FUSION SYSTEMS FOR PROFINITE GROUPS

RADU STANCU AND PETER SYMONDS

Abstract. We introduce the notion of a pro-fusion system on a pro-p group, which generalizes the notion of a fusion system on a finite p-group. We also prove a version of Alperin’s Fusion Theorem for pro-fusion systems.

1. Introduction

Profinite groups have a good theory of Sylow pro-p subgroups, so they form an obvious candidate for a generalization of fusion theory in finite groups. See for example the work of Gilotti, Ribes and Serena [9] on fusion and transfer in the context of profinite groups.

In the present work we develop a theory of fusion systems for profinite groups. We define a pro-fusion system F on a pro-p group S as an inverse limit of fusion systems on finite p-groupe (see Definition 2.7) and we study morphisms between them and also their quotients. When the fusion systems in the inverse limit are all saturated we call F a pro-saturated fusion system.

If F is a pro-saturated fusion system on a pro-p group S, then analogously to the case of profinite groups, F is an inverse limit of its quotients by open, strongly closed subgroups of S (Proposition 4.5). We also define saturation of F in terms of properties familiar from the finite case and show that these are automatically satisfied when F is pro-saturated and S is countably based (Theorem 5.2).

The main results of this work are concerned with a version of Alperin’s Fusion Theorem for pro-saturated fusion systems. We generalize the notion of an invariant subsystem on a strongly closed subgroup T of S, by extending the sets of objects to all subgroups of S; we call it a T - subsystem (Definition 6.1). A saturated T -subsystem of a pro-fusion system attempts to mimic a normal subgroup of a profinite group. We show that a morphism in a saturated T -subsystem E between two open subgroups of S can be expressed as a composition of a chain of restrictions of automorphisms in E (Theorem 6.9). This relative form of Alperin’s Theorem appears to be new even in the case of finite groups. The result for a morphism between closed subgroups is more complicated and we only prove it with additional hypotheses (Theorem 6.17). Moreover, the resulting chain of morphisms need not be finite, although it converges in the profinite topology. In the last section we give an example of a profinite group and a morphism in its associated pro-fusion system for which the length of such an Alperin chain must be infinite.

Date: April 11, 2012.

2. Terminology and basic properties

2.1. When we deal with profinite groups we always work in the category of profinite groups, so all subgroups are closed and all homomorphisms are continuous. For more information on profinite groups see, for example, [13, 15, 17]

There is a good theory of Sylow pro-p subgroups. However, there are some differences from the case of finite p-groups. One is that a proper subgroup of a pro-p group can be equal to its normalizer; the normalizer only has to be larger when the subgroup is open. Another difference is that a group can be isomorphic to one of its proper subgroups.

One result that does carry over, and which we shall use frequently, is the following: if P is a pro-p group, Q an open subgroup and N a normal subgroup such that N_{N}(Q) ≤ Q, then N ≤ Q. 2.2. Let p be a prime and S a finite p-group. A fusion system on S is a category F with objects the set of subgroups of S and morphisms which are injective group homomorphisms subject to two conditions:

(a) F contains every group homomorphism induced by conjugation by an element of S, and (b) any morphism in F factors as an isomorphism in F followed by an inclusion. A fusion system F is called saturated if it satisfies a certain extra condition based on Sylow theory (see Definition 2.16 below). For more information on fusion systems see, for example, [5, 8, 10].

Definition 2.3. Let F be a fusion system on S and G be a fusion system on T. A morphism of fusion systems between F and G is a group homomorphism α : S → T such that there exists a functor A : F → G with the property that:

(a) A(P) = α(P), ∀P ≤ S;

(b) for all φ ∈ Hom_{F}(P, Q), αφ = A(φ)α.

Notice that A is uniquely determined by α, provided that it exists. Thus a morphism of fusion systems is just a group homomorphism between the underlying p-groups that satisfies certain conditions, to wit that for each φ ∈ Hom_{F}(P, G) we have φ(P ∩ N) ≤ Q ∩ N and the induced group homomorphism \overline{φ}: P/P ∩ N → Q/Q ∩ N is, in fact, a morphism of G. We will refer to a morphism as (α, A), α or A, as convenient. 2.4. Morphisms of fusion systems can be composed in the obvious way. It is easy to see that Ker(α) is a strongly F-closed subgroup of F. Recall that Q is a strongly F-closed subgroup of S if, for all φ ∈ Hom_{F}(R, S), R ≤ Q implies φ(R) ≤ Q. Let A(F) be the collection of objects and morphisms of G that are in the image under α of objects and morphisms in F. In general, A(F) need not be a fusion system, because it may not be possible to compose morphisms. We can remedy this by defining ⟨A(F)⟩ to be the fusion system on α(S) generated by A(F). There is then a morphism of fusion systems F → ⟨A(F)⟩.

When Q is a strongly F-closed subgroup of S, F/Q is defined to be the fusion system on S/Q with morphisms all group homomorphisms φ satisfying φ ∈ Hom_{F/Q}(P Q/Q, S/Q) if and only if there exist φ̃ ∈ Hom_{F}(P Q, S) such that φ(uQ) = φ̃(u)Q for all u ∈ P. However, there might not be a morphism from F to F/Q. Define \overline{F}_{Q} to be the set of homomorphisms of subgroups of S/Q induced by morphisms in F, and let ⟨ \overline{F}_{Q}⟩ be the fusion system that it generates. There are then clearly morphisms F → ⟨\overline{F}_{Q}⟩ and F/Q → ⟨\overline{F}_{Q}⟩. If the latter is an isomorphism, there is a morphism F → F/Q and we call F/Q a quotient of F. In the saturated case all of this is well behaved; the following results are due to Puig.

Theorem 2.5 ([10]). Let F be a saturated fusion system on a finite p-group S and let N ≤ S be strongly closed. For any φ ∈ Hom_{F}(P, Q) there exists φ̃ ∈ Hom_{F}(P N, QN) such that φ̃ and φ induce the same homomorphism P N/N → QN/N.

Corollary 2.6 ([10]). Let F be a saturated fusion system on S.

(a) Let (α, A) : F → G be a morphism of fusion systems. Then A(F) is a saturated fusion system isomorphic to F/Ker(α) and (α, A) factors as a quotient followed by an inclusion.

(b) If Q is a strongly F-closed subgroup of S then F/Q is a quotient of F (i.e. the natural map from S to S/Q induces a morphism of fusion systems). Moreover, F/Q is saturated.

There is a careful discussion of these topics in [7].

We will construct a pro-fusion system by an inverse limit process on a directed system of fusion systems on finite p-groups. Recall that a partially ordered set I is directed if for all i, j ∈ I there exists k ∈ I, with k ≥ i and k ≥ j.

Definition 2.7. Suppose that we have an inverse system of fusion systems F_{i} on finite p-groups S_{i}, indexed by a directed set I. That is, for every i, j ∈ I, j ≥ i we have morphisms (f_{i,j}, F_{i,j}) : F_{j} → F_{i} such that f_{i,j}f_{j,k} = f_{i,k} whenever k ≥ j ≥ i. Set S := \underleftarrow{lim}_{i∈I} S_{i} and let f_{i} : S → S_{i} be

the induced projections and N_{i} := Ker(f_{i}). Then S is a pro-p group, and the {N_{i} | i ∈ I} form an open sub-basis for S at 1. Define F := \underleftarrow{lim}_{i∈I} F_{i} to be the category on S with objects all the

closed subgroups of S and with morphisms given by Hom_{F}(P, Q) := \underleftarrow{lim}_{i∈I} Hom_{F_{i}}(f_{i}(P), f_{i}(Q)),

for all P, Q ≤ S. We say that F is a pro-fusion system on the pro-p group S.

Notice that Aut_{F}(P) is naturally a profinite group. There is a canonical functor F_{i} : F → F_{i} sending P ≤ S to f_{i}(P) ≤ S_{i} and φ ∈ Hom_{F}(P, Q) to its image in Hom_{F}(f_{i}(P), f_{i}(Q)).

Example 2.8. If G is a profinite group with Sylow pro-p subgroup S, then F_{S}(G), the category with the closed subgroups of S as objects and as morphisms all homomorphisms induced by conjugation by the elements of G, is a pro-fusion system. For if G = \underleftarrow{lim}_{i∈I} G/N_{i} then F_{S}(G) =

\underleftarrow{lim}_{i∈I} F_{S/S∩N_{i}}(G/N_{i}).

Lemma 2.9. Every morphism in a pro-fusion system F factorizes uniquely as an isomorphism followed by an inclusion, and every endomorphism is an isomorphism.

Proof. Given φ ∈ Hom_{F}(P, Q), each F_{i}(φ) factorizes as inc_{i}θ_{i}, where θ_{i} is an isomorphism and inc_{i} is an inclusion. Thus φ factorizes as \underleftarrow{lim}_{i∈I} inc_{i} \underleftarrow{lim}_{i∈I} θ_{i}. The inverse to \underleftarrow{lim}_{i∈I} θ_{i} is \underleftarrow{lim}_{i∈I} θ_{i}^{−1}, and

\underleftarrow{lim}_{i∈I} inc_{i} is the inclusion of φ(P) in Q.

If φ is an endomorphism then so is F_{i}(φ); thus inc_{i} is the identity map, and so is \underleftarrow{lim}_{i∈I} inc_{i}. □

Sometimes we will write φ^{−1} for the inverse of the isomorphism part of this factorization of φ.

Lemma 2.10. Let F be a pro-fusion system on S and P, Q subgroups of S. Suppose that there are morphisms φ ∈ Hom_{F}(P, Q) and ψ ∈ Hom_{F}(Q, P). Then φ and ψ are isomorphisms.

Proof. By Lemma 2.9, φψ and ψφ are both isomorphisms. Thus both φ and ψ are surjective, hence isomorphisms, by Lemma 2.9 again. □

Lemma 2.11. If P is open in S and Q is isomorphic to P in F, then Q is also open and has the same index in S as P.

Proof. Since P is open in S, it must contain some N_{i}, and Q must also contain N_{i}, because N_{i} is strongly closed. Thus P/N_{i} is isomorphic to Q/N_{i} in F_{i}. □

Lemma 2.12. If F is a pro-fusion system on S then it contains every group homomorphism induced by an element of S.

Proof. Suppose that F = \underleftarrow{lim} F_{i} and that we have P, Q ≤ S and s ∈ S such that ^{s}P ≤ Q. The conjugation homomorphism c_{s} : P → Q has image c_{f_{i}(s)} : f_{i}(P) → f_{i}(Q) between subgroups of S_{i}. Since F_{i} is a fusion system, we know that c_{f_{i}(s)} ∈ Hom_{F_{i}}(f_{i}(P), f_{i}(Q)). But F_{i,j}(c_{f_{j}(s)}) = c_{f_{i}(s)}, so we obtain an element of Hom_{F}(P, Q) that is clearly equal to c_{s}. □

Thus a pro-fusion satisfies the conditions for a fusion system in 2.2. However, these are certainly not enough to ensure a good theory.

Example 2.13. Any attempt to define a pro-fusion system on an infinite pro-p group S by setting Hom(P, Q) to be all the (continuous) injective group homomorphisms from P to Q will not succeed; in other words, the resulting category cannot be expressed as an inverse limit. For if S is not all torsion then it contains a copy of the p-adic integers, which is isomorphic to a proper subgroup of itself, contradicting Lemma 2.10; if it is torsion then it contains an infinite product of cyclic groups of order p, by [18], which has the same property.

We now define the saturation of a pro-fusion system in the same way as was done for a fusion system on a finite p-group by Roberts and Shpectorov [14]. We start with some preliminary notions. For the rest of this section F is a pro-fusion system on a pro-p group S.

Definition 2.14. We say that Q is receptive in F if for all Q ≤ S and for all φ ∈ Iso_{F}(R, Q), there exists φ̃ : N_{φ} → N_{S}(Q) in F such that φ̃|_{R} = φ, where N_{φ} = {x ∈ N_{S}(R) | ∃y ∈ N_{S}(Q), φ(xux^{−1}) = yφ(u)y^{−1}, ∀u ∈ R}.

Write Aut_{S}(Q) for the image of the natural homomorphism N_{S}(Q) → Aut_{F}(Q). If K ≤ Aut_{F}(Q), write Aut_{S}^{K}(Q) for Aut_{S}(Q) ∩ K and N_{S}^{K}(Q) for the inverse image of K in N_{S}(Q).

Definition 2.15. Let K ≤ Aut_{F}(Q). We say that Q is fully K-automized in F if Aut_{S}^{K}(Q) is a Sylow pro-p subgroup of K. We say that Q is fully K-normalized in F if Q is receptive and fully K-automized in F.

When K = Aut_{F}(Q) we say that Q is fully normalized in F rather than fully Aut_{F}(Q)- normalized in F. When K = 1 we say that Q is fully centralized in F.

Definition 2.16. A pro-fusion system F is saturated if every F-isomorphism class contains a subgroup that is fully normalized in F.

Remark 2.17. We follow the treatment of Roberts and Shpectorov, because it does not require us to consider the order of N_{S}(Q)/Q or of C_{S}(Q)Q/Q, which might be infinite. All the different characterizations in the literature are known to be equivalent in the finite case.

Example 2.18. If G is a profinite group with Sylow pro-p group S then F_{S}(G) is saturated. This is a consequence of Sylow theory, and the proof is the same as in the finite case.

The development of the theory from the axioms follows the same lines as in the finite case. We just have to be careful to avoid mentioning the order of a group and not to use any hidden lemma particular to finite groups.

Lemma 2.19. Let F be a pro-fusion system on S, Q ≤ S and K ≤ Aut_{F}(Q). If Q is fully K-automized in F and L ≤ K ≤ Aut_{F}(Q), then there exists a κ ∈ K such that Q is fully κL-automized in F.

Proof. By hypothesis, Aut_{S}^{K}(Q) is a Sylow pro-p subgroup of K, so there is a κ ∈ K such that Aut_{S}^{K}(Q) contains a Sylow pro-p subgroup of ^{κ}L. Hence Aut_{S}^{^{κ}L}(Q) is a Sylow pro-p-subgroup of ^{κ}L. □

Lemma 2.20. If Q is fully K-normalized in F and φ ∈ Iso_{F}(R, Q), then there is morphism ψ : N_{S}^{^{φ^{−1}}K}(R)R → N_{S}^{K}(Q)Q in F and a χ ∈ K such that ψ|_{R} = χφ.

Proof. Consider ^{φ}Aut_{S}^{^{φ^{−1}}K}(R) ≤ Aut_{F}^{K}(Q). Since Q is fully K-automized, there is a χ ∈ Aut_{F}^{K}(Q) such that ^{χφ}Aut_{S}^{^{φ^{−1}}K}(R) ≤ Aut_{S}^{K}(Q). But ^{φ^{−1}}K = ^{(χφ)^{−1}}K, so N_{S}^{^{φ^{−1}}K}(R) ≤ N_{χφ}. Because Q is receptive, χφ extends to ψ′ : N_{χφ} → N_{S}(Q)Q. The restriction of ψ′ to N_{S}^{^{φ^{−1}}K}(R) is the morphism ψ required. □

Lemma 2.21 (cf. [5]). Let F be a pro-fusion system on S, Q ≤ S and K ≤ Aut_{F}(Q). Suppose that there is a subgroup in the F-isomorphism class of Q that is fully normalized in F. Then Q is fully K-normalized in F if and only if all φ ∈ Hom_{F}(N_{S}^{K}(Q)Q, S) satisfy φ(N_{S}^{K}(Q)) = N_{S}^{^{φ}K}(φ(Q)).

If Q is an open subgroup of S, the previous properties are equivalent to |N_{S}^{K}(Q)Q/Q| being maximal in the F-isomorphism class of Q, i.e. in the set {|N_{S}^{^{χ}K}(χ(Q))χ(Q)/χ(Q)| | χ ∈ Hom_{F}(Q, S)} (this set is bounded, by Lemma 2.11).

Proof. Suppose that Q is fully K-normalized in F and φ ∈ Hom_{F}(N_{S}^{K}(Q)Q, S). Clearly φ restricts to a map N_{S}^{K}(Q) → N_{S}^{^{φ}K}(φ(Q)). By Lemma 2.20 applied to φ^{−1}|_{φ(Q)} : φ(Q) → Q, there is a map ψ : N_{S}^{^{φ}K}(φ(Q))φ(Q) → N_{S}^{K}(Q)Q. This takes φ(Q) to Q, hence restricts to a map N_{S}^{^{φ}K}(φ(Q)) → N_{S}^{K}(Q). Thus there are morphisms in both directions between N_{S}^{K}(Q) and N_{S}^{φ̃(K)}(φ(Q)), and so, by Lemma 2.10, φ(N_{S}^{K}(Q)) = N_{S}^{^{φ}K}(φ(Q)).

Conversely, suppose that for all φ : N_{S}^{K}(Q)Q → S we have φ(N_{S}^{K}(Q)) = N_{S}^{^{φ}K}(φ(Q)). Let θ ∈ Hom_{F}(Q, S) be such that P := θ(Q) is fully normalized in F. By Lemma 2.19, there is a χ ∈ Aut_{F}(P) such that P is fully ^{χ}(^{θ}K)-normalized. Apply Lemma 2.20 to η := χθ|_{Q} to obtain a morphism ψ : N_{S}^{K}(Q)Q → N_{S}^{^{η}K}(P)P such that ψ(Q) = P and ^{η}K = ^{ψ}K. The restriction ψ|_{N_{S}^{K}(Q)} : N_{S}^{K}(Q) → N_{S}^{^{η}K}(P) must be onto, by hypothesis, hence the induced homomorphism Aut_{S}^{K}(Q) → Aut_{S}^{^{η}K}(P) must be onto. But we have arranged for Aut_{S}^{^{η}K}(P) to be a Sylow pro-p subgroup of Aut_{F}^{^{η}K}(P), so Aut_{S}^{K}(Q) must be a Sylow pro-p subgroup of Aut_{F}^{K}(Q), showing that Q is fully K-automized.

It is easy to check that Q is receptive in F. Indeed, since P is fully normalized in F, Lemma 2.20 shows that there is a ψ ∈ Hom_{F}(N_{S}(Q), N_{S}(P)) such that ψ(Q) = P. If χ ∈ Iso_{F}(R, Q), then N_{χ} ≤ N_{ψχ} and there is a ρ ∈ Hom_{F}(N_{ψχ}, N_{S}(P)) with ρ(R) = P. Moreover, ρ(N_{χ}) ≤ ψ(N_{S}(Q)), thus ψ^{−1}ρ|_{N_{χ}} extends χ.

If Q is an open subgroup of S, then Lemma 2.11 shows that for every subgroup P and φ ∈ Hom_{F}(Q, P) the image P is also an open subgroup of S and that |QN_{S}^{^{φ}K}(P)/P | is finite and bounded by |S : Q|. For any φ ∈ Hom_{F}(N_{S}^{K}(Q)Q, S) we have φ(N_{S}^{K}(Q)Q) ≤ N_{S}^{^{φ}K}(P)P and φ(N_{S}^{K}(Q)Q)/Q ≤ N_{S}^{^{φ}K}(P)φ(Q)/P. If |N_{S}^{K}(Q)/Q| is maximal then we must have equality.

Conversely, suppose that Q is fully K-normalized in F and χ ∈ Iso_{F}(Q, P). Applying Lemma 2.20 to χ^{−1} yields a ψ : N_{S}^{^{χ}K}(P)P → N_{S}^{K}(Q)Q, which restricts to an isomorphism P → Q. This induces a morphism N_{S}^{^{χ}K}(P)P/P → N_{S}^{K}(Q)Q/Q, so |N_{S}^{K}(Q)Q/Q| is maximal. □

Lemma 2.22. Let F be a pro-fusion system on S, Q ≤ S and K ≤ Aut_{F}(Q). If Q is fully Knormalized in F, then for any φ ∈ Hom_{F}(N_{S}^{K}(Q), S) the subgroup φ(Q) is fully ^{φ}K-normalized in F.

Proof. Given θ ∈ Hom_{F}(N_{S}^{^{φ}K}(φ(Q))φ(Q), S), compose it with φ. By Lemma 2.21, the map θφ : N_{S}^{K}(Q) → N_{S}^{^{θφ}K}(θφ(Q)) is onto, so θ : N_{S}^{^{φ}K}(φ(Q)) → N_{S}^{^{θφ}K}(θ(φ(Q))) is also onto. By Lemma 2.21 again, φ(Q) is fully ^{φ}K-normalized. □

When F is a pro-fusion system on S and N is an open strongly closed subgroup of S we can form the quotient system F/N in the manner of 2.4; it is a fusion system on S/N. Just as in the case of a fusion system on a finite p-group, the image of a fully F-normalized subgroup is a fully F/N-normalized subgroup.

Lemma 2.23. Let F be a pro-fusion system on a pro-p group S. Let N be an open and strongly F-closed subgroup of S and let Q be a subgroup of S containing N that is fully normalized in F. Then Q/N is fully normalized in F/N.

Proof. For simplicity, write \overline{F} := F/N and \overline{X} := XN/N for all subgroups X of S. First, note that, since N ≤ Q, we have \overline{N_{S}(Q)} = N\overline{S}(\overline{Q}). By Lemma 2.21, we only need to show that for every \overline{φ} ∈ Hom\overline{F}(N\overline{S}(\overline{Q}), \overline{S}) we have \overline{φ}(N\overline{S}(\overline{Q})) = N\overline{S}(\overline{φ}(\overline{Q})). The inclusion of the left term in the right one is clear, so it is enough to show that |\overline{φ}(N\overline{S}(\overline{Q}))| ≥ |N\overline{S}(\overline{φ}(\overline{Q}))|. Since \overline{φ} is a morphism of \overline{F}, there is a φ ∈ Hom_{F}(N_{S}(Q), S) inducing \overline{φ}. By Lemma 2.20, there is a morphism θ ∈ Hom_{F}(N_{S}(φ(Q)), N_{S}(Q)) that extends (φ|_{Q})^{−1} up to an automorphism of Q. This induces a morphism in Hom\overline{F}(N\overline{S}(\overline{φ}(\overline{Q})), N\overline{S}(\overline{Q})), and so |\overline{φ}(N\overline{S}(\overline{Q}))| ≥ |N\overline{S}(\overline{φ}(\overline{Q}))|. □

As a consequence we obtain:

Corollary 2.24. Let F be a saturated pro-fusion system on a pro-p group S, and let N be an open and strongly F-closed subgroup of S. Then F/N is saturated fusion system on S/N.

Proof. Let \overline{Q} be a subgroup of S/N and let Q be its inverse image in S. Since F is saturated, there exists a subgroup R in the F-isomorphism class of Q that is fully normalized in F. By Lemma 2.23, the subgroup \overline{R} := R/N is fully normalized in F/N. Since \overline{R} is in the F/Nisomorphism class of \overline{Q}, this implies that F/N is saturated. □

3. Morphisms of pro-fusion systems

Because we have defined a pro-fusion system as an inverse limit of fusion systems, it is incumbent upon us to deal carefully with morphisms and quotients. A morphism between two pro-fusion systems is defined exactly as for fusion systems, that is to say as a homomorphism, α, between the underlying pro-p groups that satisfies certain conditions which can be expressed by the existence of a certain functor, A, just as in Definition 2.3. We now have a category of pro-fusion systems. It is easy to verify that if F = \underleftarrow{lim}_{i∈I} F_{i} is a pro-fusion system then F really

is the inverse limit in this category.

Everything in Paragraph 2.4 is still valid for pro-fusion systems, provided that Q is taken to be open.

It might appear that this definition of morphism does not capture all the information in the inverse limit system, but we will show that, in fact, it does so in the strongest possible reasonable way. We phrase this as a continuity condition, analogous to the characterization of continuity for maps of profinite sets or groups. It is, in fact, equivalent to a morphism of the underlying pro-systems, as the reader who is familiar with such things will notice.

Definition 3.1. Let F = \underleftarrow{lim}_{i∈I} F_{i} and G = \underleftarrow{lim}_{j∈J} G_{j} be pro-fusion systems, with canonical sets

of morphisms {(f_{i}, F_{i}) : F → F_{i} | i ∈ I} and {(g_{j}, G_{j}) : G → G_{j} | j ∈ J}. A morphism (α, A) : F → G is said to be continuous if for each j ∈ J there is an i ∈ I such that the

composition F →^{(α,A)} G →^{(g_{j},G_{j})} G_{j} factors through F →^{(f_{i},F_{i})} F_{i}.

This definition appears to depend on the ways in which F and G are expressed as inverse limits (but see Corollary 3.5).

If F = \underleftarrow{lim}_{i∈I} F_{i}, we can consider the fusion systems ⟨F_{i}(F)⟩ ⊆ F_{i}. These can be assembled

into an inverse limit system and we can consider \underleftarrow{lim}_{i∈I}⟨F_{i}(F)⟩.

Lemma 3.2. Suppose that F = \underleftarrow{lim}_{i∈I} F_{i} is a pro-fusion system. Then for each i ∈ I there is a

j ≥ i such that F_{i}(F) = F_{i,j}(F_{j}).

Proof. For each j ≥ i consider X_{j} = ⟨F_{i,j}(F_{j})⟩ ⊆ F_{i}. Suppose that φ ∈ ∩_{j≥i}X_{j} and let Y_{j} = {θ ∈ F_{j} | F_{i,j}(θ) = φ}. These Y_{j} form an inverse system of finite sets, so their inverse limit is non-empty. An element of the inverse limit defines an element φ̃ ∈ F such that F_{i}(φ̃) = φ, and we conclude that F_{i}(F) = ∩_{j≥i}X_{j}. Since the sets X_{i} are finite, we must have F_{i}(F) = F_{i,j}(F_{j}) for some j ≥ i. □

Lemma 3.3. The identity map on S induces an isomorphism between F and \underleftarrow{lim}_{i∈I}⟨F_{i}(F)⟩. It is

continuous in both directions.

Proof. It is easy to see that we have an isomorphism and that the morphism \underleftarrow{lim}_{i∈I}⟨F_{i}(F)⟩ → F

is continuous.

For the other direction, consider some fixed morphism F →^{F_{i}} F_{i}. Choose j ≥ i as in Lemma 3.2 so that F_{i}(F) = F_{i,j}(F_{j}). Thus F_{i,j} has image in ⟨F_{i}(F)⟩, and so F_{i} factors through F_{j}. □

Proposition 3.4. Every morphism between pro-fusion systems is continuous.

Proof. Suppose that F = \underleftarrow{lim}_{i∈I} F_{i} and G = \underleftarrow{lim}_{j} G_{j} are a pro-fusion systems on S and T. By

Lemma 3.3, we may assume that F_{i} = ⟨F_{i}(F)⟩. Let α : S → T be a group homomorphism that, together with a functor A, induces a morphism F → G. Fix some G_{j}; since α is continuous, there is an i such that S →^{α} T →^{g_{j}} T_{j} factors through S →^{f_{i}} S_{i}. We need to construct the functor A_{i,j} associated to the factoring map S_{i} → T_{j}; it is sufficient to define this on elements of F_{i}(F).

Given φ ∈ F_{i}(F), we have φ = F_{i}(φ̃) for some φ̃ ∈ F. We define A_{i,j}(φ) = G_{j}A(φ̃); it is straightforward to check that this has the right properties. □

The continuity of morphisms has various basic consequences, such as the following ones.

Corollary 3.5. If a pro-fusion system F can be defined up to isomorphism as an inverse limit in two different ways, say as \underleftarrow{lim}_{i∈I} F_{i} or as \underleftarrow{lim}_{i∈I′} F′_{i}, then the isomorphism is continuous. Therefore, the continuity of a morphism of pro-fusion systems does not depend on the representations of the pro-fusion systems as inverse limits.

Corollary 3.6. Let F be a pro-fusion system on a a pro-p group S and let P and Q be subgroups of S. Then the topology on Hom_{F}(P, Q) depends only on the isomorphism class of F.

If (α, A) : F → G is a morphism of pro-fusion systems, then the group homomorphism Aut_{F}(P) → Aut_{G}(α(P)) induced by A is a continuous homomorphism of profinite groups and the induced map Hom_{F}(P, Q) → Hom_{G}(α(P), α(Q)) is a continuous map of profinite sets.

Proposition 3.7. For any pro-fusion system F on S we have F ≅ \underleftarrow{lim}_{Nosc}⟨\overline{F}_{N}⟩, where the limit

is taken over the set of open strongly closed subgroups of S ordered by inclusion.

Proof. There is a natural morphism F → ⟨\overline{F}_{N}⟩. The image of F in ⟨\overline{F}_{N}⟩ is \overline{F}_{N}, so the image of F in \underleftarrow{lim}_{Nosc}⟨\overline{F}_{N}⟩ is \underleftarrow{lim}_{Nosc} \overline{F}_{N}, by a standard result about profinite sets, and F is isomorphic to

its image.

Suppose that F = \underleftarrow{lim}_{i∈I} F_{i} and let N_{i} = Ker(f_{i}), for all i ∈ I. By Lemma 3.3, we may assume

that F_{i} = ⟨F_{i}(F)⟩. Let X be the set {N_{i} | i ∈ I}; then X is cofinal in the set of all open strongly closed subgroups of S, so we may replace \underleftarrow{lim}_{Nosc} by \underleftarrow{lim}_{N∈X}.

Given M ∈ X, we know that M = N_{i} for some i, so S/M ≅ f_{i}(S). By Lemma 3.2, there is a j ≥ i such that F_{i,j}(F_{j}) = F_{i}(F).

We have a commutative diagram of groups

S ⟶ S/N_{j} ⟶ S/M
∥  ^{≡}↓  ^{≡}↓
S ⟶^{f_{j}} f_{j}(S) ⟶^{f_{i,j}} f_{i}(S),

in which the arrows on the top row are the quotient maps and the vertical isomorphisms are uniquely determined. This extends to a diagram of functors

F ⟶ ⟨\overline{F}_{N_{j}}⟩ ⟶ ⟨\overline{F}_{M}⟩
∥  ^{≡}↓  ^{≡}↓
F ⟶^{F_{j}} F_{j} ⟶^{F_{i,j}} F_{i},

in which the vertical maps are isomorphisms, because the domain and codomain are both defined in the same way, as the fusion system generated by the morphisms between the subgroups of S/N_{j} or of S/M that are induced by morphisms in F. Thus the image of the morphism ⟨\overline{F}_{N_{j}}⟩ → ⟨\overline{F}_{M}⟩ is \overline{F}_{M}.

This forces \underleftarrow{lim}_{N∈X}⟨\overline{F}_{N}⟩ = \underleftarrow{lim}_{N∈X} \overline{F}_{N}. □

3.8. There are many equivalent characterizations of countably based profinite groups. The two that will be useful to us are that there are only countably many open subgroups and that the group can be expressed as an inverse limit of finite groups indexed by the set ℕ of natural numbers ordered in the usual way. Any finitely generated profinite group is countably based.

We define a pro-fusion system to be countably based if it is isomorphic to an inverse limit of fusion systems indexed by the natural numbers with the usual ordering.

Lemma 3.9. A pro-fusion system F on the pro-p group S is countably based if and only if S is countably based.

Proof. If F is countably based then so is S, directly from the definitions.

Conversely, by Proposition 3.7 we know that F ≅ \underleftarrow{lim}_{Nosc}⟨\overline{F}_{N}⟩. If S is countably based then

it has only countably many open subgroups, so it certainly has only countably many open strongly closed subgroups; enumerate them as N_{1}, N_{2}, . . ..

Recursively, choose M_{1} = N_{1} and M_{r} = M_{r−1}∩N_{r}. Then {M_{r}}_{r∈ℕ} is ordered in the same way as ℕ and is cofinal in the set of open strongly closed subgroups of S. Hence F ≅ \underleftarrow{lim}_{r∈ℕ}⟨\overline{F}_{M_{r}}⟩. □

4. Pro-saturated fusion systems

We introduce pro-saturated fusion systems and show that they are well behaved.

Definition 4.1. We say that a pro-fusion system is a pro-saturated fusion system if it is isomorphic to an inverse limit of saturated fusion systems on finite p-groups.

Lemma 4.2. Suppose that F = \underleftarrow{lim}_{i∈I} F_{i}, where the F_{i} are saturated fusion systems. Then F_{i}(F)

is a saturated fusion system for all i ∈ I, and F ≅ \underleftarrow{lim}_{i∈I} F_{i}(F).

Proof. From Lemma 3.2 we know that F_{i}(F) = F_{i,j}(F_{j}) for some j ≥ i. By Theorem 2.6, F_{i,j}(F_{j}) is a saturated fusion system and thus, by Lemma 3.3, F ≅ \underleftarrow{lim}_{i∈I} F_{i}(F). □

Lemma 4.3. Let F = \underleftarrow{lim}_{i∈I} F_{i} be a pro-saturated fusion system. Then for each φ ∈ Hom_{F}(P, Q)

and each i ∈ I there exists a φ̃ ∈ Hom_{F}(f_{i}^{−1}f_{i}(P), f_{i}^{−1}f_{i}(Q)) such that F_{i}(φ̃) = F_{i}(φ).

Proof. Let F = \underleftarrow{lim}_{i∈I} F_{i} with the F_{i} saturated. For each j ≥ i let X_{j} be the set of the morphisms

in Hom_{F_{j}}(f_{i,j}^{−1}f_{i}(P), f_{i,j}^{−1}f_{i}(Q)) with image F_{i}(φ) in F_{i}. Then each X_{j} is a finite set, and it is non-empty, by Theorem 2.5. An inverse limit of non-empty finite sets is non-empty, and an element of \underleftarrow{lim}_{j≥i} X_{j} determines a morphism φ̃ ∈ Hom_{F}(f_{i}^{−1}f_{i}(P), f_{i}^{−1}f_{i}(Q)). □

Given a pro-fusion system F on S and an open strongly closed subgroup N of S we can form the fusion system F/N just as in the finite case.

Proposition 4.4. Let F be a pro-saturated fusion system on S and let N be an open strongly F-closed subgroup of S. For any φ ∈ Hom_{F}(P, Q) there exists a φ̃ ∈ Hom_{F}(P N, QN) such that φ and φ̃ induce the same homomorphism P N/N → QN/N. As a consequence, ⟨\overline{F}_{N}⟩ = F/N, and so there is a natural morphism F → F/N.

Proof. If N is one of the N_{i}, then this is just a reformulation of Lemma 4.3. Otherwise, there is some N_{i} ≤ N, and we can use the φ̃ produced by using Lemma 4.3 relative to F_{i}.

The last part follows from the first part and the definition of F/N. □

Proposition 4.5. If F is a pro-saturated fusion system on S, then F ≅ \underleftarrow{lim}_{Nosc} F/N as N runs

through all open strongly F-closed subgroups of S ordered by inclusion.

Proof. This follows from Proposition 3.7 and Lemma 4.4. □

4.6. We denote the restriction of a pro-fusion system F on S to the open subgroups of S by F^{o}. If F is pro-saturated then we can tell whether an open subgroup N of S is strongly F-closed or not just by considering F^{o}. For there is some N_{i} ≤ N and, if P ≤ N and φ(P) ≰ N, then Proposition 4.4 shows that P N_{i} ≤ N but φ̃(P N_{i}) ≰ N. It is clear from the definition that F/N only depends on F^{o}.

From these considerations and using Proposition 4.5, we see that if F and G are both prosaturated fusion systems on S and F^{o} = G^{o} then F = G. In a sense, pro-saturated fusion systems are completely determined by what happens on open subgroups. 4.7. If F is a pro-fusion system on S such that F^{o} is saturated and N is an open strongly closed subgroup of S, then F/N is saturated, as follows directly from the definitions. If M ≤ N is also open and strongly closed, then F/N = (F/M)/(N/M), so there is a natural morphism F/M → F/N, by Theorem 2.6. We set F_{pro−sat} = \underleftarrow{lim}_{Nosc} F/N. Then F_{pro−sat} is pro-saturated

and there is a natural morphism F_{pro−sat} → F with the property that F_{pro−sat}^{o} = F^{o}.

Example 4.8. Let p be an odd prime and let S be a free pro-p group on two generators x and y. Let C be a cyclic group of order 2 and let it act on S by x ↔ y. Set G = S ⋊ C. There are two saturated fusion systems on S, F_{S}(S) and F_{S}(G): we want to mix them. Define F to be the subcategory of F_{S}(G) in which the morphisms with cyclic domain are those from F_{S}(G) and the morphisms with non-cyclic domain are those from F_{S}(S). We need to check that F is indeed a pro-fusion system.

There is a chain of open subgroups \{N_{i}|i ∈ I\} in S, each normal in G, such that S = \underleftarrow{lim}_{i∈I} N_{i}.

Set G_{i} = (S/N_{i}) ⋊ C. On S/N_{i} we define a fusion system F_{i} in an analogous way to F. Then

F = \underleftarrow{lim}_{i∈I} F_{i}.

It is easily verified that F is saturated using the fact that in a free pro-p group the normalizer of a cyclic group is cyclic (just as in the case of an abstract free group).

Also F_{pro−sat} = F_{S}(S), so F is not pro-saturated.

5. Saturation of pro-saturated fusion systems

We show that a pro-saturated fusion system is saturated, at least if it is countably based.

Theorem 5.1. If F is a pro-saturated fusion system, then F^{o} is saturated.

Proof. Suppose that we are given an open subgroup Q ≤ S. Notice that {N_{i} | N_{i} < Q} is cofinal in \{N_{i}|i ∈ I\}, so F ≃ \underleftarrow{lim}_{N_{i}<Q} F_{i}. Thus we can assume that all the N_{i} are subgroups of Q.

For each N_{i} < Q let X_{i} := {φ ∈ Hom_{F_{i}}(f_{i}(Q), S_{i}) | φ(f_{i}(Q)) is fully normalized in F_{i}}. Since F_{i} is a saturated fusion system and S_{i} is finite, X_{i} is a finite non-empty set. By Lemma 2.23, for every N_{j} < N_{i} there is a natural map X_{j} → X_{i}; thus \underleftarrow{lim}_{N_{i}<Q} X_{i} ≠ ∅. An element of the limit

determines a map ψ : Q → S such that f_{i}(ψ(Q)) is fully normalized in F_{i} for each N_{i} < Q.

We show that P = ψ(Q) is fully normalized in F. Given φ : R → P, it induces \overline{φ}_{i} : R/N_{i} → P/N_{i}, which, given that P/N_{i} is fully normalized in F_{i}, extends to ρ_{i} : N_{\overline{φ}_{i}} → N_{S/N_{i}}(Q/N_{i}). For N_{j} < N_{i} there is a natural map N_{\overline{φ}_{j}} → N_{\overline{φ}_{i}}; hence any extension ρ_{j} induces an extension ρ_{i}. Let X_{i} be the set of all possible extensions of \overline{φ}_{i} to N_{\overline{φ}_{i}}. These are finite non-empty sets and there are natural maps X_{j} → X_{i} for N_{j} < N_{i}. Thus \underleftarrow{lim}_{N_{i}<Q} X_{i} is non-empty and an element

in this set induces a map \underleftarrow{lim}_{N_{i}<Q} N_{\overline{φ}_{i}} → \underleftarrow{lim}_{N_{i}<Q} N_{S/N_{i}}(P/N_{i}), i.e. a map N_{φ} → N_{S}(P).

The image of N_{S}(P) in S/N_{i} is N_{S/N_{i}}(P/N_{i}), and this maps to a Sylow p-subgroup of Aut_{F_{i}}(P/N_{i}). Since this is true for all i ∈ I, Aut_{S}(P) is dense in a Sylow p-subgroup of Aut_{F}(P). But Aut_{S}(P) is closed, so it is a Sylow p-subgroup of Aut_{F}(P). Thus, F^{o} satisfies the saturation axioms. □

Theorem 5.2. If F is a countably based pro-saturated fusion system then F is saturated.

Before the proof we need a lemma.

Lemma 5.3. Let F be a pro-saturated fusion system on S and P ≤ S, and let P_{n} ≤ P_{n−1} ≤ ⋯ ≤ P_{1} be a non-empty sequence of open, strongly closed subgroups of S. Then there exists a subgroup Q in the F-isomorphism class of P such that QP_{i} is fully normalized in F for each i ∈ {1, …, n}.

Proof. We use induction on n. The result is true if n = 1 by Theorem 5.1. Suppose that n ≥ 2 and that there exists a subgroup R that is F-isomorphic to P and such that RP_{i} is fully normalized in F for each 2 ≤ i ≤ n. By Theorem 5.1, there exists φ ∈ Hom_{F}(RP_{1}, S) such that φ(RP_{1}) is fully normalized in F. By Lemma 2.20, there is a ψ ∈ Hom_{F}(N_{S}(RP_{1}), N_{S}(φ(RP_{1})) such that φ(RP_{1}) = ψ(RP_{1}) = ψ(R)P_{1}. Let Q := ψ(R). Since QP_{1} = (QP_{i})P_{1}, we find that N_{S}(QP_{i}) ≤ N_{S}(QP_{1}), so each N_{S}(QP_{i}) lies in the domain of ψ. Lemma 2.22 shows that each RP_{i} is fully normalized in F. □

Proof of Theorem 5.2. Since F is countably based, we can write F = \underleftarrow{lim}_{i∈ℕ} F_{i}, and the P_{i} :=

Ker(f_{i}) form a sequence of open, strongly closed subgroups of S. Given a subgroup P ≤ S, let X_{i} be the set of φ : P P_{i}/P_{i} → S/P_{i} such that φ(P)P_{i}/P_{i} is fully normalized in F/P_{i}. By Lemma 5.3, X_{i} is finite and non-empty for all i. Moreover, there is a natural map X_{i+1} → X_{i} induced by the canonical projection P P_{i}/P_{i} ≃ P/(P ∩ P_{i}) → P/P ∩ P_{i+1} ≃ P P_{i+1}/P_{i+1}. Thus \underleftarrow{lim}_{i} X_{i} is non-empty, and an element ψ ∈ \underleftarrow{lim}_{i} X_{i} induces a map ψ : P → S such that

ψ(P)P_{i}/P_{i} is fully normalized in F/P_{i} for each i. Set R := ψ(Q); we need to show that R is fully normalized in F.

First we show that every φ : Q → R extends to N_{φ}. Since RP_{i}/P_{i} is fully normalized in F_{i}, we know that \overline{φ}_{i} : QP_{i}/P_{i} → RP_{i}/P_{i} extends to a morphism N_{\overline{φ}_{i}} → N_{S/P_{i}}(R/P_{i}). Let X_{n} be the set of all maps N_{φ} → N_{S/P_{n}}(R/P_{n}) of the form N_{φ} → N_{\overline{φ}_{i}} → N_{S/P_{i}}(R/P_{i}) → N_{S/P_{n}}(R/P_{n}) for some i ≥ n. X_{n} is clearly non-empty; it is also finite, since all the maps have to have P_{n} in the kernel. Moreover, there is a natural map X_{n+1} → X_{n} induced by the projection N_{S/P_{n+1}}(R/P_{n+1}) → N_{S/P_{n}}(R/P_{n}). Thus \underleftarrow{lim}_{n} X_{n} is non-empty and an element of this limit

yields φ̃ : N_{φ} → N_{S}(R), extending φ.

Next we show that Aut_{S}(R) is a Sylow p-subgroup of Aut_{F}(R). This is done by applying the following lemma to the family N_{S/P_{i}} → Aut_{F_{i}/P_{i}}(RP_{i}/P_{i}). □

Lemma 5.4. Let {G_{i}}_{i∈I} be a directed system of finite groups and {S_{i}}_{i∈I} a directed system of finite p-groups. Let {χ_{i} : S_{i} → G_{i}}_{i∈I} be a map of directed systems such that χ_{i}(S_{i}) is a Sylow p-subgroup of G_{i} for all i. Then χ(\underleftarrow{lim}_{i∈I} S_{i}) is a Sylow p-subgroup of \underleftarrow{lim}_{i∈I} G_{i}.

Proof. Set G = \underleftarrow{lim}_{i∈I} G_{i} and S = \underleftarrow{lim}_{i∈I} S_{i}. It is easy to see that χ : S → G is injective.

Let T be a Sylow pro-p subgroup of G such that χ(S) ≤ T. For each i let p_{i} : G → G_{i} be the canonical projection. Then there exists a g_{i} ∈ G_{i} such that ^{g_{i}}p_{i}(T) ≤ χ_{i}(S_{i}). Let X_{i} be the set of all such g_{i}. There is a natural map X_{i} → X_{j} sending g_{j} to p_{i,j}(g_{i}). Hence \underleftarrow{lim}_{i∈I} X_{i} is

non-empty, and an element of it yields an element g ∈ G such that ^{g}T ≤ χ(S). Given that χ(S) ≤ T we obtain χ(S) = T. □

We do not know whether the hypothesis that the system should be countably based is necessary.

6. Invariant subsystems and a relative form of Alperin’s Fusion Theorem

Let F be a pro-fusion system on a p-group S. We introduce here the notion of a T -subsystem, where T is a strongly F-closed subgroup of S. This generalizes the notion of an F-invariant system in the literature ([1, 7, 12]), by retaining as objects in the T -subsystem all subgroups of S, not just the ones contained in T. If G is a finite group with Sylow p-subgroup S and H is a normal subgroup of G then the morphisms between the subgroups of S that are induced by conjugation by an element of H will form an (H ∩S)-subsystem of F_{S}(G). So we are attempting to model a normal subgroup.

Definition 6.1. Let F be a fusion system on a p-group S and let T be a strongly F-closed subgroup of S. A T -subsystem E of F is a subcategory of F on the same objects, with morphisms satisfying:

(a) E contains every group homomorphism induced by conjugation by an element of T,

(b) any morphism in E factors as an isomorphism of E followed by an inclusion,

(c) for all P, P′ ≤ Q ≤ S and φ ∈ Hom_{F}(Q, S) we have φ|_{P′}Hom_{E}(P, P′) = Hom_{E}(φ(P), φ(P′))φ|_{P},

(d) for all Q ≤ S and ψ ∈ Hom_{E}(Q, S) we have ψ(u)u^{−1} ∈ T, ∀u ∈ Q.

Note that the full subcategory of E on the subgroups of T is what is known as an F-invariant fusion system in the literature.

A morphism between a T_{1}-subsystem E_{1} of F_{1} and a T_{2}-subsystem E_{2} of F_{2} is a morphism (α, A) : F_{1} → F_{2} such that α(T_{1}) ≤ T_{2} and A(E_{1}) ⊆ E_{2}.

In an analogous way to the saturation of F we define the saturation of E.

Definition 6.2. We say that Q is receptive in E if for all φ ∈ Iso_{E}(R, Q) there exists φ̃ : N_{φ}^{T} → N_{T} (Q)Q in E such that φ̃|_{R} = φ, where

N_{φ}^{T} = {x ∈ N_{T} (R)R | ∃y ∈ N_{T} (Q)Q, φ(xux^{−1}) = yφ(u)y^{−1}, ∀u ∈ R} .

Definition 6.3. Let K ≤ Aut_{E}(Q). We say that Q is fully K-automized in E if Aut_{T} (Q)∩K is a Sylow p-subgroup of Aut_{E}(Q) ∩ K. We say that Q is fully K-normalized in E if Q is receptive and fully K-automized in E.

When K = Aut_{E}(Q) we say that Q is fully normalized in E rather that fully Aut(Q)- normalized in E.

Definition 6.4. A T -subsystem E of F is saturated if every E-isomorphism class contains a subgroup that is fully normalized in E.

If E is a saturated T -subsystem, then T can be recovered from E as the inverse image of Aut_{E}(S) under the conjugation map S → Aut_{F}(S), so sometimes we will not mention T. 6.5. We use the same definitions for a T -subsystem of a pro-fusion system, with the proviso that each Hom_{E}(P, Q) should be closed in Hom_{F}(P, Q). If each Hom_{E}(P, Q) is open in Hom_{F}(P, Q) we say that E is open.

The closure condition implies that if F = \underleftarrow{lim}_{i∈I} F_{i}, then E = \underleftarrow{lim}_{i∈I} F_{i}(E) as sets and hence that

E = \underleftarrow{lim}_{i∈I}⟨F_{i}(E)⟩, where ⟨F_{i}(E)⟩ denotes the f_{i}(T)-subsystem of F_{i} generated by F_{i}(E).

Example 6.6. Let G be a profinite group with Sylow pro-p subgroup S. Let H ◁ G and set T = H ∩ S. Let E_{S}(H) be the subcategory of F_{S}(G) with objects the subgroups of S and morphisms all the group homomorphisms induced by conjugation by an element of H. Then E_{S}(H) is a saturated T -subsystem of F_{S}(G).

Lemma 6.7. If Q is fully K-normalized in E and φ ∈ Iso_{E}(R, Q), then there is morphism ψ : N_{T}^{^{φ^{−1}}K}(R)R → N_{T}^{K}(Q)Q in E and an χ ∈ K such that ψ|_{R} = χφ.

Proof. This is strictly analogous to that of Lemma 2.20 and is left to the reader. □

Definition 6.8. Let F be a pro-fusion system on S, T a strongly F-closed subgroup of S, E a saturated T -subsystem of F and Q a subgroup of S. We say that Q is E-radical if Aut_{T ∩Q}(Q) = O_{p}(Aut_{E}(Q)) and that it is E-centric if C_{T} (Q′) ≤ Z(Q′) for all Q′ in the Eisomorphism class of Q. A subgroup Q ≤ S is E-essential if T ≤ Q or if Q is E-centric and S_{p} ( Aut_{E}(Q)/Aut_{T ∩Q}(Q) ) is disconnected (or empty), where, for a group G, S_{p}(G) denotes the partially ordered set of non-trivial p-subgroups of G.

Notice that an E-essential subgroup of S is also E-radical.

Theorem 6.9. Let F be a pro-fusion system on S, T a strongly F-closed subgroup of S and let E be a saturated T -subsystem of F. Suppose that P and P′ are open subgroups of S and that φ ∈ Iso_{E}(P, P′). Then there exist open subgroups of S, P = P_{0}, P_{1}, …, P_{n} = P′ and Q_{1}, Q_{2}, …, Q_{n} and morphisms φ_{i} ∈ Aut_{E}(Q_{i}) for 1 ≤ i ≤ n such that:

(a) the Q_{i} are E-essential subgroups of S and are fully normalized in E;

(b) for each i, P_{i−1}, P_{i} ≤ Q_{i} and φ_{i}(P_{i−1}) = P_{i};

(c) φ(u) = φ_{n} … φ_{2}φ_{1}(u) for all u ∈ P.

Notice that if E = F and S is finite then this is just the usual statement of Alperin’s Fusion Theorem in the context of saturated fusion systems (see [10, 5]). Even in the case of a finite group G with normal subgroup H and F = F_{S}(G) and E = E_{S}(H), it appears to be new.

Proof. The proof is modeled on the one for saturated fusion systems. We can use induction on the index |S : P |, because P is open, so this index is finite. If |S : P | = 1, then n = 1, S = Q_{1} = P_{0} = P_{1} and φ = φ_{1} ∈ Aut_{E}(S). If |S : P | > 1, let ψ ∈ Hom_{E}(P, S) be such that ψ(P) is fully normalized in E. Then φ = (ψφ^{−1})^{−1} ∘ ψ, where both ψ ∈ Hom_{E}(P, ψ(P)) and ψφ^{−1} ∈ Hom_{E}(φ(P), ψ(P)) have as image the subgroup ψ(P), which is fully normalized in E and has the same index in S as P, by Lemma 2.11. Thus we only need to decompose morphisms in E with image a subgroup of S that is fully normalized in E, because we can splice together a chain for ψ with the inverse of a chain for ψφ^{−1}.

So now we may suppose that P′ is fully normalized in E. By Lemma 6.7, there is a morphism ψ ∈ Hom_{E}(P N_{T} (P), S) and a χ ∈ Aut_{E}(P′) such that ψ|_{P} = χφ. If N_{T} (P) is not contained in P then N_{T} (P)P has strictly smaller index than P, so we can find a chain for ψ by induction. If N_{T} (P) ≤ P, then T ≤ P, as P is open, and so P is E-essential. From the last condition in the definition of a subsystem, we know that φ(u)u^{−1} ≤ T for all u ∈ P, which yields P = P′ and φ ∈ Aut_{E}(P). Therefore we are reduced to the case of decomposing an automorphism of a subgroup that is fully normalized in E.

So let P be an open subgroup of S that is fully normalized in E and χ ∈ Aut_{E}(P). There exists a χ̃ ∈ Hom_{E}(N_{χ}^{T} , T) extending χ. If P is not E-centric then there is a subgroup P′ and a θ ∈ Iso_{E}(P′, P) such that C_{T} (P′) ≰ P′. But θ extends to θ̃ : C_{T} (P′)P′ → C_{T} (P)P, and so C_{T} (P) ≰ P. Since C_{T} (P)P ≤ N_{χ}^{T}, this implies that N_{χ}^{T} is strictly bigger than P, and so χ̃ can be decomposed by induction. Thus we may assume that P is E-centric. If P happens to be E-essential, then χ already has the desired form. Otherwise, S_{p} ( Aut_{E}(P)/Aut_{T ∩P} (P) ) is connected, and so S_{p}(Aut_{E}(P))_{>Aut_{T ∩P} (P)} is also connected. Thus there exist pro-p subgroups U_{1}, …, U_{m−1} and R_{1}, …, R_{m} of Aut_{E}(P), all strictly containing Aut_{P ∩T} (P), such that U ≥ R_{1} ≤ U_{1} ≥ R_{2} ≤ U_{2} ≥ … ≤ U_{m−1} ≥ R_{m} ≤ ^{χ}U. We may suppose that the U_{i} are Sylow pro-p subgroups of Aut_{E}(P). Because U := Aut_{T} (P) is a Sylow p-subgroup of Aut_{E}(P), there exist χ_{i} ∈ Aut_{E}(P) such that U_{i} = ^{χ_{i}}U for each 1 ≤ i ≤ m − 1. Thus U and ^{χ}U are connected by the path

U ≥ R_{1} ≤ ^{χ_{1}}U ≥ R_{2} ≤ … ^{χ_{m−1}}U ≥ R_{m} ≤ ^{χ}U .

Setting χ_{0} := id and χ_{m} := χ, we have ^{χ_{i−1}}R_{i} ≤ U ≥ ^{χ_{i}}R_{i} for all 1 ≤ i ≤ m. Let θ_{i} := χ_{i}χ_{i−1}^{−1} ∈ Aut_{E}(P); then ^{θ_{i}}(^{χ_{i−1}}R_{i}) = ^{χ_{i}}R_{i} ≤ U, and hence the image of N_{θ_{i}}^{T} in Aut_{E}(P) contains ^{χ_{i−1}}R_{i}. The latter strictly contains Aut_{P ∩T} (P), so N_{θ_{i}}^{T} strictly contains P, hence has smaller index. Because P is fully normalized in E, θ_{i} extends to θ̃_{i} ∈ Hom_{E}(N_{θ_{i}}^{T}, S), which can be expressed as a chain by induction. This is true for each 1 ≤ i ≤ m, so we obtain a chain for χ = θ_{m} … θ_{1}, as required. □

The next lemma is a generalization to T -subsystems of Theorem 2.5. We follow here the proof presented in [7, proof of Proposition 5.11].

Lemma 6.10. Let F be a fusion system on a finite p-group S, let E be a saturated T -subsystem of F and let N be a strongly F-closed subgroup of T. Then for any φ ∈ Hom_{E}(P, Q) there exists φ̃ ∈ Hom_{E}(P N, QN) such that φ and φ̃ induce the same map from P N/N to QN/N.

Proof. We use induction on |S : N|. By the relative version of Alperin’s Fusion Theorem, it is enough to prove the result for φ ∈ Aut_{E}(P) when P is E-essential and fully normalized in E. Let K := Ker(Aut_{E}(P) → Aut(P N/N)); Aut_{T}^{K}(P) is thus a Sylow p-subgroup of K. Let A := Aut_{E}(P); the Frattini Argument applied to A and K yields Aut_{E}(P) = KN_{A}(Aut_{T}^{K}(P)), and so φ = χψ with χ ∈ K and ψ ∈ N_{A}(Aut_{T}^{K}(P)). Hence φ and ψ induce the same morphism in Aut(P N/N). Since P is fully normalized in E, ψ extends to N_{ψ}^{T}. We are done by induction if N_{ψ}^{T} is strictly bigger than P. Suppose that P = N_{ψ}^{T}. Because ψ ∈ N_{A}(Aut_{T}^{K}(P)), it is clear that N_{ψ}^{T} contains N_{T}^{K}(P), and hence it contains N_{N}(P), since N ≤ T. It follows that N_{N}(P) is a subgroup of P; as we are dealing with finite p-groups, this implies that N ≤ P. But in this case the result is clear. □

Definition 6.11. A T -subsystem E of a pro-fusion system F is pro-saturated if it is the inverse limit of saturated subsystems on finite groups, E_{i} ≤ F_{i}, for some expression F = \underleftarrow{lim} F_{i} (not necessarily all).

The next result is a generalization of Lemma 6.10 to pro-fusion systems.

Lemma 6.12. Let F be a pro-fusion system on a pro-p group S, let E be a pro-saturated T -subsystem of F and let N be an open strongly F-closed subgroup of T. Given any φ ∈ Hom_{E}(P, Q) we can find φ̃ ∈ Hom_{E}(P N, QN) such that φ and φ̃ induce the same morphism from P N/N to QN/N.

Proof. Let F = \underleftarrow{lim} F_{i} and E = \underleftarrow{lim} E_{i}, with E_{i} ≤ F_{i}. The result for some E_{i} can be obtained by considering the image \overline{φ}_{i} : P N_{i}/N_{i} → QN_{i}/N_{i} of φ in E_{i}. By Lemma 6.10, there exists a φ̃_{i} : P NN_{i}/N_{i} → QNN_{i}/N_{i} that has the same image as φ modulo NN_{i}/N_{i}. Let X_{i} be the set of all possible such φ̃_{i}. The X_{i} are non-empty, and they form an inverse set with respect to the functors F_{ij}, so \underleftarrow{lim}_{i∈I} X_{i} is non-empty. Take φ̃ ∈ \underleftarrow{lim}_{i∈I} X_{i} ⊆ Hom_{E}(P N, QN). □

Lemma 6.13. Let F be a pro-fusion system on S and E an open T -subsystem of F. Then for any φ ∈ Hom_{E}(P, S) there exists an open strongly F-closed subgroup N of S with the property that if θ ∈ Hom_{F}(P, S) is such that θ ≅ φ modulo N then θ ∈ Hom_{E}(P, S).

Proof. Given that Hom_{E}(P, S) is an open subset of Hom_{F}(P, S), there is an i ∈ I such that (F_{i}|_{Hom_{F} (P,S)})^{−1}F_{i}(φ) ⊆ Hom_{E}(P, S). Let N = N_{i}. □

6.14. We say that F has a countable convergent sequence of open pro-saturated subsystems if there exist open subsystems F = E^{1} ≥ E^{2} ≥ ⋯ such that E^{i} is a pro-saturated T^{i}-subsystem and S = \underleftarrow{lim}_{i∈ℕ} S/T^{i}. By the last condition in Definition 6.1 it follows that ∩_{i}E^{i} is the set of

inclusions between the subgroups of S.

Example 6.15. If F = F_{S}(G) is the fusion system associated to a countably based profinite group G and \{H_{i}\}_{i∈ℕ} is a sequence of open subgroups such that G = \underleftarrow{lim}_{i∈ℕ} G/H_{i}, then the

E_{i} := E_{S}(H_{i}) have the above property.

Lemma 6.16. Let F be a saturated fusion system on S with a countable convergent sequence of open pro-saturated T^{i}-subsystems E^{i}. Then, for any open strongly F-closed subgroup N of S and any P ≤ S, there exists an i such that if θ ∈ Hom_{E^{i}}(P, S) then θ ≃ id_{P} modulo N.

Proof. Consider the image of Hom_{E^{i}}(P, S) in Hom_{F/N}(P N/N, S/N). Since ⋂_{i≥1} Hom_{E^{i}}(P, S) = {inc_{P} } and Hom_{F/N}(P N/N, S/N) is finite, we must have, by compactness, that the image of Hom_{E^{i}}(P, S) is {inc_{P N/N}} for large enough i. □

Theorem 6.17. Let F be a pro-fusion system on a pro-p group S that has a countable convergent sequence of open pro-saturated T^{i}-subsystems E^{i} and let P, P′ be two closed subgroups of S. Then for each φ ∈ Iso_{F}(P, P′) there exist subgroups P = P_{0}, P_{1}, P_{2}, … and open subgroups Q_{1}, Q_{2}, Q_{3}, … of S and morphisms φ_{i} ∈ Aut_{F}(Q_{i}), i ≥ 1 such that:

(a) for each j there exists an n_{j} such that for any i ≥ n_{j}, φ_{i} ∈ Aut_{E^{j}}(Q_{i}) and Q_{i} is E^{j}-essential and fully normalized in E^{j}; (b) P_{i−1}, P_{i} ≤ Q_{i}, φ_{i}(P_{i−1}) = P_{i};

(c) φ(u) = … φ_{i}φ_{i−1} … φ_{2}φ_{1}(u)for all u ∈ P and the infinite composition converges.

Proof. We construct the Alperin chain by successive approximation. Recall that E^{1} = F, and start with R_{0} = P and θ_{0} = φ. Given θ_{i} ∈ Iso_{E^{i}}(R_{i}, P′), we show that there is a morphism ψ_{i+1} ∈ Iso_{E^{i}}(R_{i}, R_{i+1}), given by an Alperin chain in E^{i} as in the theorem, and a morphism θ_{i+1} ∈ Iso_{E^{i+1}}(R_{i+1}, P′) such that θ_{i} = θ_{i+1}ψ_{i+1}. Thus φ = θ_{i}ψ_{i} ⋯ ψ_{1}. We consider the infinite chain ⋯ ψ_{3}ψ_{2}ψ_{1} and replace each ψ_{i} by its Alperin chain to obtain the Alperin chain required.

The first two conditions are satisfied, with n_{j} set equal to the sum of the lengths of the chains for ψ_{i}, …, ψ_{1}. Given an open strongly F-closed subgroup N of S, choose j as in Lemma 6.16. If i ≥ j then φ_{i} ⋯ φ_{1} = φ_{i} ⋯ φ_{N_{j}+1}ψ_{j} ⋯ ψ_{1} = φ_{i} ⋯ φ_{N_{j}+1}θ_{j}^{−1}φ. But φ_{i} ⋯ φ_{N_{j}+1}θ_{j}^{−1} ∈ E_{j}, and so φ_{i} ⋯ φ_{N_{j}+1}θ_{j}^{−1} ≃ id_{P′} modulo N. Thus φ ≃ φ_{i} ⋯ φ_{1} modulo N, which shows that the infinite composition converges.

Now for the construction. Let inc_{P′} denote the inclusion of P′ in S; Lemma 6.13 allows us to choose an open strongly F-closed subgroup N of S such that if χ ≃ inc_{P′} modulo N then χ ∈ Hom_{E^{i+1}}(P, S). Replacing N by N ∩ T^{i+1} if necessary, we may assume that N ≤ T^{i+1}. Given θ_{i} ∈ Hom_{E^{i}}(R_{i}, P′), apply Lemma 6.12 to θ_{i} to obtain a θ̃_{i} ∈ Hom_{E^{i}}(R_{i}N, P′N) such that θ_{i} and θ̃_{i} are equal modulo N. Set ψ_{i+1} := θ̃_{i}|_{R_{i}}, R_{i+1} := ψ_{i+1}(R_{i}) and θ_{i+1} := θ_{i}ψ_{i+1}^{−1}. Since R_{i}N is an open subgroup of S, we can apply Theorem 6.9 relative to E^{i} to obtain an Alperin chain in E_{i} for θ̃_{i} and thus also for ψ_{i+1}. Moreover, θ_{i+1}^{−1} = inc_{P′} modulo N; hence θ_{i+1}^{−1} ∈ Iso_{E^{i+1}}(P′, R_{i+1}), and so θ_{i+1} ∈ Iso_{E^{i+1}}(R_{i+1}, P′) □

Lemma 6.18. Suppose that for some φ ∈ Iso_{F}(P, P′) there is a finite Alperin chain in F as in the conclusion of the preceding theorem, although it is not required to satisfy condition (a). Then there is such a chain with all of the Q_{i} F-essential and fully normalized in F.

Proof. Since the Q_{i} are open subgroups, we can apply Theorem 6.9 to each φ_{i} ∈ Aut_{F}(Q_{i}) in order to decompose it as a finite composition of restrictions of automorphisms of F-essential F-normalized subgroups. Now splice these chains together. □

Remark 6.19. The proof of the preceding lemma will not work if the original chain is infinite, because of problems with convergence.

7. The length of an Alperin chain

It really is necessary to allow the possibility of an infinite chain in the statement of Theorem 6.17, as we will now show with an example.

Let G be a finite group with Sylow p-subgroup S. Let P, P′ ≤ S be two subgroups that are conjugate by an element g ∈ G but are not conjugate by any element of N_{G}(S). For example, if G = Gl_{3}(𝔽_{p}) and S consists of matrices of the form ( \begin{matrix}1 & ∗ & ∗ \\ 0 & 1 & ∗ \\ 0 & 0 & 1 \end{matrix} ), P those of the form ( \begin{matrix}1 & ∗ & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} ), P′

those of the form ( \begin{matrix}1 & 0 & 0 \\ 0 & 1 & ∗ \\ 0 & 0 & 1 \end{matrix} ) and g = ( \begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix} ).

Now consider the cartesian product of infinitely many copies of G, indexed by some set I. We will denote this by ∏ G_{i}. The subgroups ∏ P_{i} and ∏ P′_{i} are conjugate by ∏ g_{i}.

An open subgroup of ∏ S_{i} is of the form X = Y × ∏_{i∉J} S_{i}, for some finite set J ⊂ I. Thus an automorphism of X will have as i-coordinate an element of N_{G}(S_{i}) for all but finitely many i. The same will be true of a composition of finitely many such automorphisms. In particular, such a composition cannot be equal to ∏ g_{i}.

The rest of this section contains various facts about the length of an Alperin chain.

Let F be a pro-fusion system on a pro-p group S. Let P, P′ ≤ S and φ ∈ Iso_{F}(P, P′). By an Alperin chain in F for φ we mean a sequence of subgroups of S, P = P_{0}, P_{1}, …, P_{n} = P′, a sequence of open subgroups Q_{1}, …, Q_{n} and morphisms φ_{i} ∈ Aut_{F}(Q_{i}) such that P_{i−1}, P_{i} ≤ Q_{i}, φ_{i}(P_{i−1}) = P_{i} and φ = φ_{n} ⋯ φ_{1}. By a chain of essential subgroups we mean such a chain with the extra condition that all the Q_{i} are F-essential.

The length of the chain is defined to be the number of subgroups Q_{i} that are not equal to S. It is known that a chain can be rearranged so that all the Q_{i} that are equal to S appear at the end and all their automorphisms can thus be merged into one (see, for example, [16, 2.8]). Thus not counting the Q_{i} equal to S will only change the length of a shortest chain by at most one.

Definition 7.1. The Alperin length of φ is the minimum length of a chain for φ as defined above; we denote it by Alp_{F}(φ). If no such (finite) chain exists, we write Alp_{F}(φ) = ∞. If we require a chain of essential subgroups we write Alp_{F}^{ess}(φ). If we relax the condition that the Q_{i} be open then we write Alp_{F}^{clo}(φ).

Lemma 7.2. We have Alp_{F}^{ess}(φ) ≥ Alp_{F}(φ) ≥ Alp_{F}^{clo}(φ) and Alp_{F}^{ess}(φ) ≥ Alp_{F}^{clo,ess}(φ) ≥ Alp_{F}^{clo}(φ). If Alp_{F}(φ) < ∞ then Alp_{F}^{ess}(φ) < ∞.

Proof. The first statement is clear from the definitions and the second is the content of Lemma 6.18. □

For simplicity we will mostly, in what follows, deal only with pro-fusion systems associated to profinite groups. In this case a fixed choice of Sylow pro-p subgroup S is implicit and we write Alp_{G}(φ) for Alp_{F_{S}(G)}(φ).

Lemma 7.3. If we have a collection of profinite groups {G_{i}|i ∈ I}, possibly infinite, with Sylow pro-p subgroups S_{i}, pro-p subgroups P_{i}, P′_{i} ≤ S_{i} and morphisms φ_{i} ∈ Iso_{G_{i}}(P_{i}, P′_{i}) and form the cartesian product, then ∑_{i} Alp_{G_{i}}(φ_{i}) ≥ Alp_{∏ G_{i}}(∏ φ_{i}) ≥ sup{Alp_{G_{i}}(φ_{i})} and Alp_{∏ G_{i}}^{clo}(∏ φ_{i}) = sup_{i}{Alp_{G_{i}}(φ_{i})}.

Proof. An Alperin chain in ∏ G_{i} for ∏ φ_{i} can be projected to a chain in any G_{i} for φ_{i}; this gives the inequality with sup in either case. For the other inequality, we may assume that sup_{i}{Alp_{G_{i}}(φ_{i})} (or its closed version) is finite, equal to N, say. For each i, let φ_{i} = φ_{i}^{n_{i}} ⋯ φ_{i}^{1} be an Alperin chain of minimal length, with φ_{i}^{j} ∈ Aut(Q_{i}^{j}), and with the Q_{i}^{j} open subgroups if that is the case we are treating.

In the closed case, extend each chain to have length N by adding copies of the identity morphism on S_{j}. Then the ∏_{i} φ_{i}^{j} ∈ Aut_{∏_{i} G_{i}}(∏_{i} Q_{i}^{j}) form an Alperin chain of length N.

In the open case we may assume that ∑_{i} Alp_{G_{i}}(φ_{i}) < ∞. Thus Alp_{G_{i}}(φ_{i}) is only non-zero for finitely many i, say for i ∈ K, and φ_{i} ∈ Aut_{G_{i}}(S_{i}) for i ∉ K. For i ∈ K, extend each φ_{i}^{j} to φ_{i}^{j} × 1 ∈ Aut_{∏_{i} G_{i}}(Q_{i}^{j} × ∏_{j≠i} S_{j}). These form an Alperin chain of length ∑_{i} Alp_{G_{i}}(φ_{i}), for which the composition is equal to ∏_{i∈K} φ_{i} × ∏_{i∉K} 1. We can obtain ∏_{i} φ_{i} as composition by adding the term ∏_{i∈K} 1 × ∏_{i∉K} φ_{i} ∈ Aut_{∏_{i} G_{i}}(∏_{i} S_{i}), which does not count towards the length. □

Proposition 7.4. If we have a collection of profinite groups G_{i} with Sylow pro-p subgroups S_{i}, pro-p subgroups P_{i}, P′_{i} ≤ G_{i} and morphisms φ_{i} ∈ Iso_{G_{i}}(P_{i}, P′_{i}) and form the cartesian product, then Alp_{∏ G_{i}}^{ess}(∏ φ_{i}) = ∑ Alp_{G_{i}}^{ess}(φ_{i}).

Proof. The inequality ∑_{i} Alp_{G_{i}}^{ess}(φ_{i}) ≥ Alp_{∏ G_{i}}^{ess}(∏ φ_{i}) is proved just as in the open case of the preceding lemma.

For the reverse inequality, suppose that we have a chain for ∏_{i} φ_{i} in terms of θ^{j} ∈ Aut_{∏_{i} G_{i}}(Q^{j}), 1 ≤ j ≤ N. By Corollary 7.7 below, Q^{j} is of the form E_{i(j)} × ∏_{i≠i(j)} S_{i} for some i(j) and some open essential subgroup E_{i(j)} of S_{i(j)}. Thus θ^{j} = ∏_{i} θ_{i}^{j} with θ_{i}^{j} ∈ Aut_{G_{i}}(S_{i}) if i ≠ i(j) and θ_{i}^{j} ∈ Aut_{G_{i}}(E_{i}) if i = i(j).

For fixed i, the θ_{i}^{j} form an Alperin chain for φ_{i}. By our convention that automorphisms of the Sylow pro-p subgroup do not count towards the length, each j only counts towards the length of the chain for one i. The inequality follows. □

We say that p divides the order of a profinite group if a Sylow pro-p subgroup is non-trivial.

Lemma 7.5. Let A and B be profinite groups and let P be a pro-p subgroup of A × B that is either finite or is open in a Sylow pro-p subgroup of A × B.

(1) If P is radical then P = Q × R, where Q is radical in A and R is radical in B.

(2) If P is essential then P = Q × R, where Q is essential in A and R is essential in B. Furthermore, p does not divide both |N_{A}(Q)/Q| and |N_{B}(R)/R|.

(3) If P is open in a Sylow pro-p subgroup of A × B and is also essential, then P = Q × R, where either Q is essential in A and R is a Sylow pro-p subgroup of B, or R is essential in B and Q is a Sylow pro-p subgroup of A.

Proof. Denote the projections by π_{A} and π_{B}. We have P ≤ π_{A}P × π_{B}P; the inclusion is open, by the hypotheses on P.

If P = π_{A}P × π_{B}P then it is easy to verify that π_{A}P is radical in A and π_{B}P is radical in B. If P ≠ π_{A}P × π_{B}P then P ≠ N_{π_{A}P ×π_{B}P} (P) ≤ O_{p}(N_{A×B}(P)), contradicting the assumption

that P is radical.

For the second part, we know from the first part that P = Q × R. Since Q is radical, it is also centric in A; thus p divides |N_{A}(Q)/Q| if and only if it divides |N_{A}(Q)/C_{A}(Q)Q|. The same applies to R. The result follows from the next lemma applied to N_{A×B}(Q×R)/C_{A×B}(Q× R)(Q × R) ≅ N_{A}(Q)/C_{A}(Q)Q × N_{B}(R)/C_{B}(R)R.

For the third part, suppose that p does not divide |N_{A}(Q)/Q| and let S be a Sylow pro-p subgroup of A that contains Q. Then p does not divide |N_{S}(Q)/Q|, so N_{S}(Q) ≤ Q and thus S ≤ Q, since Q is open in S. □

Lemma 7.6. The poset S_{p}(A × B) is connected whenever both |A| and |B| are divisible by p. If |A|, say, is not divisible by p, then S_{p}(A × B) is connected if and only if S_{p}(B) is connected.

Proof. If both |A| and |B| are divisible by p, choose non-trivial pro-p subgroups X ≤ A and Y ≤ B. We will show that any non-trivial pro-p subgroup P ≤ A × B is connected to X × 1.

If π_{B}P ≠ 1 then we can use the chain P ≤ π_{A}P × π_{B}P ≥ 1 × π_{B}P ≤ X × π_{B}P ≥ X × 1.

If π_{A}P = 1 then P = π_{A}P × 1 ≤ π_{A}P × Y ≥ 1 × Y ≤ X × Y ≥ X × 1.

If |A| is not divisible by p then S_{p}(A × B) ≡ S_{p}(B), and the second part follows. □

Corollary 7.7. An open essential subgroup of ∏_{i} S_{i} ≤ ∏_{i} G_{i} is of the form E_{j} × ∏_{i≠j} S_{i} for some j and an open essential subgroup E_{j} of S_{j}.

Proof. Let P be the open essential subgroup. For each j, apply Lemma 7.5 to G_{j} × ∏_{i≠j} G_{i} to see that the projection of P to its j-coordinate is both a factor of P and essential. Thus P = ∏_{i} P_{i}, with the P_{i} essential. If P_{k}, say, is not a Sylow pro-p subgroup of S_{k} and j ≠ k, then apply Lemma 7.5 to (G_{j} × G_{k}) × ∏_{i≠j,k} G_{i} to see that P_{j} × P_{k} is essential and hence, by the lemma again, P_{j} = S_{j}. □

Theorem 7.8. Let G_{i} be an infinite collection of finite groups, each with p-subgroups P_{i}, P′_{i} ≤ G_{i} and φ_{i} ∈ Iso_{G_{i}}(P_{i}, P′_{i}), and suppose that Alp_{G_{i}}(φ_{i}) ≥ 1. Then Alp_{∏ G_{i}}(∏ φ_{i}) = Alp_{∏ G_{i}}^{ess}(∏ φ_{i}) = ∞.

Proof. We must have Alp_{G_{i}}^{ess}(φ_{i}) ≥ 1, so Alp_{∏ G_{i}}^{ess}(∏ φ_{i}) ≥ ∑ 1 = ∞, by Proposition 7.4. The same must be true for Alp_{∏ G_{i}}(∏ φ_{i}), by Proposition 7.2. □

This provides another proof of the fact that it is necessary to allow infinite chains in Theorem 6.17. A direct approach to showing that infinite chains are necessary, even for Alp^{clo}, would be to find a sequence of finite groups G_{i} and P, P′ ≤ S_{i}, φ_{i} ∈ Iso(P, P′) such that Alp_{G_{i}}(φ_{i}) → ∞ as i → ∞, and then apply Lemma 7.3 to their product. It seems to be generally believed that such a sequence exists; however, it seems difficult to find a lower bound for Alp_{G}(φ) in examples.

Notice that if such a sequence does not exist at the prime p then there exists an integer n such that, at the prime p, Alp_{G}(φ) ≤ n for all finite groups G and all morphisms φ. In this case we will say that Alp is uniformly bounded at p.

Sometimes it is convenient to consider all the morphisms in a profusion system F as a set, Mor(F) (morphisms with different domains or codomains are considered to be different).

Lemma 7.9. If F ≅ \underleftarrow{lim}_{i} F_{i} then Mor(F) ≅ \underleftarrow{lim}_{i} Mor(F_{i}).

Proof. There is clearly a natural injective map Mor(F) → \underleftarrow{lim}_{i} Mor(F_{i}). An element of the

codomain is a compatible collection of morphisms φ : P_{i} → Q_{i}; these combine to define a morphism φ : \underleftarrow{lim}_{i} P_{i} → \underleftarrow{lim}_{i} Q_{i}. □

We will now regard Mor(F) as a profinite set in this way.

The ordered set of the morphisms in an Alperin chain for φ of length at most n can be considered as an element of Mor(F)^{n} (shorter chains can be extended by the identity map). This element completely determines the Alperin chain.

Proposition 7.10. If F = \underleftarrow{lim} F_{i}, then Alp_{F}^{clo}(φ) ≤ sup\{Alp_{F_{i}}(F_{i}(φ))\}.

Proof. If sup{Alp_{F_{i}}(F_{i}(φ_{i}))} = ∞ there is nothing to prove, so assume that sup{Alp_{F f_{i}}(F_{i}(φ_{i}))} = n.

Let C_{i} be the set of all Alperin chains in F_{i} for F_{i}(φ) of length n, considered as a subset of Mor(F_{i})^{n}. The sets C_{i} are non-empty, by hypothesis, and if j ≥ i there is a natural map C_{j} → C_{i}. Thus we can form \underleftarrow{lim}_{i} C_{i} and it is non-empty; an element of it yields an Alperin

chain in closed subgroups in F of length n for φ. □

Corollary 7.11. It is always possible to find a finite Alperin chain in closed subgroups for any profinite group and any morphism at the prime p, if and only if Alp is uniformly bounded at p. If there is a uniform bound then it also applies to profinite groups.

Proof. This is a consequence of Proposition 7.10 and Lemma 7.3. □

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LAMFA-CNRS UMR 7352, Univ. de Picardie, 33, Rue Saint-Leu, 80039 Amiens CX 1, France

  E-mail address: radu.stancu@u-picardie.fr

School of Mathematics, University of Manchester, Manchester M13 9PL, United Kingdom

  E-mail address: Peter.Symonds@manchester.ac.uk