the course of proving this last result in the group case for p = 2, Goldschmidt reduces to the situation in which a putative counterexample G has a weakly embedded 2-local subgroup. Then his post-thesis classification [5] of such groups gives a contradiction. However, any weakly embedded 2-local M controls 2-fusion, and so the 2-subgroup O_{2}(M) will show up as a normal subgroup in the fusion system, a shadow of the weakly embedded phenomenon. This allows the corresponding fusion result to hold for an arbitrary prime.

Date: May 11, 2009.

We note that Theorem 2 has the following corollary in the category of groups.

Theorem 3. Let P be a nonabelian Sylow p-subgroup of a finite group G. Suppose that P has nilpotence class at most n(p − 1) + 1 and that G has no nontrivial strongly closed abelian p-subgroup. Then Z(P) has exponent at most p^{n}.

Proof. We can form the saturated fusion system F_{P} (G), and Theorem 2 then says that ℧^{n}(Z(P)) is strongly F-closed (see Proposition 1 below), that is, strongly closed in P with respect to G. Thus, ℧^{n}(Z(P)) must be trivial. □

Using a recent theorem of Flores and Foote [4], in which they apply the Classification of Finite Simple Groups to describe all finite groups having a strongly closed p-subgroup, we get the following direct generalization of Goldschmidt’s main theorem.

Corollary 2. Let P be a nonabelian Sylow p-subgroup of a finite simple group G. If P has nilpotence class at most n(p − 1) + 1, then Z(P) has exponent at most p^{n}.

Proof. Suppose to the contrary that A := ℧^{n}(Z(P)) ≠ 1. Then by Theorem 2, A is a nontrivial strongly closed abelian subgroup of P. By inspection of the simple groups arising in the conclusion of the main theorem in [4], either P is abelian or Z(P) has exponent p. Since P is nonabelian, we must have n ⩾ 1 and the corollary follows. □

However, if the hypotheses of Corollary 2 are weakened slightly to assume only that F^{∗}(G) is simple, then the statement is false for all odd primes p, as the following example shows. Let H = PSL(2, q) with q = r^{p} for some prime power r and with the p-part of q − 1 equal to p^{e}. Let σ be a field automorphism of F_{q} of order p and G = H⟨σ⟩. If P is a Sylow p-subgroup of G, then P has nilpotence class 2, while Z(P) has exponent p^{e−1}, and we may take e as large as we like.

Recall the Thompson subgroup J(P), defined as the group generated by the abelian subgroups of P of maximum order. We also prove the following

Theorem 4. Let F be a saturated fusion system on P. Then

F = ⟨ C_{F}(℧^{1}(Z(P)), N_{F}(J(P)) ⟩.

2. Definitions and notation

We collect in this section the necessary information on fusion systems. Since there are by now many good sources of this knowledge [2], in particular in background sections of papers [3,7] to which this one is similar, we will content ourselves to be brief.

Let P be a finite p-group. A category on P is a category F with objects the subgroups of P and whose morphism sets Hom_{F}(Q, R) consist of injective group homomorphisms subject to the requirement that every morphism in F is a composition of an isomorphism in F and an inclusion.

Let F be a category on the p-group P. Let Q and R be subgroups of P. We write Aut_{F}(Q) for Hom_{F}(Q, Q), Hom_{P} (Q, R) for the set of group homomorphisms in F from Q to R induced by conjugation by elements of P, and Out_{F}(Q) for Aut_{F}(Q)/ Aut_{Q}(Q).

We say Q is

• fully F-normalized if |N_{P} (Q)| ⩾ |N_{P} (Q′)| for all Q′ which are F-isomorphic to Q,

• fully F-centralized if |C_{P} (Q)| ⩾ |C_{P} (Q′)| for all Q′ which are F-isomorphic to Q,

• F-centric if C_{P} (Q′) ⩽ Q′ for all Q′ which are F-isomorphic to Q, and

• F-radical if O_{p}(Out_{F}(Q)) = 1.

For a morphism φ : Q → P in F, let

N_{φ} = { x ∈ N_{P} (Q) | ∃y ∈ N_{P} (φ(Q)), ∀z ∈ Q, φ(xzx^{−1}) = yφ(z)y^{−1}}

Note that we have QC_{P} (Q) ⩽ N_{φ} for all φ : Q → P in F.

A saturated fusion system on P is a category F on P whose morphism sets contain all group homomorphisms induced by conjugation by elements of P, and which satisfies the following two axioms.

• (Sylow axiom) Aut_{P} (P) is a Sylow p-subgroup of Aut_{F}(P), and

• (Extension axiom) for every isomorphism φ : Q → Q′ with Q′ fully F-normalized, there exists a morphism φ̃ : N_{φ} → P such that φ̃ |_{Q} = φ.

For the remainder of the paper, F will denote a saturated fusion system on the finite p-group P, even though we will often drop the adjective “saturated”.

For Q ⩽ P, we define the following local subcategories of F. The normalizer N_{F}(Q) of Q in F is the category on N_{P} (Q) such that for any R_{1}, R_{2} ⩽ N_{P} (Q), Hom_{N_{F} (Q)}(R_{1}, R_{2}) consists of those φ : R_{1} → R_{2} in F for which there is an extension φ̃ : QR_{1} → QR_{2} of φ in F such that φ̃(Q) = Q. The centralizer C_{F}(Q) of Q in F is the category on C_{P} (Q) such that for any R_{1}, R_{2} ⩽ C_{P} (Q), Hom_{C_{F} (Q)}(R_{1}, R_{2}) consists of those φ : R_{1} → R_{2} in F for which there is an extension φ̃ : QR_{1} → QR_{2} of φ in F such that φ̃ |_{Q} = id_{Q}. Lastly, we define N_{P} (Q)C_{F}(Q) as we do the normalizer of Q, but only allow those φ : R_{1} → R_{2} whose extensions φ̃ restrict to automorphisms in Aut_{P} (Q).

If Q is fully F-normalized, then N_{F}(Q) is a saturated fusion system. And if Q is fully F-centralized, then both C_{F}(Q) and N_{P} (Q)C_{F}(Q)) are saturated fusion systems.

A characteristic functor is a mapping from finite p-groups to finite p-groups which takes Q to a characteristic subgroup W (Q) of Q such that for any group isomorphism φ : Q → Q′, φ(W (Q)) = W (Q′). We say that a characteristic functor is positive provided W (Q) ≠ 1 whenever Q ≠ 1. The center functor, sending a finite p-group P to its center, is a positive characteristic p-functor.

A conjugation family for F is a set C of nonidentity subgroups of P such that F is generated by compositions and restrictions of morphisms in Aut_{F}(Q) as Q ranges over C. Alperin’s fusion theorem for saturated fusion systems says that the set of F-centric, F-radical subgroups is a conjugation family for F, and we call this the Alperin conjugation family.

Recall that a subgroup W of P is said to be weakly F-closed if for each φ ∈ Hom_{F}(W, P), φ(W) = W. The subgroup W is strongly F-closed if for each subgroup W′ of W and each φ ∈ Hom_{F}(W′, P), φ(W′) ⩽ W. We say W is normal in F if F = N_{F}(W), and denote by O_{p}(F) the largest such subgroup of P.

3. Proofs

The following proposition is slightly misstated in [1, Proposition 1.6], where a normal W is claimed to be contained in every radical subgroup. For this reason, we state a correct version here, but the proof in [1] goes through with little modification.

Proposition 1. Let F be a fusion system on P and W ⩽ P. The following are equivalent. (a) W is normal in F.

(b) W is strongly F-closed and is contained in every F-centric, F-radical subgroup of P.

(c) W is weakly F-closed and is contained in every subgroup of some conjugation family for F.

Lemma 1. Suppose P has nilpotence class at most n(p − 1) + 1. If Q is a subgroup of P with C_{P} (℧^{n}(Z(Q))) = Q, then Q = P.

Proof. This is Corollary 6 in [6]. □

Proposition 2. Let W be a characteristic subfunctor of the center functor such that W (P) ⩽ W (Q) for all Q ⩽ P with C_{P} (Q) ⩽ Q. Then for any fusion system F on P, either there exists a proper F-centric subgroup Q of P such that C_{P} (W (Q)) = Q, or W (P) is normal in F.

Proof. Suppose there is no proper F-centric subgroup Q of P with C_{P} (W (Q)) = Q. We will show that W (P) is weakly closed in F. In this case, W (P) ⩽ Z(P) is contained in every F-centric subgroup of P, hence in every member of an Alperin conjugation family for F. Thus, by Proposition 1, W (P) is in fact normal in F.

Let Q be a fully F-normalized, F-centric subgroup of P. Then by hypothesis, W (P) ⩽ W (Q). Let α ∈ Aut_{F}(Q). By Alperin’s fusion theorem, it suffices to show that W (P) is invariant under α. We do this by induction on |P : Q|. If Q = P, then α(W (P)) = W (P) since W (P) is a characteristic subgroup of P, so suppose that Q < P. Then C_{P} (W (Q)) > Q. Let β : W (Q) → R be an isomorphism in F with R fully F-normalized. Then by the extension axiom, β extends to a map β̃ : C_{P} (W (Q)) → P. By induction and Alperin’s fusion theorem, we have that β(W (P)) = β̃(W (P)) = W (P). But βα|_{W (Q)} also extends to C_{P} (W (Q)), and βα(W (P)) = W (P) by the same reasoning. Therefore α(W (P)) = β^{−1}βα(W (P)) = W (P), and this completes the proof. □

We are now ready to prove Theorem 2.

Theorem 5. Suppose P has nilpotence class at most n(p − 1) + 1 and F is a fusion system on P. Then ℧^{n}(Z(P)) is normal in F.

Proof. Let W = ℧^{n}Z. If C_{P} (Q) ⩽ Q ⩽ P, then Z(P) ⩽ Z(Q) and so W (P) = ℧^{n}(Z(P)) ⩽ ℧^{n}(Z(Q)) = W (Q). Thus W satisfies the hypotheses of Proposition 2, and Lemma 1 says that there is no proper subgroup of P with C_{P} (W (Q)) = Q. Therefore by Proposition 2, ℧^{n}(Z(P)) is normal in F. □

Theorem 1 now follows immediately from Theorem 2. The following example generalizes a remark of Goldschmidt’s in [6], and shows that the bound on the exponent of Z(P) given in Theorem 1 is sharp.

Example 1. Let p be an odd prime, let G = SL(p + 1, q) with |q − 1|_{p} = p^{n}, and let P be a Sylow p-subgroup of G. Then P is a isomorphic to C_{p^{n}} ≀ C_{p}. Let x be the wreathing element, a p-cycle permutation matrix, generating the C_{p} on top. Then P′ = [P, P] is isomorphic to p − 1 copies of C_{p^{n}}. Let P_{0} = ⟨P′, x⟩. As Z(P) has exponent p^{n}, the bound in Theorem 1 is sharp provided the class of P is n(p − 1) + 1. For this it suffices to show that P_{0} has class n(p − 1), that is, P_{0} is of maximal class.

By an inductive argument, we quickly reduce to the case where n = 2. Suppose n = 2 and let a_{1}, …, a_{p−1} be generators for the p − 1 cyclic groups of order p^{2}. Then x sends a_{i} to a_{i+1} for 1 ⩽ i ⩽ p − 2 and a_{p−1} to a_{1}^{−1} ⋯ a_{p−1}^{−1}. Factoring by Ω_{1}(P′) we have that [P′/Ω_{1}(P′), x; p − 1] = 1 so that [P′, x; p − 1] ⩽ Ω_{1}(P′). By direct computation,

[a_{1}, x; p − 1] = ∏_{k=0}^{p−2} a_{k+1}^{(−1)^{k}\binom{p−1}{k}−1}.

The sum of the exponents of the a_{i} in [a_{1}, x; p − 1] is

−p + 1 + ∑_{k=0}^{p−2}(−1)^{k}\binom{p − 1}{k} = −p + 1 + (1 − 1)^{p−1} − \binom{p − 1}{p − 1} = −p.

This means that [a_{1}, x; p − 1] lies outside the sum-zero submodule (which is the unique maximal submodule) for the action of x on Ω_{1}(P′), and so [P′, x; p−1] = Ω_{1}(P′). It follows that P_{0} has class 2(p − 1), as claimed.

Therefore P has class n(p − 1) + 1 while Z(P) has exponent p^{n}, and so the bound of Theorem 1 is sharp.

We now turn to the proof of Theorem 4. We will need a version of the Frattini argument due to Onofrei and Stancu [8, Proposition 3.7].

Proposition 3. Let F be a fusion system on P and suppose Q ⩽ P is normal in F. Then

F = ⟨ P C_{F}(Q), N_{F}(QC_{P} (Q)) ⟩.

Lemma 2. Suppose P is a p-group, Q ⊴ P, and C_{P} (℧^{1}(Z(Q))) = Q. Then J(P) ⩽ Q.

Proof. This is Lemma 8 in [6]. □

The Thompson ordering on subgroups of P is defined by

Q ⩽_{P} Q′ iff |N_{P} (Q)| ⩽ |N_{P} (Q′)| or |N_{P} (Q)| = |N_{P} (Q′)| and |Q| ⩽ |Q′|.

We are now ready to prove

Theorem 6. Let F be a fusion system on P. Then

F = ⟨ C_{F}(℧^{1}(Z(P)), N_{F}(J(P)) ⟩.

Proof. Write F′ = ⟨ C_{F}(℧^{1}(Z(P))), N_{F}(J(P)) ⟩. Since each F-centric subgroup of P contains Z(P), it suffices by Alperin’s fusion theorem to prove that N_{F}(Q) ⊆ F′ for all Q ⩽ P with Z(P) ⩽ Q. We do this by induction on the Thompson ordering. If Q = P, then N_{F}(Q) ⊆ N_{F}(J(P)) ⊆ F′, since J(P) is a characteristic subgroup of P, so suppose that Q <_{P} P with Z(P) ⩽ Q and that N_{F}(Q′) ⊆ F′ for all Q′ >_{P} Q with Z(P) ⩽ Q′.

First we reduce to the case where Q is fully F-normalized. Suppose Q is not fully F-normalized. By [7, Lemma 2.2], there exists α : N_{P} (Q) → P such that α(Q) is fully F-normalized. Note that α(Q) >_{P} Q, and since R >_{P} Q for every R ⩽ P with |N_{P} (Q)| ⩽ |R|, we have by induction and Alperin’s fusion theorem that α is in F′. Also note that α(N_{P} (Q)) ⩽ N_{P} (α(Q)); we still denote by α the induced morphism N_{P} (Q) → N_{P} (α(Q)). Let φ : R_{1} → R_{2} be a morphism in N_{F}(Q), and let φ̃ be an extension to QR_{1} ⩽ N_{P} (Q). Then αφ̃α^{−1} : α(Q)α(R_{1}) → α(Q)α(R_{2}) restricts to an automorphism of α(Q), whence is contained in F′ by induction. But α is in F′, so φ is in F′ too. Thus N_{F}(Q) ⊆ F′, so henceforth we assume Q is fully F-normalized.

For brevity, set W = ℧^{1}(Z(Q)), N = N_{P} (Q), and C = C_{N}(W). Then C ⊴ N, so that N_{P} (C) ⩾ N. Suppose first that C = Q. Then by Lemma 2, we have J(N) ⩽ Q. As J(N) ⊴ N_{P} (N), either J(N) >_{P} Q or N = P. In the first case, since Z(P) ⩽ J(N) and J(N) = J(Q) is a characteristic subgroup of Q, we apply induction to get N_{F}(Q) ⊆ N_{F}(J(N)) ⊆ F′. In the second case we have J(P) ⩽ Q, so J(P) = J(Q), and hence N_{F}(Q) ⊆ N_{F}(J(P)) ⊆ F′ here as well.

Assume now that C > Q. Then C >_{P} Q because C ⊴ N. Looking to see that W ⊴ N_{F}(Q), we apply Proposition 3 in this normalizer to get

N_{F}(Q) = ⟨ NC_{N_{F} (Q)}(W), N_{N_{F} (Q)}(C) ⟩.

Since C contains Z(P), we have by induction that N_{N_{F} (Q)}(C) ⊆ N_{F}(C) ⊆ F′, so to complete the proof, it suffices to show that NC_{N_{F} (Q)}(W) ⊆ C_{F}(℧^{1}(Z(P))). To see this, let R_{1}, R_{2} ⩽ N, and let φ : R_{1} → R_{2} be a morphism in NC_{N_{F} (Q)}(W). Then there exists x ∈ N such that φ extends to an F-map φ̃ : W R_{1} → W R_{2} with φ̃ |_{W} = c_{x}, the conjugation map induced by x. But since Q contains Z(P), it follows that W = ℧^{1}(Z(Q)) ⩾ ℧^{1}(Z(P)), and so φ̃ |_{℧^{1}(Z(P))} = c_{x} |_{℧^{1}(Z(P))} = id_{℧^{1}(Z(P))}. Therefore, φ ∈ C_{F}(℧^{1}(Z(P))), as was to be shown. We conclude that N_{F}(Q) ⊆ F′ and the result follows. □

Remark 1. In [3, Theorem 4.1], the authors prove in part that for any fusion system F on P, ℧^{1}(Z(P)) ∩ Z(N_{F}(J(P))) ⩽ Z(F) by reducing to the group case. Theorem 4 gives a reduction-free proof of this fact.

4. Acknowledgements

We would like to thank Ron Solomon for encouraging us to take up this work.

References

1. Carles Broto, Natàlia Castellana, Jesper Grodal, Ran Levi, and Bob Oliver, Subgroup families controlling p-local finite groups, Proc. London Math. Soc. (3) 91 (2005), no. 2, 325–354.

2. Carles Broto, Ran Levi, and Bob Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), no. 4, 779–856 (electronic).

3. Antonio Díaz, Adam Glesser, Nadia Mazza, and Sejong Park, Glauberman and Thomson’s theorems for fusion systems, Proc. Amer. Math. Soc. 137 (2009), 495–503.

4. Ramón Flores and Richard Foote, Strongly closed subgroups of finite groups, preprint (2009).

5. David M. Goldschmidt, Weakly embedded 2-local subgroups of finite groups, J. Algebra 21 (1972), 341– 351.

6. , On the 2-exponent of a finite group, J. Algebra 319 (2008), 616–620.

7. Radha Kessar and Markus Linckelmann, ZJ-theorems for fusion systems, Trans. Amer. Math. Soc. 360 (2008), 3093–3206.

8. Silvia Onofrei and Radu Stancu, A characteristic subgroup for fusion systems, preprint (2008).

Department of Mathematics, The Ohio State University, Columbus, OH 43210

  E-mail address: jlynd@math.ohio-state.edu

Department of Mathematical Sciences, University of Aberdeen, Aberdeen, UK AB24 3UE

  E-mail address: s.park@abdn.ac.uk