CONTROL OF THE FUSION IN FULL FROBENIUS SYSTEMS

RADU STANCU^{∗}

Abstract. Let P be a p-group and F a full Frobenius system over P. The aim of this paper is to give necessary and sufficient conditions on a subgroup Q of P for the normalizer N_{F} (Q) to be F itself. This generalizes a result of Gilotti and Serena on finite groups.

1. Introduction

Let p be a prime number, G a finite group and F_{p}(G) the Frobenius category of G related to p. It’s a category whose objects are the nontrivial p-subgroups of G and whose morphisms are the morphisms given by conjugation by the elements of G. This category contains the p-local information of G. One can prove that the Frobenius category is equivalent to its full subcategory whose objects are the subgroups of a Sylow p-subgroup of G.

For founding new information on the character table of simple finite groups, Richard Brauer, in the ’40s, had the original idea to study group representations over fields of characteristic p, where p is a prime number dividing the order of G. This was a successful approach which gave a lot of deep results. This new mathematical tool was cold modular representation theory. One of the aims of this theory is to obtain a classification of the irreducible representations over a field k of characteristic p. As in the case of finite groups, we can organize all the information in a category called the Brauer category. Its definition will be given in the next section.

In the ’90s, Puig, convinced of the richness of the p-local structures, as the Frobenius category or the Brauer category, gave an axiomatic description. The notes in which Puig introduced these new concepts have remained in the handwritten state, since recently, when the author wrote a paper [Pu2]. The new concepts that Puig introduced are the full Frobenius systems over a p-group P.

In a recent preprint, Broto, Levi and Oliver [BLO] identified and studied a certain class of spaces which in many ways behave like p-completeed classifying spaces of finite groups. They showed that these spaces occur as the “classifying spaces” of full Frobenius systems. In fact, in the paper, they talk about saturated fusion system and the definition they gave is slightly different from those of Puig for full Frobenius systems, but they proved that the two definition are equivalent. In our paper we’ll adopt the Puig’s definition.

When P is a Sylow p-subgroup of a finite group G and Q a subgroup of P, Serena and Gilotti [GS] found necessary and sufficient conditions on Q for the normalizer N_{G}(Q) to control the p-fusion in G. Recall that a subgroup H of G is said to control p-fusion in G if the canonical inclusion of H in G induces an equivalence of categories between F_{p}(H) and F_{p}(G). Let F be a full Frobenius system over a p-group P. As a generalization of Gilotti and Serena’s result, in this paper we will give necessary and sufficient conditions on a subgroup Q of P for the normalizer N_{F}(Q) to be F itself.

Date: 20th September 2002.

∗ This work is a part of the Ph.D. thesis at the University of Lausanne, under the supervision of Prof. Jacques Thévenaz.

The paper is organized as follows. The second section is dedicated to the definition and some properties of the Brauer category. In Section 3 we will present the notion of full Frobenius system and we will prove that the Brauer subcategory under a maximal Brauer pair is a full Frobenius system. Section 4 gives a generalization of a theorem of Alperin to full Frobenius systems, using essential p-subgroups. In Section 5 we give properties on a p-group for not to be realized as essential p-subgroup in a finite group and we find examples of such p-groups. The main result of this paper lies in Section 7 where we give a generalization of a theorem of Gilotti and Serena. Using this theorem, we find in Section 8 families resistant of p-groups, which are p-groups P such that, for any full Frobenius system F over P, the normalizer N_{F}(Q) is F itself.

2. Brauer category

To give a representation (irreducible) of a group G over a field k of characteristic p is equivalent to give a (simple) module over the group algebra kG. We recall that a group algebra over a field k is a vector space over k having as base the elements of G with the multiplication given by the multiplication in G.

The group algebra kG can be decomposed in direct product of indecomposable

algebras kG ≃ ∏_{j=1}^{n} B_{j}, unique up to algebra isomorphism. To any B_{j} it corresponds

an idempotent b_{j} from the centre Z(kG) such that b_{j} is projected over 1 in B_{j} and over zero in the others factors. In other words, we have an orthogonal decomposition 1 = ∑_{j=1}^{n} b_{j}. We have the isomorphism B_{j} ≃ kGb_{j}. We say that kGb_{j} is a block algebra and b_{j} is a block idempotent, or simply a block. The blocks are very important, as all the representation theory over k is naturally decomposing over the block algebras.

If a block algebra kGb is isomorphic to a matrix algebra we say that b is a block of defect zero. And this happens for all the blocks of kG if p doesn’t divide the order of G. When p divides the order of G, we have to consider blocks with higher complexity degrees. The first invariant that measures this complexity is the defect group of the block. It is a p-subgroup of G, unique up to conjugation, whose definition will be given latter. The defect group of a block is trivial if and only if the block is of defect zero. At the other extremity is the case where the defect group of a block b is a Sylow p-subgroup of G. Let b_{0} be the unique block that contains the trivial representation of dimension 1. In this case the defect group of b_{0} is a Sylow p-subgroup of G. We call b_{0} the principal block of kG.

We canonically associate to any block of kG families of p-subgroups of G and we study the morphisms given by the elements of G between this p-subgroups of G. For this, we introduce the concept of Brauer pair. The fundaments of all the theory are in an article of Alperin and Broué of 1979 [AB]. A very good reference for Brauer pair is the Thevenaz’s book [Th].

Definition 2.1. Let k be a field of characteristic p and G a finite group. A Brauer pair of kG is a pair (P, e), where P is a p-subgroup of G and e is a block of kC_{G}(P), which means a primitive idempotent of Z(kC_{G}(P)).

The group G acts by conjugation on the set of Brauer pairs. Indeed, if (P, e) is a Brauer pair of kG, then, for all g ∈ G, ^{g}e is a block of kC_{G}(^{g}P) = k^{g}(C_{G}(P)). We can then define ^{g}(P, e) := (^{g}P, ^{g}e). We note N_{G}(P, e) the normalizer of the pair (P, e). It’s the set of elements g ∈ N_{G}(P) such that e = ^{g}e. Moreover, it’s a subgroup of N_{G}(P) containing P C_{G}(P). Definition 2.2. The Brauer category of kG is a category whose objets are the Brauer pairs of kG and whose morphisms are the morphisms between the Brauer pairs given by conjugation by the elements of G.

On the set of Brauer pairs we put a partial order. For doing this, we have to define the Brauer morphism. We note (kG)^{P} the set of elements of kG fixed by the action of P. Definition 2.3. Let P be a p-subgroup of G. The Brauer morphism related to P is a k-linear surjective map

br_{P} : (kG)^{P} ⟶ kC_{G}(P) ,

sending every element of C_{G}(P) on itself and to 0 every sum of elements of a nontrivial P -conjugacy class. Remark 2.4. The Brauer morphism is well defined as the elements of C_{G}(P) together with the sums of elements of a nontrivial P -conjugacy class, for all the nontrivial P -conjugacy classes, represent a k-basis of (kG)^{P}.

The order relation ≤ on the Brauer pairs is generated by the relations between two Brauer pairs (Q, f) and (P, e), when Q ⊴ P. In this case, we say that (Q, f) ≤ (P, e) if f ∈ (kC_{G}(Q))^{P} and e occurs in the decomposition of br_{P} (f) in kC_{G}(P) and we write (Q, f) ⊴ (P, e). In the general case, for a subgroup Q of P, we say that (Q, f) ≤ (P, e) if there exists a subnormal series Q = Q_{0} ⊴ Q_{1} ⊴ … ⊴ Q_{n} = P and some blocks f_{i} of kC_{G}(Q_{i}) for i = 0, …, n such that f_{0} = f, f_{n} = e and (Q_{i−1}, f_{i−1}) ⊴ (Q_{i}, f_{i}) for all i = 1, …, n. When the order relation is strict, we write <.

The next proposition clarifies the structure of the poset of Brauer pairs of kG. Proposition 2.5 ([Th], Cor. 40.9, Cor. 40.11). Let (P, e) be a Brauer pair of kG and Q a subgroup of P.

a) There exists a unique block f of kC_{G}(Q) such that (Q, f) ≤ (P, e).

b) The map Q ↦ (Q, f) is an isomorphism between the poset of subgroups of (P, e) and the poset of Brauer pairs contained in P (ordered by inclusion).

c) N_{P} (Q, f) = N_{P} (Q).

So, if (Q, f) < (P, e) and g is the unique block of kC_{G}(N_{P} (Q)) such that (N_{P} (Q), g) ≤ (P, e) then (Q, f) ⊲ (N_{P} (Q), g) ≤ (P, e) and (N_{P} (Q), g) ≠ (Q, f). Definition 2.6. Let b be a block of kG. We say that the Brauer pair (P, e) is associated to b if (1, b) ≤ (P, e).

By the last proposition, there exists a unique block b of kG with this property, so the notion is well defined. The order relation on the set of Brauer pairs, allows us to find, for any block b of kG, maximal pairs associated to b. The first arguments of these pairs are the defect groups of the block b. But, we already said that the defect group is unique up to isomorphism. We have even more. We have the equivalent for the Sylow’s theorem. Proposition 2.7 ([Th], Prop. 40.13). The maximal Brauer pairs of G associated to b are conjugated in G.

This proposition has as immediate consequence the fact that all the defect groups of b are conjugated by elements of G.

Let Q be a p-subgroup of G. One useful property of p-groups is the fact that all the maximal p-subgroups normalizing Q are conjugated under N_{G}(Q). This property remain valid for the Brauer pairs. Proposition 2.8 ([Th], Prop. 40.15). Let (Q, f) be a Brauer pair of kG. Then a) (Q, f) is also a Brauer pair for kN_{G}(Q, f). Moreover the set of Brauer pairs of G normalizing (Q, f) coincide with the set of Brauer pairs of N_{G}(Q, f) containing (Q, f).

b) all the Brauer pairs of G, maximal with the respect of normalizing (Q, f), are conjugated in N_{G}(Q, f).

So, in the in the same way that we can consider the Frobenius category of N_{G}(Q) as a subcategory of the Frobenius category of G, we can also consider the Brauer category of kN_{G}(Q, f) as a subcategory of the Brauer category of kG.

Let (Q, f) be a Brauer pair of kG and let N ≤ G. If QC_{G}(Q) ≤ N, then (Q, f) is also a Brauer pair of kN. Indeed, we have C_{N}(Q) = C_{G}(Q) and Q is a p-subgroup of N. Definition 2.9. Let (P, e) be a maximal pair of kG and (Q, f) ≤ (P, e). Note N := N_{P} (Q) and let f_{N} be the unique block of kC_{G}(N) such that (N, f_{N}) ≤ (P, e). We say that the pair (Q, f) is completeely normalized in (P, e) with respect to kG if (N, f_{N}) is a maximal Brauer pair of kN_{G}(Q, f).

3. Full Frobenius systems

All the results and the proofs in this section are due to Puig [Pu2]. Let us start with a more general definition. Definition 3.1. A category C over P is a category whose objets are the subgroups of P and whose set of morphisms between the subgroups Q and R of P, is a set C(Q, R) of injective group morphisms from Q to R, that contains the morphisms by conjugation by the elements of P.

A full Frobenius system F over P is a category over P whose morphisms satisfy the three following axioms. A1 If Q and R are subgroups of P, then, for any morphism ϕ ∈ F(Q, R) there exists a morphism ψ ∈ F(ϕ(Q), Q), such that ψϕ(u) = u for all u ∈ Q.

If Q ≤ R, the inclusion of Q in R is also in F(Q, R) as it is induced by conjugation by 1. In particular, the morphisms in a full Frobenius system F over P are determined by the sets F(Q, P), where Q runs over all the subgroups of P. An other important remark is the fact that if we have a morphism ϕ ∈ F(Q, R) and T ≤ Q then the restriction of ϕ to T is a morphism of F(T, R). Indeed, let ι ∈ F(T, Q) be the inclusion of T in Q. Then ϕι is a morphism in F(T, R). It’s the restriction of ϕ to T.

Before writing down Axiom 2 we give some notations. Let Q be a subgroup of P. We denote by F(Q) := F(Q, Q) the set of automorphisms of Q which belong to F. By Axiom A1, F(Q) is a group. Let R ≤ P and denote by I_{R}(Q) := N_{R}(Q)/C_{R}(Q) the set of morphisms of Q, given by the conjugation by the elements of R. As all the morphisms by conjugation by the elements of P are in F, it is straight forward that I_{R}(Q) ≤ F(Q). Finally, put I(Q) := I_{Q}(Q), which is the group of inner automorphisms of Q. A2 The group I(P), of inner automorphisms of P, is a Sylow p-subgroup of F(P).

If G is a finite group and S a Sylow p-subgroup of G, then we say that a subgroup Q of S is fully normalized, respectively fully centralized in S if N_{S}(Q), respectively C_{S}(Q) is a Sylow p-subgroup of N_{G}(Q), respectively C_{G}(Q). We give now a definition equivalent to the notions of fully normalized and fully centralized in a p-group for the full Frobenius systems. For doing this in the same time for the two notions, we denote by N_{P}^{K}(Q) the K-normalizer of Q in P, which is the inverse image of K in N_{P} (Q), where K is a subgroup of Aut(Q). For any morphism ϕ ∈ F(Q·N_{P}^{K}(Q), P), we have ϕ(N_{P}^{K}(Q)) ≤ N_{P}^{^{ϕ}K}(ϕ(Q)), where ^{ϕ}K = {ϕθϕ^{−1}|θ ∈ K} is a subgroup of Aut(ϕ(Q)).

Definition 3.2. We say that Q is K-fully normalized in F if, for any morphism ϕ ∈ F(Q · N_{P}^{K}(Q), P), we have ϕ(N_{P}^{K}(Q)) = N_{P}^{^{ϕ}K}(ϕ(Q)).

If K = 1 or K = Aut(Q), we say that Q is fully centralized or, respectively, fully normalized in F. Moreover, we have the next property.

Lemma 3.3 ([Pu2], 2.3.2). Let R be a subgroup of Q · N_{P}^{K}(Q) containing Q and ϕ ∈ F(R, P), satisfying that, for any other morphism ϕ′ ∈ F(R, P), we have

|N_{P}^{^{ϕ′} K}(ϕ′(Q))| ≤ |N_{P}^{^{ϕ}K}(ϕ(Q))| . (∗)

Then ϕ(Q) is ^{ϕ}K-fully normalized in F.

Proof. Note Q′ := ϕ(Q) and K′ := ^{ϕ}K. With these notations, for any morphism ψ ∈ F(Q′ · N_{P}^{K′}(Q′), P), we have a morphism ϕ′ ∈ F(R, P) mapping v ∈ R to ψ(ϕ(v)). Moreover, note Q′′ := ϕ′(Q) and K′′ := ^{ϕ′}K. By (∗), we obtain that |N_{P}^{K′′}(Q′′)| ≤ |N_{P}^{K′}(Q′)|. But we always have ψ(N_{P}^{K′}(Q′)) ≤ N_{P}^{K′′}(Q′′). As

|ψ(N_{P}^{K′}(Q′))| = |N_{P}^{K′}(Q′)| ≥ |N_{P}^{K′′}(Q′′)| ,

we conclude that ψ(N_{P}^{K′}(Q′)) = N_{P}^{K′′}(Q′′). So Q′ is K′-fully normalized. □

In particular, there always exists a morphism ϕ ∈ F(Q, P) such that ϕ(Q) is ϕK-fully normalized in F.

We can give now the axiom allowing us to have the equivalent of Sylow’s theorem, in the field of full Frobenius systems. A3 For any subgroup Q of P, for any subgroup K of Aut(Q) and for any morphism ϕ ∈ F(Q, P), such that ϕ(Q) is ^{ϕ}K-fully normalized in P, there exists a morphism ψ ∈ F(Q · N_{P}^{K}(Q), P) and χ ∈ K, such that ψ(u) = ϕ(χ(u)) for all u ∈ Q.

The approach between Axiom A3 and the Sylow’s theorem is a little bit hidden. We see that being K-fully normalized for a subgroup Q of P is equivalent to say that Q is the best placed between all its images by morphisms in F(Q, P). The Sylow’s theorem is equivalent to say that we can bring any p-subgroup of a group G, by conjugation by an element of G, inside a given Sylow p-subgroup of G. In the full Frobenius systems, Axiom A3 allow us to bring the K-normalizer in P of a image of Q by a morphism of F, inside N_{P}^{K}(Q), if Q is K-fully normalized in F.

By the way, the following property, which will be very useful in the following, shows us that, behind the fact that a group is K-fully normalized is hidden, somewhere, a Sylow p-subgroup.

Proposition 3.4 ([Pu2], Prop. 2.7). Let Q be a subgroup of P and K a subgroup of Aut(Q). Then Q is K-fully normalized in F if and only if Q is fully centralized in F and K ∩ I_{P} (Q) is a Sylow p-subgroup of K ∩ F(Q).

Proof. If Q is fully centralized in F and ϕ ∈ F(Q·N_{P}^{K}(Q), P), then the restriction of ϕ to Q · C_{P} (Q) is in F(Q · C_{P} (Q), P), so we have ϕ(C_{P} (Q)) = C_{P} (ϕ(Q)). If, moreover, K ∩ I_{P} (Q) = N_{P}^{K}(Q)/C_{P} (Q) is a Sylow p-subgroup of K ∩ F(Q), we obtain

|N_{P}^{K}(Q)| = |C_{P} (Q)| · |K ∩ I_{P} (Q)| =

= |C_{P} (Q)| · |K ∩ F(Q)|_{p} = |C_{P} (ϕ(Q))| · |^{ϕ}K ∩ F(ϕ(Q))|_{p} ≥

≥ |C_{P} (ϕ(Q))| · |^{ϕ}K ∩ I_{P} (ϕ(Q))| = |N_{P}^{^{ϕ}K}(ϕ(Q))| .

where by |G|_{p} we denote the order of a Sylow p-subgroup of G. But we always have that ϕ(N_{P}^{K}(Q)) ≤ N_{P}^{^{ϕ}K}(ϕ(Q)), so |N_{P}^{K}(Q)| = |ϕ(N_{P}^{K}(Q))| ≤ |N_{P}^{^{ϕ}K}(ϕ(Q))|. In this way we obtain ϕ(N_{P}^{K}(Q)) = N_{P}^{^{ϕ}K}(ϕ(Q)) and so Q is K-fully normalized in F.

Suppose now that Q is K-fully normalized in F. First, we’ll prove that Q is fully centralized in F. Let ϕ ∈ F(Q · C_{P} (Q), P). By Axiom A1, there exists ϕ^{−1} ∈ F(ϕ(Q), Q) such that ϕ^{−1}ϕ(u) = u for all u ∈ Q. Now, the group Q is K-fully normalized, so, by Axiom A3, applied to ϕ^{−1}, it follows that there exists ρ ∈ F(ϕ(Q) · N_{P}^{^{ϕ}K}(ϕ(Q)), P) and χ ∈ ^{ϕ}K such that ρ(v) = ϕ^{−1}(χ(v)) for all v ∈ ϕ(Q). Moreover, we have ρ(C_{P} (ϕ(Q))) ≤ C_{P} (ρϕ(Q)) = C_{P} (Q), as ρ(ϕ(u)) = ϕ^{−1}(χ(ϕ(u)) ∈ Q for all u ∈ Q. This gives |C_{P} (ϕ(Q))| ≤ |C_{P} (Q)|, so ϕ(C_{P} (Q)) = C_{P} (ϕ(Q)). As this result is true for all ϕ ∈ F(Q·C_{P} (Q), P), we obtain that Q is fully centralized in F.

We show now, by induction on the index |P : Q|, that K ∩ I_{P} (Q) is a Sylow p-subgroup of K ∩ F(Q). For P = Q, by Axiom A2 the group I(P) is a Sylow p-subgroup of F(P) and, as I(P) is a normal subgroup of Aut(P), we obtain that K ∩ I(P) is a Sylow p-subgroup of K ∩ F(P) for all K ≤ Aut(P).

Let now Q < P and let us prove our assertion first in the case where K = Aut(Q), so, when Q is fully normalized. Put R := N_{P} (Q) and let J be the subgroup of Aut(R) stabilizing Q. It is straight forward that N_{P}^{J}(R) = N_{P} (Q) ∩ N_{P} (R) = R. Moreover, R is J-fully normalized in F. Indeed, as Q is fully normalized, for all ϕ ∈ F(R, P), we have ϕ(R) = N_{P} (Φ(Q)). Also, ^{ϕ}J is the subgroup of Aut(ϕ(R)) stabilizing ϕ(Q) so N_{P}^{^{ϕ}J}(ϕ(R)) = N_{P} (ϕ(Q)) ∩ N_{P} (ϕ(R)) = ϕ(R). We obtain that, for all ϕ ∈ F(R · N_{P}^{J}(R), P) = F(R, P), we have N_{P}^{^{ϕ}J}(ϕ(R)) = ϕ(R) = ϕ(N_{P}^{J}(R)), so R is J-fully normalized in F. As Q is a proper subgroup of R, by induction hypothesis, we obtain that J ∩ I_{P} (R) = I_{R}(R) is a Sylow p-subgroup of J ∩ F(R).

We have clearly I_{P} (Q) = I_{R}(Q). For showing that I_{R}(Q) is a Sylow p-subgroup of F(Q), it suffices to prove that I_{R}(Q) is a Sylow p-subgroup of N_{F(Q)}(I_{R}(Q)), because, if a p-subgroup of a finite group G is a Sylow p-subgroup of its normalizer, then it is a Sylow p-subgroup of G.

Let us show now that I_{R}(Q) is a Sylow p-subgroup of N_{F(Q)}(I_{R}(Q)). For this, we’ll show that any morphism ϕ ∈ N_{F(Q)}(I_{R}(Q)) can be lifted in J ∩ F(R), which means that the group morphism f : J ∩ F(R) ⟶ N_{F(Q)}(I_{R}(Q)), given by the restriction to Q of the morphisms in J ∩ F(R), is surjective. As, moreover, f(I(R)) = I_{R}(Q), it’s following that I_{R}(Q) is a Sylow p-subgroup of N_{F(Q)}(I_{R}(Q)). We used here the fact that the image of a Sylow p-subgroup of J ∩ F(R) by the surjective morphism f is a Sylow p-subgroup of N_{F(Q)}(I_{R}(Q)).

We’ll show now that any morphism ϕ ∈ N_{F(Q)}(I_{R}(Q)) can be lifted in J ∩F(R). Take ϕ ∈ N_{F(Q)}(I_{R}(Q)) so we have ϕ(Q) = Q and ^{ϕ}(I_{R}(Q)) = I_{R}(Q). Moreover, R contains Q · C_{P} (Q), so N_{P}^{I_{R}(Q)}(Q) = R. We have already seen that Q is fully centralized in F. We have also that Q is I_{R}(Q)-fully normalized in F. Indeed, for any morphism η ∈ F(Q · N_{P}^{I_{R}(Q)}(Q), P) = F(R, P), we have C_{P} (η(Q)) = η(C_{P} (Q)), as Q is fully normalized and so C_{P} (η(Q)) ≤ η(R). This gives that η(Q)C_{P} (η(Q)) ≤ η(R) so, N_{P}^{I_{η(R)}(η(Q))}(η(Q)) = η(R) = η(N_{P}^{I_{R}(Q)}(Q)). It follows that Q is I_{R}(Q)-fully normalized in F. By Axiom A3 applied to the morphism ϕ, there exists ψ ∈ F(Q · N_{P}^{I_{R}(Q)}(Q), P) = F(R, P) and χ ∈ I_{R}(Q) such that ψ(u) = ϕχ(u), for all u ∈ Q. Let χ̃ be a preimage of χ in I_{R}(R). We have then that ψχ^{−1} is a preimage of ϕ in F(R, P). In fact ψχ^{−1} ∈ F(R). Indeed, ψ and χ̃ normalize I_{R}(Q) and ψχ^{−1}(C_{P} (Q)) = C_{P} (ψχ^{−1}(Q))) = C_{P} (Q). So, R/C_{P} (Q) = I_{R}(Q) = ^{ψχ^{−1}}I_{R}(Q) = ψχ^{−1}(R)/C_{P} (Q) which gives that that R = ψχ^{−1}(R) and, so, ψχ^{−1} ∈ F(R). This finishes the the proof in the case where K = Aut(Q).

In the general case, let θ ∈ F(Q, P) such that θ(Q) is fully normalized in F. We have seen that such a morphism θ always exists. Now, θ(Q) is fully normalized, so by the proof in the case K = Aut(Q), applied to θ(Q) in the place of Q, we obtain that I_{P} (θ(Q)) is a Sylow p-subgroup of F(θ(Q)). As ^{θ}(K ∩ F(Q)) is a subgroup of ^{θ}(F(Q)) = F(θ(Q)), there exists ρ ∈ F(θ(Q)) such that a Sylow p-subgroup of ρθ(K ∩ F(Q)) is contained in I_{P} (θ(Q)). So ^{ρθ}K ∩ I_{P} (θ(Q)) is a Sylow p-subgroup of ^{ρθ}(K ∩ F(Q)).

We take now θ′ := ρθ, Q′ := θ(Q), K′ := ^{θ′}K and we apply Axiom A3 to θ^{′−1}, as Q is K-fully normalized. So there exists ψ ∈ F(Q′ · N_{P}^{K′}(Q′), P) and χ′ ∈ K′ such that ψ(u) = θ^{′−1}(χ′(u)) for all u ∈ Q′. Let χ := ^{θ^{′−1}}χ′, which is an element of K. We have then ψ^{−1} = θ′χ^{−1} which implies that ^{ψ^{−1}}K = K′. But ψ(N_{P}^{K′}(Q′)) ≤ N_{P}^{K}(Q) and, as Q′ is fully centralized, we obtain that ψ(C_{P} (Q′)) = C_{P} (Q), so ψ(K′ ∩ I_{P} (Q′)) ≤ K ∩ I_{P} (Q). Moreover, K′ ∩ I_{P} (Q′) is a Sylow p-subgroup of K′ ∩ F(Q′) and ϕ(K′ ∩ F(Q′)) = K ∩ F(Q). We deduce that K ∩ I_{P} (Q) is a Sylow p-subgroup of K ∩ F(Q). □

Let Q a subgroup of P and K a subgroup of Aut(Q). We define, based on F, a category N_{F}^{K}(Q) over N_{P}^{K}(Q), whose objets are the subgroups of N_{P}^{K}(Q) and whose morphisms between two subgroups R and T of N_{P}^{K}(Q) are the morphisms ϕ ∈ F(R, T), satisfying that there exists a morphism ψ ∈ F(Q·R, Q·T) and χ ∈ K such that χ(u) = ψ(u) for all u ∈ Q and ψ(v) = ϕ(v) for all v ∈ R; in other words, all the morphisms which can be extended to Q · R and such that this extension acts over Q by a morphism of K. In the case where K = Aut(Q), we denote this category, simply, N_{F}(Q).

Proposition 3.5 ([Pu2], Prop. 2.8). Let F be a full Frobenius system over P. If Q ≤ P is K-fully normalized in F, then, N_{F}^{K}(Q) is a full Frobenius system over N_{P} (Q).

For the proof of this proposition we’ll need the next technical lemma:

Lemma 3.6 ([Pu2], Lemma 2.9). Let F be a full Frobenius system over P, Q a subgroup of P and K a subgroup of Aut(Q). Let R be a subgroup of N_{P}^{K}(Q), L a subgroup of Aut(R) and M the subgroup of Aut(Q · R) stabilizing Q and R and acting on them by the morphisms in K, respectively L.

Then N_{P}^{M}(Q · R) = N_{P}^{K}(Q) ∩ N_{P}^{L}(R) and, if R is fully normalized in N_{F}^{K}(Q), the group Q · R is M-fully normalized in F.

Proof. The first equality is straight forward as we have u ∈ N_{P}^{M}(Q · R) if and only if u ∈ P stabilize Q and R and u acts on Q, respectively R by elements of K, respectively L which is equivalent with u ∈ N_{P}^{K}(Q), respectively u ∈ N_{P}^{L}(R), or, in other words, with u ∈ N_{P}^{K}(Q) ∩ N_{P}^{L}(R). We remark that this is independent on the existence of F.

For the second part of the lemma, let’s put T := Q · R, P′ := N_{P}^{K}(Q) and F′ := N_{F}^{K}(Q). Let’s consider ϕ ∈ F(T · N_{P}^{M}(T), P). We want to show that ϕ(N_{P}^{M}(Q)) = N_{P}^{^{ϕ}M}(ϕ(T)). Note Q′ := ϕ(Q), T′ := ϕ(T) and K′ := ^{ϕ}K. Now, we have ϕ|_{Q} ∈ F(Q, P), so, by Axiom A3 applied to ϕ^{−1}, using that Q is K-fully normalized in F, there exists ψ ∈ F(Q′ · N_{P}^{K′}(Q′), P) and χ′ ∈ F(Q′) such that ψ(u) = ϕ^{−1}χ′(u) for all u ∈ Q′. In particular, ψ(N_{P}^{K′}(Q)′) ≤ N_{P}^{K}(Q). Now, we have ϕ(T ·N_{P}^{M}(T)) ≤ ϕ(T)·N_{P}^{^{ϕ}M}(T′) ≤ Q′ ·N_{P}^{K′}(Q′) so, if we note η := ψϕ, we obtain η ∈ F(T ·N_{P}^{M}(T), QP′) = F(Q·R·N_{P}^{M}(T), QP′). Moreover η(R·N_{P}^{M}(T)) ≤ P′, which implies that η|_{R·N_{P}^{M} (T)} ∈ F′(R · N_{P}^{M}(T), P′), by the very definition of F′.

By a simple computation, we have N_{P}^{M}(T) = N_{P}^{K}(Q) ∩ N_{P}^{L}(R) = N_{P′}^{L}(R) (1) and, using the fact that R is L-fully normalized in F′, we obtain η(N_{P′}^{L}(R)) = N_{P′}^{^{η}L}(η(R)) (2). Moreover, we remark that η(Q) = Q and ^{η}K = K and ^{η}M is a subgroup of the automorphisms group of Q · η(R) stabilizing Q and η(R) and acting on Q, respectively η(R) by morphisms of K, respectively η(L). So, N_{P}^{^{η}M}(Q · η(R)) = N_{P}^{K}(Q) ∩ N_{P}^{^{η}L}(η(R)) = N_{P′}^{^{η}L}(η(R)) (3), by the first part of the lemma. From (1), (2) and (3) we deduce that η(N_{P}^{M}(T)) = N_{P}^{^{η}M}(Q · η(R)). So, we have

ψ(ϕ(N_{P}^{M}(T))) = η(N_{P}^{M}(T)) = N_{P}^{^{η}M}(Q · η(R)) ≥ ψ(N_{P}^{^{ϕ}M}(ϕ(T)))

We also have that |ϕ(N_{P}^{M}(T))| ≥ |N_{P}^{^{ϕ}M}(ϕ(T))|. Now, as ϕ(N_{P}^{M}(T)) ≤ N_{P}^{^{ϕ}M}(ϕ(T)), we obtain that ϕ(N_{P}^{M}(T)) = N_{P}^{^{ϕ}M}(ϕ(T)).

This is true for all ϕ ∈ F(T · N_{P}^{M}(T), P), so T is M-fully centralized in F. □

Proof of Proposition 3.5. As in the preceding lemma put F′ := N_{F}^{K}(Q) and P′ := N_{P}^{K}(Q). We will verify Axioms A1, A2 and A3 for F′. Axiom A1 is trivially satisfied.

Let’s verify now Axiom A2. Clearly P′ is fully normalized in F′, and taking N the subgroup of Aut(Q·P′) which stabilizes Q and P′ and acts on Q by morphisms of K, as Q is K-fully normalized, by the lemma 3.6, the group Q · P′ is N-fully normalized in F. By Proposition 3.4, the group N ∩I_{P} (Q·P′) is a Sylow p-subgroup of N ∩ F. But, by the definition of F′, any morphism of F′(P′) lifts in a morphism of F(Q·P′)∩N. So the restriction morphism N ∩F(Q·P′) → F′(P′) is surjective. Moreover this morphism maps N ∩ I_{P} (Q · P′) surjectively on I(P′). Thus I(P′) is a Sylow p-subgroup of F′(P′) and Axiom A2 is satisfied.

For verifying Axiom A3, let R ≤ P′, let L a subgroup of Aut(R) and ϕ ∈ F′(R, P′) such that ϕ(R) is ^{ϕ}L-fully normalized in F′. By the definition of F′ we have that there exists ψ ∈ F(Q · R, Q · P′) and χ ∈ K such that ψ(v) = ϕ(v), for all v ∈ R and ψ(u) = χ(u), for all u ∈ Q. Let’s take again the notations in the lemma 3.6: let T := Q · R and M be the subgroup of Aut(T) stabilizing Q and R and acting on Q, respectively R by morphisms of K, respectively L. As Q is K-fully normalized in F and ψ(R) is ^{ψ}L-fully normalized in F′, by the lemma 3.6, we obtain that ψ(T) is M-fully normalized in F.

We apply Axiom A3 to ψ ∈ F(T, P) and we obtain that there exists ρ ∈ F(T · N_{P}^{M}(T), P) and ν ∈ M such that ρ(w) = ψ(ν(w)), for all w ∈ T. In particular, we have ρ(u) = ψν(u) and so ρ determines an element of K, as it was already the case for ψ and ν. Now, R and N_{P}^{M}(T) are subgroups of N_{P}^{K}(Q) and as ρ determines an element of K, the images ρ(R) and ρ(N_{P}^{M}(T)) remain subgroups of N_{P}^{K}(Q). On the other hand, the action of ν over R determines an element λ of L and, by the lemma 3.6, we have N_{P}^{M}(T) = N_{P′}^{L}(R). We obtain, in consequence, that the restriction of ρ to R · N_{P′}^{L}(R) is a morphism of F′(R · N_{P′}^{L}(R), P′). Moreover we have ρ(v) = ψ(ν(v)) = ϕ(λ(v)) for all v ∈ R, so Axiom A3 is also satisfied. □

We will show now that the Brauer full subcategory under a maximal pair is a full Frobenius system.

Proposition 3.7. Let G be a finite group and k a field of characteristic p. Let (P, e) be a maximal Brauer pair of kG. Then the Brauer full subcategory under (P, e) is a full Frobenius system over P.

Proof. Axiom A1 is trivially satisfied. For Axiom A2, we need to show that P/Z(P) is a Sylow p-subgroup of N_{G}(P, e)/C_{G}(P). In other words, we need to show that the order of N_{G}(P, e)/P C_{G}(P) is not divisible by p. But this is a direct consequence of the first Brauer’s theorem [Th, Thm. 40.14].

Now, let’s verify Axiom A3. Let (Q, f) a Brauer pair K-fully normalized in (P, e). Put N := N_{P}^{K}(Q) and let f_{N} be the unique block of kC_{G}(Q · N) such that (N, f_{N}) ≤ (P, e) and b the unique block of kG such that (1, b) ≤ (P, e). We’ll first prove that (Q · N, f_{N}) is a maximal Brauer pair of k(Q · N_{G}^{K}(Q, f)).

Let (S, f_{S}) be a Brauer pair of k(Q · N_{G}^{K}(Q, f)) containing (Q · N, f_{N}). We have C_{G}(S) = C_{N_{G}^{K}(Q,f)}(S) as C_{G}(S) ≤ C_{G}(Q) ≤ N_{G}^{K}(Q, f), so (S, f_{S}) is also a Brauer pair for G. Let (P′, e′) be a maximal Brauer pair of kG containing (S, f_{S}). Now, both pairs (P, e) and (P′, e′) contain (Q, f) so they are associated to b. By Proposition 2.7 there exists an element g ∈ G such that (P, e) = ^{g}(P′, e′). As a consequence we have ^{g}(S, f_{S}) ≤ (P, e) and the conjugation by g gives us a morphism from Q·N to P. As ^{g}S ≤ ^{g}Q·N_{G}^{^{g}K}(^{g}Q, ^{g}f) using that (Q, f) is K-fully normalized we obtain that ^{g}S = ^{g}(Q · N). We have, thus, Q · N = S and (Q · N, f_{N}) is a maximal pair of k(Q · N_{G}^{K}(Q, f)).

Let’s put ourselves in the hypothesis of Axiom A3. Let (Q, f), ^{g}(Q, f) ≤ (P, e) be such that ^{g}(Q, f) is ^{g}K-fully normalized in P. Let (Q · N, f_{N}) be constructed as before, N′ := N_{P}^{^{g}K}(^{g}Q) and f_{N′} the unique block of kC_{G}(^{g}Q · N′) such that (^{g}Q · N′, f_{N′}) ≤ (P, e). We have seen that in this case (^{g}Q · N′, f_{N′}) is a maximal Brauer pair of k(^{g}Q·N_{G}^{^{g}K}(^{g}Q, ^{g}f)). But ^{g}(Q·N, f_{N}) is also a Brauer pair of k(^{g}Q· N_{G}^{^{g}K}(^{g}Q, ^{g}f)). This Brauer pair is contained in a maximal pair, so, by Proposition 2.7, there exists h ∈ ^{g}Q·N_{G}^{^{g}K}(^{g}Q, ^{g}f) such that ^{hg}(Q·N) ≤ ^{g}Q·N′. We quotient by ^{g}Q and we deduce that there exists an element \overline{h} ∈ N_{G}^{^{g}K}(^{g}Q, ^{g}f)/N _{^{g}Q}^{^{g}K}(^{g}Q), such that \overline{h}N/N _{^{g}Q}^{^{g}K}(^{g}Q) ≤ N′/N _{^{g}Q}^{^{g}K}(^{g}Q). Taking h′ to be a representative modulo N _{^{g}Q}^{^{g}K}(^{g}Q) of \overline{h} in N_{G}^{^{g}K}(^{g}Q, ^{g}f), we obtain that conj(h′g) is a morphism from Q·N to P such that

conj(h′g)(u) = conj(g)conj(g^{−1}h′g)(u) for all u ∈ Q .

This is the wanted form in Axiom A3 because we have conj(g^{−1}h′g) ∈ K. In this way Axiom A3 is also satisfied.

So we can conclude that the Brauer full subcategory, under a maximal pair (P, e), is a full Frobenius system over P. □

As the full subcategory F_{p}(G)_{≤S} under of a Sylow p-subgroup S of G of the Frobenius category F_{p}(G) is equivalent to the Brauer full subcategory under a maximal pair associated to the principal block the first is also a full Frobenius system over S.

4. Alperin’s theorem

In this section we give the proof of Alperin’s theorem on local control, in the full Frobenius systems. The principal ideas of the proof are from [Th]. In all the section P is a finite p-group, Q a subgroup of P and F a full Frobenius system over P. For a finite group G we denote by S_{p}(G) the Quillen complex of G, whose vertices are the objects in F_{p}(G) and whose simplices are given by chains of groups ordered by inclusion.

Before writing down the theorem, we give two definitions. Definition 4.1. Let Q be a subgroup of P. We say that Q is centric if C_{P} (ϕ(Q)) ⊂ ϕ(Q), for all ϕ ∈ F(Q, P). Remark 4.2. A centric subgroup of P is fully centralized in F. Indeed, for any morphism ϕ ∈ F(Q · C_{P} (Q), P) = F(Q, P), we have

ϕ(C_{P} (Q)) = ϕ(Z(Q)) = Z(ϕ(Q)) = C_{P} (ϕ(Q)) .

We give now a generalization to full Frobenius systems of the notion of essential p-subgroup of a finite group, introduced by Puig in 1976, in his thesis [Pu1]. Definition 4.3. Let Q be a subgroup of P. We say that Q is an essential subgroup of F if Q is centric and S_{p}(F(Q))_{>I(Q)} is disconnected.

If ϕ ∈ F(P), we say that ϕ is a maximal morphism; if ϕ ∈ F(E), where E is an essential subgroup of F, we say that ϕ is an essential morphism. We are now able to enounce the Alperin’s theorem, in the full Frobenius systems. Theorem 4.4. Any morphism ϕ ∈ F(Q, P) can be written as the composition of restrictions to Q of essential morphisms, followed by a maximal morphism. More precisely, there exists a positive integer n and, for all 1 ≤ i ≤ n, essential subgroups E_{i} of F fully normalized in F and morphisms ψ_{i} ∈ F(E_{i}), ψ_{n+1} ∈ F(P) such that we have

ϕ(u) = ψ_{n+1}ψ_{n} … ψ_{1}(u), for all u ∈ Q .

Proof. The proof is given by induction on the index |P : Q|. If |P : Q| = 1, then P = Q and ϕ ∈ F(P). Suppose now that |P : Q| > 1. Let ψ ∈ F(Q, P), such that ψ(Q) is fully normalized in F.

Now, ψ ∈ F(Q, ψ(Q)) and ψϕ^{−1} ∈ F(ϕ(Q), ψ(Q)) are morphisms mapping to ψ(Q), which is a subgroup of P fully normalized in F. So, it suffices to find the decomposition of the theorem for an isomorphism which maps to a subgroup of P fully normalized in F because, if we find this decomposition for ψ and for ψϕ^{−1}, we find one for ϕ. It is true that the essential and maximal morphisms generally will not be in a good order, but we’ll see at the end of the proof that we can interchange the places of maximal and essential morphisms.

So we reduce ourselves to the case where we want to decompose a morphism ψ ∈ F(Q, P), such that ψ(Q) is fully normalized in F. By Axiom A3 applied to ψ, we have a morphism ρ ∈ F(N_{P} (Q), P) and χ ∈ F(Q) such that ρ|_{Q} = ψχ. Now |P : N_{P} (Q)| < |P : Q| so, by induction, ρ has the wanted form. So, for decomposing ψ, it suffices to decompose the morphism ψχψ^{−1} ∈ F(ψ(Q)).

In this way, we reduce ourselves, by changing if necessarily the notations, to the case where we have to decompose χ ∈ F(Q) for a subgroup Q of P fully normalized in F. By Proposition 3.4, Q is fully centralized in F, and, applying Axiom A3 to χ, there exists χ′ ∈ F(QC_{P} (Q), P) such that χ′(u) = χ(u), for all u ∈ Q. If Q is not centric, then |P : Q| > |P : QC_{P} (Q)| and we obtain the decomposition of χ′ by induction. So we can suppose that Q is centric.

If Q is an essential subgroup of F, then χ ∈ F(Q) and χ is of the wanted form, as Q is fully normalized in F. If not, S_{p}(F(Q))_{>I(Q)} is connected. Let T = I_{P} (Q). It’s a Sylow p-subgroup of F(Q), by Proposition 3.4, as Q is fully normalized in F. By the connexity of S_{p}(F(Q))_{>I(Q)}, there exists a path

T ≥ S_{1} ≤ ^{χ_{1}}T ≥ S_{2} ≤ … ^{χ_{n−1}}T ≥ S_{n} ≤ ^{χ}T

between T and ^{χ}T, where χ_{i} ∈ F(Q) and S_{i} > I(Q) for all i = 1, …, n. In other words, putting χ_{0} = id and χ_{n} = χ, we have, for all 1 ≤ i ≤ n − 1, ^{χ_{i−1}}S_{i} ≤ T ≥ ^{χ_{i}}S_{i}. Put K := ^{χ_{i−1}}S_{i}. As K ≤ T = I_{P} (Q), we obtain K ∩ I_{P} (Q) = K = K∩F(Q), so, by Proposition 3.4, Q is K-fully normalized. By Axiom A3, applied to χ_{i}χ_{i−1}^{−1}, there exists ρ ∈ F(QN_{P}^{K}(Q), P) and σ ∈ K such that χ_{i}χ_{i−1}^{−1}σ(u) = ρ(u) for all u ∈ Q. As I(Q) is a proper subgroup of K, the group Q is a proper subgroup of N_{P}^{K}(Q) and, so, σ is the restriction of a morphism by conjugation by an element of F(N_{P}^{K}(Q)) (given by the conjugation by a preimage of σ in N_{P}^{K}(Q)). We decompose, by induction, ρ and σ as in the theorem and, by consequent, χ_{i}χ_{i−1}^{−1}. So

χ = (χ_{n}χ_{n−1}^{−1})(χ_{n−1}χ_{n−2}^{−1}) … (χ_{2}χ_{1}^{−1})

is also decomposing.

The last step is to see that the composition of a maximal morphism with an essential morphism is equal to the composition of an essential morphism with a maximal morphism, which gives that we can put all the maximal morphisms at the left in the decomposition. Indeed, if ν ∈ F(E), where E is an essential subgroup of F, fully normalized in F and θ ∈ F(P), we have νθ(u) = θ(θ^{−1}νθ)(u), for all u ∈ θ^{−1}(E) where (θ^{−1}νθ) ∈ F(θ^{−1}(E)) and θ^{−1}(E) is a essential subgroup of F, fully normalized in F, as the image of E by a morphism μ ∈ F(R, P), for all R containing N_{P} (E), is a group fully normalized in F. Indeed we have μ(N_{P} (E)) = N_{P} (μ(E)) as E is fully normalized, so |N_{P} (μ(E))| is maximal. □

5. Are there any essential p-subgroups?

Concerning the group of automorphisms of a p-group Q, Martin [Ma] and then, Henn and Priddy [HP] proved that, for almost all p-group Q, the group of automorphisms, Aut(Q), is a p-group. On the other hand, Gorenstein and Lyons [GL] and Aschbacher [As] found the complete list of finite groups having a strongly p-embedded subgroup. We recall that a strongly p-embedded subgroup of a finite group G is a stabilizer of a connected component of S_{p}(G). It is true that the fact that the list is complete is based entierly on the classification of the simple finite groups.

So, we remark that there are very restrictive conditions imposed by the existence of a strongly p-embedded subgroup and also that in general, if Q is a p-group, Out(Q) is a p-group. This push us to search families of p-groups that can never be essential p-subgroups. The first example for p = 2 are the quaternion groups, the dihedral groups and the semi-dihedral groups as their groups of automorphisms are 2-groups.

In the following we give a sufficient condition for a p-group not to be able to be realized as essential p-subgroup. For this, we need some properties of the Frattini subgroup of a p-group, which is the subgroup generated by commutators and p-th powers. We denote the Frattini subgroup of Q by Φ(Q).

The Frattini subgroup of Q is caracteristic in Q. The quotient of Q by Φ(Q) is an elementary (as we quotient by the p-th powers) abelian (as we quotient by the commutators) p-group of rank n (where n is the minimal cardinal of a system of generators of Q). So we can look at Q/Φ(Q) as a vector space of dimension n over the field F_{p}. Consequently, Aut(Q/Φ(Q)) can be seen as the group GL(n, F_{p}) of the nonsingular matrices of dimension n, with coefficients in F_{p}. We have the next property.

Proposition 5.1. Let ϕ : Aut(Q) ⟶ Aut(Q/Φ(Q)) be the canonical map induced by the projection Q ⟶ Q/Φ(Q). Then Kerϕ is a p-group.

Proof. A result of Burnside, whose prove can be found in Gorenstein’s book [Gr, Thm. 5.1.4], says that any p′-automorphism of Q that induces the identity on Q/Φ(Q) is, in fact, the identity on Q. So Ker(ϕ) doesn’t contain any nontrivial p′-element, which implies that Ker(ϕ) is a p-group. □

Lemma 5.2. Let p be a prime number, Q a p-group, Φ(Q) its Frattini subgroup, n the rank of the elementary abelian group Q/Φ(Q) and B the multiplicative group of the nonsingular lower triangular matrices, of dimension n, with coefficients in F_{p}. Suppose that the image of the canonical morphism ϕ : Aut(Q) ⟶ Aut(Q/Φ(Q)) induced by the canonical projection is isomorphic to a subgroup of B. Then Q cannot be realized as essential p-subgroup in a full Frobenius system.

Proof. We note Imϕ the image and Kerϕ the kernel of the morphism ϕ. We have that Imϕ is isomorphic to a subgroup of B, so it has a unique Sylow p-subgroup. As Imϕ ≃ Aut(Q)/Kerϕ and Kerϕ is a p-group, Aut(Q) has also a unique Sylow p-subgroup and it is the same for Out(Q), as it is the quotient of Aut(Q) by the group of the inner automorphisms of Q, which is a p-group. On the other hand, F(Q)/I(Q) is a subgroup of Out(Q) so it has a unique Sylow p-subgroup. So, S_{p}(F(Q)/I(Q)) is connected and Q can never be realized as essential p-subgroup in a full Frobenius system. □

This result helps us to find others families of p-groups, that cannot be realized as essential p-subgroups, for example the direct product of two cyclic p-groups of different order or the non-abelian metacyclic p-subgroup, for p odd.

Proposition 5.3. Let p a prime number and α_{1} ≠ α_{2} two positive integers. Then Q := C_{p^{α_{1}}} × C_{p^{α_{2}}} cannot be realized as an essential p-subgroup in a full Frobenius system.

Proof. Let x and y be two generators of Q of orders p^{α_{1}}, respectively p^{α_{2}} and σ an automorphism of Q. We have σ(x) = x^{a}y^{b} and σ(y) = x^{c}y^{d}. By the lemma 5.2, a sufficient condition, for Q not to be able to be realized as essential p-subgroup, is that the image of the group of automorphisms of Q in Aut(Q/Φ(Q)) be embedded in the group of the lower triangular matrices. So it suffices to show that, for any σ ∈ Aut(Q) as before, we have that p divide b or p divide c.

Suppose, without loss of generality, that α_{1} > α_{2}. Now, the order of σ(y) is the same as the order of y, so equal to p^{α_{2}}. So σ(y)^{p^{α_{2}}} = 1. As y^{p^{α_{2}}} = 1, we deduce that x^{cp^{α_{2}}} = 1. This implies that p divide c as the order of x is greater than p^{α_{2}}. □

Definition 5.4. We say that a p-group is metacyclic if it’s the extension of a cyclic p-group by another cyclic p-group. If there exists such a split extension we say that the metacyclic p-group is split, otherwise we say that it’s non-split.

In other words, a p-group Q is metacyclic if it has a cyclic normal subgroup such that the quotient by this group is also cyclic. It is well known (see [Hu]) that a presentation by generators and relations of a metacyclic p-group is given by

< x, y|x^{p^{m}} = 1, y^{p^{n}} = x^{p^{q}}, ^{y}x = x^{p^{l}+1} >

Then we say that Q is of the type (m, n, q, l).

Using Burnside’s work [Bu], Dietz [Dz] classified the presentations of non-abelian metacyclic p-subgroup for p odd, and obtained that Q is non-split if and only if q ≠ m and l < q < n. In the following we will prove that the non-abelian metacyclic p-groups, cannot be realized as essential p-subgroups.

Proposition 5.5. Let p be an odd prime and Q =< x, y > be a non-abelian metacyclic p-group. Then Q cannot be realized as essential p-subgroup.

Proof. Suppose that Q is of type (m, n, q, l) and chose x and y such that q = m if Q is split. Let σ be an automorphism of Q. We have σ(x) = x^{a}y^{b} and σ(y) = x^{c}y^{d}. By the lemma 5.2, a sufficient condition, for Q not to be able to be realized as essential p-subgroup, is that the image of the group of automorphisms of Q in Aut(Q/Φ(Q)) be embedded in the group of the lower triangular matrices. So it suffices to show that, for all σ ∈ Aut(Q), as before, p divides b or p divides c. The relations in Q give that

\begin{matrix}(1) & (σ(x))^{p^{m}} = 1 , \\ (2) & (σ(y))^{p^{n}} = (σ(x))^{p^{q}} , \\ (3) & ^{σ(y)}σ(x) = (σ(x))^{p^{l}+1} . \end{matrix}

We compute now, using the generators, the powers of σ(x) and of σ(y), and σ(y)σ(x), by putting σ(x) = x^{a}y^{b} and σ(y) = x^{c}y^{d}. These developments will be useful to us in the following. Let r and s be two positive integers.

(σ(x))^{s} = (x^{a}y^{b})^{s} = x^{a ∑_{i=0}^{s−1}(p^{l}+1)^{ib}}y^{bs} ,
(σ(y))^{r} = (x^{c}y^{d})^{r} = x^{c ∑_{i=0}^{r−1}(p^{l}+1)^{id}}y^{dr} ,
σ(y)σ(x) = (x^{c}y^{d})(x^{a}y^{b})(x^{c}y^{d})^{−1} = x^{c}y^{d}x^{a}y^{b}y^{−d}x^{−c}
= x^{c+a(p^{l}+1)^{d}−c(p^{l}+1)^{b}}y^{b} .

We distinguish two cases.

If l ≥ n, then Q is necessarily split and, replacing in the relation (2) with the computed developments and using the fact that q = m, by the choice we have done before, we obtain

x^{c ∑_{i=0}^{p^{n}−1}(p^{l}+1)^{id}}y^{dp^{n}} = (σ(x))^{p^{m}} .

But (σ(x))^{p^{m}} = 1, so y^{dp^{n}} = 1 and x^{c ∑_{i=0}^{p^{n}−1}(p^{l}+1)^{id}} = 1. The second equality is equivalent to c ∑_{i=0}^{p^{n}−1}(p^{l} + 1)^{id} ≡ 0 (mod p^{m}). As (p^{l} + 1)^{d} ≡ 1 (mod p), the sum ∑_{i=0}^{p^{n}−1}(p^{l} + 1)^{id} is exactly divisible by p^{n} (i.e. divisible by p^{n} and non-divisible by p^{n+1}). Now, Q is non-abelian, so l < m, which implies, as l ≥ n, that n < m. This forces c to be a multiple of p.

If l < n, we replace in the relation (3) by the computed developments and we obtain

x^{c+a(p^{l}+1)^{d}−c(p^{l}+1)^{b}}y^{b} = x^{a ∑_{i=0}^{p^{l}}(p^{l}+1)^{ib}}y^{b(p^{l}+1)} .

This implies that y^{b(p^{l}+1)}y^{−b} ∈< x >, which is equivalent to bp^{l} ≡ 0 (mod p^{n}). And so b is a multiple of p.

These computations show us that, in the two cases, the image of the group of automorphisms of Q in Aut(Q/Φ(Q)) is embedded in a group isomorphic to the group of the lower triangular matrices, as, if Q is a metacyclic non-cyclic p-group, then {x, y} is a minimal generator system. The proposition follows. □

6. Main result

As in the last section, F is a full Frobenius system over a p-group P. The notion of control of the p-fusion in the case of p-groups become in the full Frobenius systems the equality of full Frobenius systems over P as we deal directly with the morphisms. Let us define, in general, the notion of subsystem of a full Frobenius system. Definition 6.1. Let F be a full Frobenius system over P. A subsystem F′ of F is a full Frobenius system over a subgroup Q of P such that set of morphisms of F′ is a subset of the set of morphisms of F.

We will give necessary and sufficient conditions on a subgroup Q of P, completeely normalized in F, for N_{F}(Q) to be equal to F. If F = F_{p}(G)_{≤P}, where P is a Sylow p-subgroup of a finite group G, the fact that F = N_{F}(Q) is equivalent to the control of the p-fusion in G by N_{G}(Q). Indeed, N_{F}(Q) = F_{p}(N_{G}(Q))_{≤N_{P} (Q)}, as all morphism in F_{p}(N_{G}(Q))_{≤N_{P} (Q)} between two subgroups R and T of N_{P} (Q) given by the conjugation by an element h of N_{G}(Q) can be regarded as a morphism by conjugation by h between Q · R and Q · T, given that ^{h}Q = Q. The inclusion in the other sense is obvious. In the case the Frobenius categories, the control of the p-fusion in G by N_{G}(Q) became F_{p}(G)_{≤P} = F_{p}(N_{G}(Q))_{≤N_{P} (Q)}. So it’s natural to wonder if a subsystem F′ of a full Frobenius system F is equal to F.

Let’s start with some definitions. Definition 6.2. We say that Q is strongly closed in F if ϕ(R) ≤ Q, for any subgroup R of Q and any morphism ϕ ∈ F(R, P). Definition 6.3. We say that Q is weakly closed in F if ϕ(Q) = Q, for any morphism ϕ ∈ F(Q, P). Remark 6.4. In fact, Q is strongly closed in F if and only if F(R, P) = F(R, Q), for any subgroup R of Q, and weakly closed in F if and only if F(Q, P) = F(Q). It’s clear that, if Q is strongly closed in F, then Q is weakly closed in F. Definition 6.5. A central series for Q is a series 1 = Q_{0} ⊲Q_{1} ⊲Q_{2} ⊲…⊲Q_{n} = Q such that Q_{i+1}/Q_{i} ≤ Z(Q/Q_{i}). If we have equality in the last relation then the series is called upper central.

For a full Frobenius system F over P we can define the quotient \overline{F} of F by a subgroup Q of P, strongly closed in F. Definition 6.6. Let F be a full Frobenius system over P and Q a subgroup of P strongly closed in F. We define \overline{F} := F/Q as the category over P/Q which objets are the subgroups of P/Q and which morphisms are those induced by F.

In fact if G is a finite group having P as Sylow p-subgroup, and Q is a subgroup of P normal in G, then Q is strongly closed in F_{p}(G)_{≤P} and, moreover F_{p}(G)_{≤P} /Q = F_{p}(G/Q)_{≤P/Q}. The fact that Q is strongly closed in F_{p}(G)_{≤P} is straight forward, as any morphism in F_{p}(G)_{≤P} given by conjugation by an element g of G mapping from a subgroup R of Q extends in a morphism by conjugation by g which maps from Q and we have ^{g}R ≤ ^{g}Q = Q as G normalizes Q. The equality F_{p}(G)_{≤P} /Q = F_{p}(G/Q)_{≤P/Q} comes simply from the fact that the objets and the morphisms of the two categories are trivially the same.

Let’s prove now that F/Q is a full Frobenius system. This result is also due to Puig. Note : P → P/Q the canonical projection. Proposition 6.7 ([Pu2], Prop. 2.15). Let F be a full Frobenius system over P and Q a subgroup of P strongly closed in F. Then \overline{F} := F/Q is a full Frobenius system.

Proof. We will verify the three axioms for \overline{F}. For verifying Axiom A1 it suffices to show that, for all \overline{ϕ} ∈ \overline{F}(\overline{R}, \overline{T}), the morphism \overline{ψ} : \overline{ϕ}(\overline{R}) → \overline{R} defined by \overline{ψ}(\overline{ϕ}(\overline{u})) = u, for all \overline{u} ∈ \overline{R}, is a morphism of \overline{F}(\overline{ϕ}(\overline{R}), \overline{R}). But, by the definition of \overline{F}, we have ϕ ∈ F(R, T). By Axiom A1 applied to ϕ, there exists ψ ∈ F(ϕ(R), R) such that ψ(ϕ(u)) = u, for all u ∈ R, so \overline{ψ} ∈ \overline{F}(\overline{ϕ}(\overline{R}), \overline{R}) has the wanted properties.

By construction of \overline{F}, the quotient morphism F(P) ⟶ \overline{F}(\overline{P}) is surjective and the image of I(P) by this morphism is I(\overline{P}). By Axiom A2 applied to P the group I(P) is a Sylow p-subgroup of F(P), so I(\overline{P}) is a Sylow p-subgroup of \overline{F}(\overline{P}), which gives that Axiom A2 is satisfied in \overline{F}.

Let’s verify now Axiom A3. Consider \overline{ϕ} ∈ \overline{F}(\overline{R}, \overline{P}), ϕ ∈ F(R, P) a representative of \overline{ϕ} in F and \overline{K} a subgroup of Aut(\overline{R}). We denote by K the inverse image of \overline{K} in the stabilizer of Q in Aut(R). We remark that, even if K isn’t necessarily sent surjectively on \overline{K}, the group N\overline{K} ; \overline{P}(\overline{R}) is the image of N_{P}^{K}(R). Put

\overline{K}′ := \overline{ϕ}\overline{K}, \overline{R}′ := \overline{ϕ}(\overline{R}), K′ := ^{ϕ}K and R′ := ϕ(R). Suppose that \overline{R}′ is \overline{K}′-fully normalized. As N\overline{K} ; \overline{P}(\overline{R}) is the image of N_{P}^{K}(R), it follows that N\overline{K}′ ; \overline{P}(\overline{R}′) is the

image of N_{P}^{K′}(R′), so \overline{F}(\overline{R}′ · N\overline{K}′ ; \overline{P}(\overline{R}′), \overline{P}) is the image of F(R′ · N_{P}^{K′}(R′), P). So

for all θ ∈ F(R′ · N_{P}^{K′}(R′), P) we obtain \overline{θ} ∈ \overline{F}(\overline{R}′ · N\overline{K}′ ; \overline{P}(\overline{R}′), \overline{P}) and, as \overline{R}′ is \overline{K}′-

fully normalized, we deduce that \overline{θ}(N\overline{K}′ ; \overline{P}(\overline{R}′)) = N \overline{θ} \overline{K}′ ; \overline{P}(\overline{θ}(\overline{R}′)), and so θ(N_{P}^{K′}(R′)) = N_{P}^{^{θ}K′}(θ(R′)). We deduce that R′ is K′-fully normalized.

By Axiom A3 applied to ϕ, there exists ρ ∈ F(RN_{R}^{K}(P), P) and χ ∈ K such that ρ(u) = ϕ(χ(u)) for all u ∈ R. As Q is strongly closed in F, we obtain ρ(Q) = Q, so \overline{ρ} is a morphism of \overline{F}(\overline{R} · N\overline{K} ; \overline{P}(\overline{R}), \overline{P}). Moreover χ(Q) = Q as χ ∈ K and so \overline{χ} ∈ \overline{K}. At the end we obtain that \overline{ρ}(\overline{u}) = \overline{ϕ}(\overline{χ}(\overline{u})) for all \overline{u} ∈ \overline{R} and Axiom A3 is verified for \overline{F}. □

Remark 6.8. Let F be a full Frobenius system and Q a normal subgroup of P. Then N_{P} (Q) = P, so Q is fully normalized in R. But Q isn’t necessarily strongly closed in F. But, Q is strongly closed in N := N_{F}(Q). Indeed, for any subgroup R ≤ Q and any morphism ϕ ∈ N (R, N_{P} (Q)) there exists a morphism ψ ∈ F(Q · R, P) = F(Q, P) such that ψ(u) = ϕ(u), for all u ∈ R and ψ(Q) = Q. By consequence, we have ϕ(R) = ψ(R) ≤ ψ(Q) = Q and as this is true for all the subgroups of Q, the former is strongly closed in N.

We continue with two technical lemmas.

Lemma 6.9. Let Q a p-group, T a normal subgroup of Q and : Q → Q/T the canonical projection. If the series given by Q = Q_{n} ≥ Q_{n−1} ≥ … ≥ Q_{1} is a central series of Q then \overline{Q} = \overline{Q_{n}} ≥ \overline{Q_{n−1}} ≥ … ≥ \overline{Q_{1}} is a central series of \overline{Q}. If, moreover, Q_{1} = T, then the converse is also true.

Proof. For the first part of the lemma we use the property that \overline{Z(Q)} ≤ Z(\overline{Q}) and \overline{Z(Q/Q_{i})} ≤ Z(\overline{Q}/\overline{Q_{i}}). So the relations \overline{Q}_{1} ≤ Z(\overline{Q}) and \overline{Q_{i+1}}/\overline{Q_{i}} ≤ Z(\overline{Q}/\overline{Q_{i}}) are always satisfied, given that \overline{Q}_{1} ≤ \overline{Z(Q)} and \overline{Q_{i+1}}/\overline{Q_{i}} = \overline{Q_{i+1}/Q_{i}} ≤ \overline{Z(Q/Q_{i})}. We deduce that \overline{Q} = \overline{Q_{n}} ≥ \overline{Q_{n−1}} ≥ … ≥ \overline{Q_{1}} is a central series of \overline{Q}. For the converse, T is central in Q, so Q_{1} = T ≤ Z(Q). The others verifications are straight forward, using the fact that Q/Q_{i} ≃ \overline{Q}/\overline{Q_{i}} and Q_{i+1}/Q_{i} ≃ \overline{Q_{i+1}}/\overline{Q_{i}}, for all i = 1, …, n − 1. □

Lemma 6.10. Let P be a p-group, F a full Frobenius system over P and T a subgroup of P strongly closed in F. Let \overline{F} := F/T, Q ≤ P and denote by : P → P/T the canonical projection. If Q is strongly, respectively weakly closed in F, then \overline{Q} is strongly, respectively weakly closed in \overline{F}. If Q contains T and \overline{Q} is weakly closed in \overline{F}, then Q is weakly closed in F.

Proof. By definition, Q is strongly closed in F if and only if for all R ≤ Q, we have F(R, P) = F(R, Q). This implies that \overline{F}(\overline{R}, \overline{P}) = \overline{F}(\overline{R}, \overline{Q}), as the morphisms in \overline{F} are induced by the morphisms in F. As any subgroup of \overline{Q} has a preimage in Q, we have the equivalence with the fact that \overline{Q} is strongly closed in \overline{F}. The fact that Q is weakly closed in F, by the same argument, implies the fact that \overline{Q} is weakly closed in \overline{F}.

Suppose now that T ≤ Q and \overline{Q} is weakly closed in \overline{F}. Let ϕ ∈ F(Q, P). It induces a morphism \overline{ϕ} ∈ \overline{F}(\overline{Q}, \overline{P}). As \overline{Q} is weakly closed in \overline{F} it follows that \overline{ϕ} ∈ \overline{F}(\overline{Q}). This makes that ϕ ∈ F(Q), so Q is weakly closed in F. □

In the middle of the ’80s, Gilotti and Serena [GS] gave necessary and sufficient conditions for the normalizer of a p-subgroup of a finite group G to control the p-fusion in G. In the last part of this section we’ll generalize this result to full Frobenius systems. This is the main result of our paper.

Theorem 6.11. Let P be a finite p-group, Q a subgroup of P and F a full Frobenius system over P. Then N_{F}(Q) = F if and only if Q is strongly closed in F and Q admits a central series Q = Q_{n} ≥ Q_{n−1} ≥ … ≥ Q_{1} where Q_{i} is weakly closed in F, for all 1 ≤ i ≤ n − 1.

Proof. Put N := N_{F}(Q).

⟹ Suppose that N = F. Let R be a subgroup of Q and ϕ ∈ F(R, P). As N = F, we obtain that ϕ ∈ N (R, P), so the morphism ϕ extends to a morphism ψ ∈ F(Q). In this way, ϕ(R) = ψ(R) ≤ ψ(Q) = Q, which gives that Q is strongly closed in F. Consider the upper central series Q = Q_{n} ≥ Q_{n−1} ≥ … ≥ Q_{1} and, for a 1 ≤ i ≤ n − 1, let ϕ_{i} ∈ F(Q_{i}, P). As before, we have ϕ_{i} ∈ N (Q_{i}, P). But, Q_{i} is characteristic in Q, so Q_{i} = ϕ_{i}(Q_{i}), as ϕ can be lifted in a morphism in F(Q). In this way Q_{i} is weakly closed in F, for all 1 ≤ i ≤ n − 1.

⟸ Suppose that the group Q is strongly closed in F and that it admits a central series Q = Q_{n} ≥ Q_{n−1} ≥ … ≥ Q_{1} > 1 where Q_{i} is weakly closed in F for all 1 ≤ i ≤ n − 1. By the fact that Q_{i} is weakly closed in F we have Q_{i} ⊲ P, for all 1 ≤ i ≤ n. In particular, N_{P} (Q) = P, so, by Proposition 3.5, N is a full Frobenius system over P. Put T := Q_{1} ∩ Z(P). As Q_{1} is normal in P, we have that Q intersects nontrivialy the centre Z(P), so T isn’t trivial.

We first prove that F(T, P) = N (T, P). Indeed, let ϕ ∈ F(T, P). By the fact that Q is strongly closed in F we have ϕ(T) ≤ Q. As Q_{1} ≤ Z(Q), we obtain that T, ϕ(T) ≤ C_{P} (Q_{1}) and, so, that Q_{1} ≤ C_{P} (T). But T is fully centralized in F, as C_{P} (T) = P. So there exists a morphism χ ∈ F(C_{P} (ϕ(T)), P) such that χ|_{ϕ(T)} = ϕ^{−1}. In this way χ|_{Q_{1}} ∈ F(Q_{1}, P) and, by the weak closure of Q_{1} in F, we have χ|_{Q_{1}} ∈ F(Q_{1}). As η(Q_{1}) = Q_{1}, for all η ∈ F(Q_{1}, P), we deduce that Q_{1} is fully centralized in F. Now, Q ≤ C_{P} (Q_{1}), as Q_{1} ≤ Z(Q). In this way, χ|_{Q_{1}} can be extended, by Axiom A3, to χ̃ ∈ F(C_{P} (Q_{1}), P). We obtain, by restriction to Q, that χ̃|_{Q} ∈ F(Q, P), so, as Q is strongly closed F, we have χ̃|_{Q} ∈ F(Q). This implies that χ̃|_{T} is a morphism in N (T, P). In conclusion, ϕ = χ^{−1}|_{T} = χ^{−1}|_{T}, so ϕ is in N (T, P).

Let’s go back now to the general case. By the Alperin’s theorem it suffices to show that F(U) = N (U), for U centric, fully normalized in F. The proof is done by induction on the number of morphisms in F. If F = {id} is a full Frobenius system over P, then P is necessarily abelian as F(P) contains the inner automorphisms of P and in this case the theorem is trivial.

Let C := N_{F}^{{id}}(T) and ϕ ∈ F(U). As C_{P} (T) = P, the group T is fully centralized in F, so, by Proposition 3.5, we obtain that C is a full Frobenius system over P. Put N′ := N_{C}(Q) and, as Q is also fully normalized in C, it follows that N′ is a full Frobenius system over P. We have clearly N′(R, P) = N (R, P) ∩ C(R, P) for any subgroup R of P. We distinguish two cases.

Case 1: F ≠ C. As U is centric, it contains Z(P), so, a fortiori, T. In this way, we have ϕ|_{T} ∈ F(T, P) and, by the first part of the proof, ϕ|_{T} ∈ N (T, P). Also, by Axiom A3 applied to ϕ|_{T}, using the fact that T is fully centralized in P and C_{P} (T) = P, we can lift ϕ|_{T} in ϕ̃ ∈ F(P) such that ϕ̃|_{T} = ϕ|_{T}. Moreover, ϕ̃|_{Q} ∈ F(Q) as Q is strongly closed in F. We obtain in this way ϕ̃^{−1}ϕ ∈ F(U, P) ∩ C(T). By the induction hypothesis applied to C, we have ϕ̃^{−1}ϕ ∈ N′(U, P). As N′(U, P) ≤ N (U, P) we obtain that ϕ̃^{−1}ϕ ∈ N (U, P). But, ϕ̃ ∈ F(P) = N (P), which implies that ϕ ∈ N (U, P) and finishes the proof in this case.

Case 2: F = C. In this case T is strongly closed in F. Indeed let R ≤ T and ϕ ∈ F(R, P). As F = C the morphism ϕ extends to ψ ∈ F(T) which fixes T point by point and so ϕ ∈ F(R). In this way T is strongly closed in F.

Let’s take \overline{F} := F/T. By Proposition 3.5, \overline{F} is a full Frobenius system over \overline{P}. Moreover, by the lemma 6.9, \overline{Q} = \overline{Q_{n}} ≥ \overline{Q_{n−1}} ≥ … ≥ \overline{Q_{1}} is a central series of \overline{Q} and, by the lemma 6.10, it follows that \overline{Q} is strongly closed in \overline{F} and \overline{Q}_{i} is weakly closed in \overline{F}, for all 1 ≤ i ≤ n − 1. As T isn’t trivial, by induction hypothesis, we have \overline{F} = N\overline{F}(\overline{Q}) =: \overline{N}, so \overline{ϕ} ∈ \overline{N} (\overline{U}). We lift \overline{ϕ} in ψ ∈ N (U) as the canonical projection of N (U) on \overline{N} (\overline{U}) is surjective. Moreover, for all x ∈ U we have ψϕ^{−1}(x)x^{−1} ∈ T. Put K := {θ ∈ F(U)|θ(x)x^{−1} ∈ T, for all x ∈ U}. Now, K is a subgroup of Aut(U) which centralizes U/T by construction and T by the fact that any morphism of F centralizes T, so K centralizes the quotients of the normal series 1 ◁ T ◁ U. This implies that K is a p-group. Moreover, K is a normal subgroup of F(U), as, for all χ ∈ F(U) and θ ∈ K, we have

(^{χ}θ)(x)x^{−1} = χ(θ(y)y^{−1}) ∈ χ(T) = T where y = χ^{−1}(x), for all x ∈ U .

As U is fully normalized in F, by Proposition 3.4, the group I_{P} (U) is a Sylow p-subgroup of F(U). Moreover, K is a normal subgroup of F(U), so K ∩ I_{P} (U) is a Sylow p-subgroup of K. It follows that K = I_{P}^{K}(U) as K is a p-group. As ψϕ^{−1} ∈ K, there exists u ∈ P such that ψϕ^{−1}(x) = ^{u}x, for all x ∈ U. But any conjugation by an element of P is in F and, moreover, given that Q is normal in P, we obtain conj(u)(Q) = Q. This makes that conj(u) ∈ N (U). In this way, we have ϕ = conj(u^{−1})ψ ∈ N (U), which proves the theorem. □

7. Resistant groups

The generalization of the theorem of Gilotti and Serena gives us a necessary end sufficient condition for the resistance of a p-group in the field of the full Frobenius systems.

Definition 7.1. We say that a p-group P is resistant in the field of full Frobenius systems if for all full Frobenius system F over P, we have F = N_{F}(P).

For a p-group P to be resistant it has and it suffices, by Theorem 6.11, for all full Frobenius system F over P, to posses a central series composed of subgroups weakly closed in F. The condition that P is strongly closed in F is trivially satisfied. So, we have that the abelian p-groups are trivially resistant in the full Frobenius systems as the upper central series is reduced to P. For finding new families of resistant groups we’ll need the following lemma.

Lemma 7.2. Let p be an odd prime, Q an elementary abelian p-group of order p^{2} and H a subgroup of the group of automorphisms of Q. Suppose that H has a nontrivial Sylow p-subgroup R and let P be a p-group such that the short sequence 1 ⟶ Q ⟶ P ⟶ R ⟶ 1 be exact.

a) If, for any morphism ϕ ∈ N_{H}(R), there exists an automorphism ψ of P, such that ψ(u) = ϕ(u), for all u ∈ Q and ψρψ^{−1} = ϕρϕ^{−1}, for all ρ ∈ R, then we can extend the short sequence 1 ⟶ Q ⟶ P ⟶ R ⟶ 1 to a short sequence 1 ⟶ Q ⟶ L ⟶ H ⟶ 1.

b) If H has at least two distinct Sylow p-subgroups and we can extend the short sequence 1 ⟶ Q ⟶ P ⟶ R ⟶ 1 to a short sequence 1 ⟶ Q ⟶ L ⟶ H ⟶ 1 then P is not isomorphic to C_{p^{2}} ×C_{p}.

Proof. a) For verifying that the short sequence 1 ⟶ Q ⟶ P ⟶ R ⟶ 1 extends to the short sequence 1 ⟶ Q ⟶ L ⟶ H ⟶ 1, it suffices to prove [Br, pp. 84-85] that the cohomology class h(P), determined by P in H^{2}(R, Q), is H-stable, i.e. for all ϕ ∈ H we have

res_{R ∩^{ϕ}R}^{R}h(P) = res_{R ∩^{ϕ}R}^{^{ϕ}R}c_{ϕ}^{∗}(h(P)) (∗) .

Here res is the restriction in cohomology and c_{ϕ}^{∗} is the morphism induced in cohomology by the conjugation by ϕ. If R ≠ ^{ϕ}R then R ∩ ^{ϕ}R = 1, as R is a p-group of order p, and the relation (∗) is clearly is satisfied. Suppose that R = ^{ϕ}R. By the hypothesis, there exists an automorphism ψ of P such that ψ(u) = ϕ(u), for all u ∈ Q and ψρψ^{−1} = ϕρϕ^{−1} for all ρ ∈ R. This implies that h(P) = c_{ϕ}^{∗}(h(P)) and (∗) is once again satisfied. Indeed, let 1 ⟶ Q ⟶ P′ ⟶ R ⟶ 1 be a exact short sequence representing the class c_{ϕ}^{∗}(h(P)) ∈ H^{2}(R, Q). By the definition of c_{ϕ}^{∗} (see [Br, p. 80]), there exists an isomorphism f : P ⟶ P′ such that the next diagram commutes

h(P)  1  Q  P  R  1 c_{ϕ}^{∗}(h(P))  1  Q  P′  R  1 w  w_{_{ϕ} u}  w_{_{f} u}  w_{_{conj(ϕ)} u} w  w  w  w

On the other hand, the condition on the existence of ψ gives us the next commutative diagram.

h(P)  1  Q  P  R  1 h(P)  1  Q  P  R  1 w  w_{_{ϕ} u}  w_{_{ψ} u}  w_{_{conj(ϕ)} u} w  w  w  w

Taking g := fψ^{−1} : P ⟶ P′ we obtain

h(P)  1  Q  P  R  1 c_{ϕ}^{∗}(h(P))  1  Q  P′  R  1 w  w_{_{id} u}  w_{_{fψ^{−1}} u}  w_{_{id} u} w  w  w  w

which gives that h(P) = c_{ϕ}^{∗}(h(P)).

b) The proof of the second part of the lemma is due to Jacques Thévenaz. It is based on the fact that H^{2}(SL(2, F_{p}), Q) = 0, as we already showed in [St]. Now, by the first part of the proof, h(P) extends to a cohomology class in H^{2}(H, Q). Moreover H contains SL(2, F_{p}), given that H has at least two Sylow p-subgroups, and any two Sylow p-subgroups of GL(2, F_{p}) generate SL(2, F_{p}). So, h(P) extends to a cohomology class in H^{2}(SL(2, F_{p}), Q), which implies that h(P) is trivial. We obtain that P isn’t isomorphic to C_{p^{2}} × C_{p} as the cohomology class induced by C_{p^{2}} × C_{p} in H^{2}(Q, R) isn’t trivial (in other words, C_{p^{2}} × C_{p} isn’t a semi-direct product of C_{p} × C_{p} by C_{p}). □

First we prove a similar result to those obtained in the the field of Frobenius categories in [St], for the generalized extraspecial p-groups.

Theorem 7.3. Let P be an extraspecial generalized p-group. Then P is resistant in the field of full Frobenius systems, excepting the case where P = E × A with A elementary abelian and E dihedral of order 8 (when p = 2) or of order p^{3} and exponent p (when p is odd).

Proof. If P is such a p-group, the upper central series is simply 1 ⊲ Φ(P) ⊲ P. So P is resistant if and only if Φ(P) is weakly closed in F, for all full Frobenius system F over P. By the Alperin’s theorem, any morphism in F(Φ(P), P) can be decomposed in a composition of maximal morphisms and of essential morphisms. As the maximal morphisms stabilise Φ(P), we need only to see that we have the same for the essential morphisms.

Let ϕ ∈ F(Q), where Q is an essential subgroup of F, fully normalized in F. As Q is centric, it contains Z(P) and Q/Z(P) is a maximal isotropic subspace of P/Z(P) for the symplectic form β : P/Z(P) × P/Z(P) → Φ(P), defined by β(x, y) = [x, y] (see [St] for more details). Moreover, if Q is not elementary abelian, then Φ(Q) = Φ(P). Any automorphism of Q preserves Φ(Q) and so ϕ stabilizes Φ(P). We have the same property for ϕ if |P/Z(P)| > p^{2}. Indeed, |P : C_{P} (ϕ(Φ(P)))| ≤ p, as the centralizer of a subgroup of order p of P is of index at most p in P, and we deduce that C_{P} (ϕ(Φ(P)))/Z(P) can’t be contained in a maximal isotropic subspace of P/Z(P) with respect to β. In this way C := C_{P} (ϕ(Φ(P))) can’t be elementary abelian. So Φ(C) isn’t trivial and we have Φ(C) = Φ(P). In the mean time, as C_{P} (Φ(P)) = P, it follows that Φ(P) is fully centralized. By Axiom A3 applied to ϕ, there exists a morphism ψ ∈ F(C, P) such that ψ^{−1}|_{Φ(P)} = ϕ. We deduce that ϕ(Φ(P)) is the Frattini subgroup of C, so ϕ(Φ(P)) = Φ(P).

So we can restrict our research to the case where Q is elementary abelian and |P : Z(P)| = p^{2}. In this case Q is of index p in P and we have P ≃ E ×A where E is an extraspecial p-group of order p^{3} and A a p-group elementary abelian. Moreover E ≃ P/A contains a subgroup isomorphic to Q/A ≃ C_{p} × C_{p}, as Q is elementary abelian, so E can’t be isomorphic to Q_{8}. The only case that remains to exclude is the case E ≃ C_{p^{2}} ×C_{p}, for p odd. Let K = (P/Q) regarded as subgroup of F(Q/A).

If K = ^{ϕ}K then N_{P}^{^{ϕ}K}(Q) = P, so Q is ^{ϕ}K-fully normalized in F. By Axiom A3, applied to ϕ, there exists ψ ∈ F(P) and χ ∈ K such that ψ(u) = ϕ(χ(u)), for all u ∈ Q. Now, ψ and χ fix Φ, so it is same for ϕ.

It remains to verify the case where K ≠ ^{ϕ}K. We regard K and ^{ϕ}K as p-subgroups of Aut(Q/A) ≃ GL(2, F_{p}). In this context K and ^{ϕ}K generate SL(2, F_{p}).

We show now that the extension 1 ⟶ Q/A ⟶ E ⟶ K ⟶ 1 satisfies the hypothesis of the lemma 7.2. Indeed, for any morphism ϕ′ ∈ F(Q) fixing K, we have K = ^{ϕ′}K, so Q is ^{ϕ}K-fully normalized in F. By Axiom A3, applied to ϕ′, there exists ψ′ ∈ F(P) and χ′ ∈ K such that ϕ′(u) = ψ′(χ′)^{−1}(u), for all u ∈ Q. But K = P/Q so χ′ lifts in the conjugation by a preimage of χ′ in P, which is an element of F(P). As a conclusion, ϕ′ lifts in F(P).

In this way, the extension 1 ⟶ Q/A ⟶ E ⟶ K ⟶ 1 satisfies the hypothesis of the lemma 7.2 so, by this lemma, it can be extended to the extension 1 ⟶ Q/A ⟶ L ⟶< K, ^{ϕ}K >⟶ 1. Moreover, K and ^{ϕ}K are two distinct Sylow p-subgroups of < K, ^{ϕ}K > so, by the same lemma 7.2, E isn’t isomorphic to C_{p^{2}} ×C_{p}.

The remaining cases are really exceptions to the theorem, as we proved in [St]. □

The second class of resistant p-groups is given by the metacyclic p-groups, for p odd. Proposition 7.4. Let P a metacyclic p-group, with p odd. Then P is resistant in the field of full Frobenius systems.

Proof. Consider that P is given by generators and relations

P :=< u, v|u^{p^{m}} = 1, v^{p^{n}} = u^{p^{q}}, ^{v}u = u^{p^{l}+1} > .

In other words, P is of the type (m, n, q, l) and corresponds to an extension:

1  C_{p^{m}}  P  C_{p^{n}}  1 w  v  w  w  w .

If P is abelian then it is trivially resistant. If not, P is necessarily non-cyclic. In the case where P is a non-cyclic metacyclic p-group, for p odd, Nadia Mazza [Mz] recently proved that P has an unique subgroup Q isomorphic to C_{p} × C_{p}. This implies that if P has a subgroup isomorphic to C_{p}^{α} × C_{p}^{α}, for a positive integer number α, this subgroup is unique.

We will find a central series of P composed of subgroups strongly closed in F by induction on the order of P. The case where |P | ≤ p^{2} is trivial as P is abelian.

If we manage to find 1 ≠ P_{1} ≤ Z(P) weakly closed in F, then we can apply the induction hypothesis to \overline{F} := N_{F}(P_{1})/P_{1}, which is a full Frobenius system over P/P_{1}, by Proposition 6.7, given that P_{1} is strongly closed in N_{F}(P_{1}). Indeed, P/P_{1} is a metacyclic p-group of order strictly inferior to P. By induction hypothesis P/P_{1} admits a central series 1 = \overline{P}_{1} ⊲ \overline{P}_{2} ⊲ … ⊲ \overline{P}_{n} = P/P_{1} with \overline{P}_{i} weakly closed in \overline{F}. Let π : P ⟶ P/P_{1} be the canonical projection. Consider the subnormal series of P, given by 1 = P_{0}⊲P_{1}⊲P_{2}⊲…⊲P_{n} = P with P_{i} = π^{−1}(\overline{P}_{i}) for 1 ≤ i ≤ n. By the second part of the lemma 6.9, this is a central series of P. Moreover, by the lemma 6.10, P_{i} is weakly closed in N := N_{F}(P_{1}), for all 2 ≤ i ≤ n. We will prove now that P_{i} is also weakly closed in F, for all 2 ≤ i ≤ n. Indeed, let ϕ ∈ F(P_{i}, P). We have ϕ|_{P_{1}} ∈ F(P_{1}, P). But P_{1} is weakly closed in F, which gives that ϕ(P_{1}) = P_{1} and, in consequence that ϕ ∈ N (P_{i}, P). Now, P_{i} is weakly closed in N, so ϕ(P_{i}) = P_{i} and we obtain that P_{i} is weakly closed in F.

Let’s search now a nontrivial central subgroup P_{1} of P which is weakly closed in F. Recall that Q is the unique subgroup of P isomorphic to C_{p} × C_{p}.

If Q ≤ Z(P) then we take P_{1} := Q which is weakly closed in F, as it is unique. If not we have necessarily M :=< u^{p^{m−1}} >≤ Z(P) and Z(P) ≃ C_{p^{l}}. We want to see that < u^{p^{m−1}} > is normalized by the maximal and essential morphisms of P. The maximal morphisms normalize Z(P) and, a fortiori, < u^{p^{m−1}} > as it is the unique subgroup of order p of Z(P).

Let now E be a subgroup of P, centric and fully normalized in P, candidate to be essential in F. By elimination of metacyclic p-groups which can never be realized as essential subgroups (see Proposition 5.3 and Proposition 5.5), we arrive to the fact that E is isomorphic to C_{p^{α}} × C_{p^{α}}, where α is a positive integer.

Let Φ(E) be the Frattini subgroup of E. By Proposition 5.1, the kernel of the canonical application ϕ : F(E) ⟶ Aut(E/Φ(E)) is a p-group. But, as E is an essential subgroup in F and I(E) = 1, the complex S_{p}(F(E)) is disconnected. So, O_{p}(F(E)) is trivial, which gives that the kernel of ϕ is also trivial. We deduce that a Sylow p-subgroup of F(E) is of order p, as Aut(E/Φ(E)) ≃ GL(2, F_{p}).

On the other hand, E is the unique subgroup of P isomorphic to C_{p^{α}} × C_{p^{α}}, so it is characteristic in P. In this way N_{P} (E) = P and E is fully normalized in F. As, moreover, E is centric, P/E is a Sylow p-subgroup of F(E). We have seen that a Sylow p-subgroup of F(E) is of order p, which implies that |P : E| = p.

So P is a metacyclic subgroup, of order p^{2α+1}, having a subgroup E isomorphic to C_{p^{α}} × C_{p^{α}} and a centre Z(P) cyclic. This is equivalent to the fact that the fact that u doesn’t commute with v^{p^{n−1}} and that u^{p^{m−α}} commutes with v^{p^{n−α}}.

But v^{p^{n−1}}uv^{−p^{n−1}} = u^{(p^{l}+1)^{p^{n−1}}} and if we want that u^{(p^{l}+1)^{p^{n−1}}} ≠ u then necessarily n + l − 1 < m. On the other hand,

u^{p^{m−α}} = v^{p^{n−α}}u^{p^{m−α}}v^{−p^{n−α}} = u^{p^{m−α}(p^{l}+1)^{p^{n−α}}}

and, using the fact that (p^{l} + 1)^{p^{n−α}} − 1 is exactly divisible by p^{l+n−α}, it follows that (m − α) + (l + n − α) ≥ m. As m + n = 2α + 1, we obtain that 1 + l ≥ m. In fact, we have m = l + 1, as l < m. Let’s come back now to the inequality n + l − 1 < m in which we replace m = l + 1 in such way that we obtain n < 2. So n = 1 which gives that α = 1 and m = 2. We reach the conclusion that P have to be a p-group extraspecial of order p^{3} and exponent p^{2}, case where we have seen, in the last theorem that any morphism in F(E) fixes the centre of P.

So, if the centre of P is cyclic, < u^{p^{m−1}} > is weakly closed in P with respect to F and we can take P_{1} =< u^{p^{m−1}} >. □

We remark that the p-groups we found to be resistant in the Frobenius category remain resistant in the full Frobenius systems. Is it just the hazard or is there something deeper lying beneath?

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   E-mail address: Radu.Stancu@math.unige.ch

Section de Mathématiques, Université de Genève, CP 240, CH-1211 Genève 24