ALMOST ALL GENERALIZED EXTRASPECIAL p-GROUPS ARE RESISTANT

RADU STANCU^{∗}

Abstract. A p-group P is called resistant if, for any finite group G having P as a Sylow p-subgroup, the normalizer N_{G}(P) controles p-fusion in G. The aim of this paper is to prove that any generalized extraspecial p-group P is resistant, excepting the case when P = E × A where A is elementary abelian and E is dihedral of order 8 (when p = 2) or is of order p^{3} and exponent p (when p is odd). This generalizes a result of Green and Minh.

1. Introduction

Let G be a finite group and H a subgroup of G. Two elements of H are said to be fused in G if they are conjugate in G but not in H. We are interested in p-groups P such that for any finite group G, having P as a Sylow p-subgroup, the p-fusion is controled only by the normalizer N_{G}(P) of P (that is any two elements of P which are fused in G are fused in N_{G}(P)). We will prove in Proposition 2.8 that this is equivalent to the requirement that any such group G does not contain essential p-subgroups (Definition 2.2). Following the terminology suggested by Jesper Grodal, we will call such a group resistant.

In fact, by a theorem of Mislin [Mi], the notion of resistant group is equivalent to what Martino and Priddy [MP] call Swan group. We recall that P is a Swan group if, for any G as before, the mod − p cohomology ring H^{∗}(G) is isomorphic to the mod − p cohomology ring H^{∗}(N_{G}(P)).

In a recent preprint [GM], Green and Minh proved that almost all extraspecial p-groups are Swan groups. In our paper we find the same result for generalized extraspecial p-groups (Definition 3.2) and give a proof avoiding cohomological methods.

2. Essential Groups and Alperin’s Theorem

Let F_{p}(G) be the Frobenius category of a finite group G. We recall that the objects in this category are the non-trivial p-subgroups of G and the morphisms are the group homomorphisms given by the conjugation by elements of G. For a subgroup H of G we denote by F_{p}(G)_{≤H} the full subcategory of F_{p}(G) containing the non-trivial p-subgroups of H.

A natural question is: What is the minimal information needed to completely characterise these morphisms? For a Sylow p-subgroup P of G, Alperin showed in [Al] that these morphisms are locally controled, i.e. by normalizers N_{G}(Q) for Q a subgroup of P, running over all tame intersections and such that C_{P} (Q) = Z(Q).

Date: February 16, 2001.

∗ This work is a part of a doctoral thesis in preparation at the University of Lausanne, under the supervision of Prof. Jacques Thévenaz.

Recall that Q is a tame intersection if there exists a Sylow p-subgroup R of G such that Q = P ∩ R and N_{P} (Q), N_{R}(Q) are Sylow p-subgroups of N_{G}(Q). Nine years later Puig [Pu] refined this and required Q to be an essential p-subgroup of G (Definition 2.2). The answer to the above question will be given in a theorem of Alperin. Before stating this theorem we need some preliminary definitions. From now on P will denote a Sylow p-subgroup of G, Q a subgroup of P and, for g ∈ G,^{g}Q the subgroup obtained from Q by conjugation by g. We note W̃_{G}(Q) := N_{G}(Q)/Q and W_{G}(Q) := N_{G}(Q)/QC_{G}(Q).

Definition 2.1. We say that Q is p-centric if Z(Q) = C_{P} (Q).

We remark that this definition is equivalent to the condition that Z(Q) is a Sylow p-subgroup of C_{G}(Q). In the literature a p-centric subgroup is also called p-self-centralizing [Th, p. 324].

Consider now the Quillen complex S_{p}(H) of a finite group H whose vertices are the objects in F_{p}(H) and whose simplices are given by chains of groups.

Definition 2.2. We say that Q is an essential subgroup of G if the Quillen complex S_{p}(W̃_{G}(Q)) is disconnected and C_{G}(Q) does not act transitively on the connected components.

One can find in [Th, Thm. 48.8] that

Proposition 2.3. Q is an essential p-subgroup of G if and only if Q is p-centric and S_{p}(W_{G}(Q)) is disconnected.

The proof has been done in a more general case. In the terminology and notation of [Th, Thm. 48.8] it suffices to replace local pointed groups by p-subgroups, N_{>Q} by S_{p}(N_{G}(Q))_{>Q} and OG by G. In most of the proofs of this paper we will use this proposition as an alternative definition of essential groups.

Remarks 2.4. 1) If P is a Sylow p-subgroup of a group G, then P is not an essential subgroup of G because S_{p}(W̃_{G}(P)) is empty, hence connected.

2) If a Sylow p-subgroup P of G does not contain essential subgroups of G then G does not contain essential subgroups. This comes from the fact that any essential subgroup is embedded in a Sylow p-subgroup of G. As the Sylow p-subgroups are conjugate by elements of G and the conjugation sends an essential subgroup to another, if one Sylow p-subgroup contains essential subgroups of G then any Sylow p-subgroup does.

We can now state Alperin’s theorem in a version improved by Puig [Pu] and Goldschmidt [Go]:

Theorem 2.5 (Alperin). Any morphism in the category F_{p}(G)_{≤P} is a composite of a sequence of morphisms of two types:

i) conjugation by an element of N_{G}(E) where E is an essential subgroup of G contained in P,

ii) conjugation by an element of N_{G}(P), followed by an inclusion. Precisely if ^{g}Q < P then there exists a decomposition g = g_{1}g_{2} … g_{n}g_{n+1}c, where

a) g_{i} ∈ N_{G}(E_{i}) for E_{i} essential, i = 1, …, n,

b) g_{n+1} ∈ N_{G}(P),

c) c ∈ C_{G}(Q).

The proof of this theorem is given in [Th, Thm 48.3] in a more general form. As before by replacing local pointed groups by p-subgroups and L_{G}(b)_{≤P} by F_{p}(G)_{≤P} we obtain the proof in our case.

Definition 2.6. We say that a subgroup H of a group G controls p-fusion in G if (|G : H|, p) = 1 and for any g ∈ G and any Q, such that Q and ^{g}Q are contained in H, there exists h ∈ H and c ∈ C_{G}(Q) such that g = hc.

Remark 2.7. One can also define the same thing by asking that the inclusion H ↪ G induces an equivalence of categories F_{p}(H) ≃ F_{p}(G).

The notions of control of fusion and essential p-subgroups are strongly linked. The next proposition shows one of the aspects of this link.

Proposition 2.8. The normalizer N_{G}(P) controls p-fusion in G if and only if there are no essential p-subgroups in G.

Proof : The ’if’ part of our proposition is given by Alperin’s theorem. For the ’only if’ part let Q be p-subgroup of G candidate for being essential. By Proposition 2.3 and Remarks 2.4 we can suppose that Q is a p-centric proper subgroup of P. Moreover Q is subgroup of any Sylow p-subgroup of N_{G}(Q). Let T := N_{P} (Q) and R a Sylow p-subgroup of N_{G}(Q) containing T. Another Sylow p-subgroup R′ of N_{G}(Q) will be conjugate to R by an element h ∈ N_{G}(Q). As Q = ^{h}Q < N_{G}(P), conjugation by h is a morphism in F_{p}(N_{G}(P)). By the fact that N_{G}(P) controls p-fusion in G, h = gc with g ∈ N_{G}(P) and c ∈ C_{G}(Q). We have h, c ∈ N_{G}(Q) so g = hc^{−1} ∈ N_{G}(P) ∩ N_{G}(Q) ≤ N_{G}(T). We have T ≤ R and gT ≤ ^{g}R so T ≤ R ∩ ^{g}R. As Q is a proper subgroup of T (the normalizer in a p-group of proper p-subgroup is bigger then the latter) and C_{P} (Q) = Z(Q) (Q is p-centric) the image of T in W_{G}(Q) is a nontrivial group and this means that we have a vertex v_{T} in S_{p}(W_{G}(Q)) corresponding to T. Now the vertices corresponding to R and ^{g}R in S_{p}(W_{G}(Q)) are in the same connected components as both are in the same connected component as v_{T}. On the other hand ^{h}R and ^{g}R correspond to the same vertex in S_{p}(W_{G}(Q)) as they are conjugate by c ∈ C_{G}(Q). This implies that S_{p}(W_{G}(Q)) is connected and so Q cannot be an essential subgroup of G.

3. Generalized Extraspecial Groups

From now on C_{n} will denote the cyclic group of order n.

Definition 3.1. A p-group P is called extraspecial if its Frattini subgroup, Φ, has order p and Φ(P) = [P, P] = Z(P) ≃ C_{p}.

Definition 3.2. A p-group P is called generalized extraspecial if its Frattini subgroup, Φ, has order p, Φ(P) = [P, P] ≃ C_{p} and Z(P) ≥ Φ(P).

The word generalized comes from the fact that we don’t require Z(P) to be cyclic of order p.

Consider V := P/Φ(P). It is an elementary abelian p-group, so we can see it as a vector space over F_{p}. Identifying the group C_{p} with F_{p} we can define a symplectic bilinear form:

\begin{matrix}β : V × V & → F_{p} \\ (x, y) & ↦ [x, y] \end{matrix}

We decompose V = Kerβ ⊕ U, so β|_{U×U} will be a nondegenerate symplectic form. In fact U ≃ P/Z(P). By the classification of the nondegenerate symplectic forms [Di, I.16], U has an even dimension, say 2n and admits a symplectic basis.

Now we study separately the cases when p is odd and when p = 2.

Case p odd: We can define a linear form λ on U by λ(x) = x^{p}. There are two possibilities. If λ is zero this means that all the elements of P have order p and we take any symplectic basis {u_{1}, …, u_{n}, v_{1}, …, v_{n}} of U. If λ is nonzero then we take u_{1} to be a preimage of 1 and we take all other elements of the basis in Kerλ.

Case p = 2: The application ρ : U → F_{p} given by ρ(x) = x^{2} is a quadratic form with associated bilinear form β. This means that ρ(xy)+ρ(x)+ρ(y) = β(x, y). For a symplectic basis {u_{1}, …, u_{n}, v_{1}, …, v_{n}} we have β(u_{i}, v_{i}) = 1. There are two possibilities depending on the existence of a pair (u_{i}, v_{i}) such that ρ(u_{i}) = ρ(v_{i}) = 1. If there is such a pair, we suppose, without loss of generality, that i = 1 and we choose the others elements of the basis such that ρ(u_{i}) = ρ(v_{i}) = 0, i > 1. If not, then we choose all the basis elements satisfying ρ(u_{i}) = ρ(v_{i}) = 0, ∀i.

Once the basis of U is constructed, we take {e_{1}, …, e_{n}, f_{1}, …, f_{n}} to be a choice of representatives of {u_{1}, …, u_{n}, v_{1}, …, v_{n}} in P. We have now a generating system of P = < e_{1}, …, e_{n}, f_{1}, …, f_{n}, Z >, where Z is the centre of P. We call such a system a symplectic generating system of P.

Remark 3.3. Let u, v ∈ U and let e, f ∈ P be representatives of u and v respectively. Then there exists z ∈ Φ(P) of order p such that

1. if β(u, v) = 1 then [e, f] = z,

2. if β(u, v) = 0 then [e, f] = 1,

3. if λ(u) = 1 (or ρ(u) = 1) then e^{p} = z,

4. if λ(u) = 0 (or ρ(u) = 0) then e^{p} = 1,

If all the elements in P are of order p, we say that that P is of exponent p, otherwise we say that P is of exponent p^{2}. Let S ⊂ P be a set; we denote by < S > the subgroup of P generated by S.

Lemma 3.4. Let P be a generalized extraspecial p-group and let z ∈ P be a generator of Φ(P). Then the centre of P is isomorphic < z > ×A or < t > ×A, where A is an elementary abelian p-group and t^{p} = z.

Proof : Let Z be the centre of P and Z = C_{1} × C_{2} × … × C_{n} be a decomposition of Z in cyclic p-groups (such a decomposition exists as Z is abelian). By changing indices if necesary, we can suppose that z ∈ C_{1} and, consequently, Φ(P) ≤ C_{1}. Let x ∈ C_{j}, for j ≠ 1. We have x^{p} ∈ Φ(P) as the p^{th} powers of any element of P are in Φ(P). So x^{p} ∈ Φ(P) ∩ C_{j} ≤ C_{1} ∩ C_{j} = {1}. This means that C_{j} is cyclic of order p. Take now y ∈ C_{1}; we have y^{p} ∈ Φ(P) and, therefore, y is of order p or p^{2}. Consider A := C_{2} × … × C_{n}. We obtain that Z = C_{1} × A with C_{1} cyclic of order p or p^{2} and A an elementary abelian p-group. This completes our proof.

We recall that an isotropic vector subspace of U with respect to β is a subspace on which β is identically zero. A maximal isotropic subspace of U has a dimension equal to half of the dimension of U. As dim U = 2n, the dimension of a maximal isotropic subspace of U is n.

Definition 3.5. A subgroup Q of P is called maximal isotropic if there exists a maximal isotropic subspace Ũ of U such that Q is generated by the representatives of a basis of Ũ and the centre of P.

For example < e_{1}, …, e_{n}, Z > and < f_{1}, …, f_{n}, Z > are maximal isotropic subgroups of P.

Lemma 3.6. A p-centric subgroup Q of P contains a maximal isotropic subgroup and has a generating system Q = < e_{i_{1}}, …, e_{i_{m}}, f_{1}, …, f_{n}, Z(P) > with 0 ≤ m ≤ n and [e_{i_{j}}, f_{i_{j}}] = z ∈ Z(P), 1 ≤ j ≤ m. This system can be completed to a symplectic generating system of P, P = < e_{1}, …, e_{n}, f_{1}, …, f_{n}, Z(P) >.

Proof A p-centric subgroup of P clearly contains the centre Z := Z(P) of P. Suppose that Q does not contain a maximal isotropic subgroup of P. In this case V := Q/Z, considered as vector space, does not contain a maximal isotropic subspace of U = P/Z whith respect to β. This means that there exists u ∈ U \ V with β(u, x) = 0, ∀x ∈ V. By taking a representaive e of u in P we have e ∈ P \ Q and e commutes with all the elements of Q. So e ∈ Q \ C_{P} (Q) ≠ ∅ which is a contradiction with the fact that Q is p-centric.

Let {v_{1}, …, v_{n}} be a basis of a maximal isotropic subspace of V. If λ (or ρ) over this isotropic subspace we can choose our basis such that λ(v_{1}) = 1 (or ρ(v_{1}) = 1) and λ(v_{i}) = 0 (ρ(v_{i}) = 0), ∀1 < i ≤ n. We complete this basis of V to a basis {u_{i_{1}}, …, u_{i_{m}}, v_{1}, …, v_{n}} of V with β(u_{i_{j}}, v_{i_{j}}) = 1. This basis of V can be completed to a symplectic basis of U. The representatives of these basis of V and U, together with Z, give generating systems for Q and P as we claimed in the lemma.

4. Resistant Groups

Definition 4.1. A p-group P is called resistant if for any finite group G such that P is a Sylow p-subgroup of G, the normalizer N_{G}(P) controls p-fusion in G.

Here is now the main result of this paper.

Theorem 4.2. Let P be a generalized extraspecial p-group. Then P is resistant excepting the case when P = E ×A where A is elementary abelian and E is dihedral of order 8 (when p = 2) or is of order p^{3} and exponent p (when p is odd).

Corollary 4.3. If P satisfies the conditions of the theorem then P is a Swan group.

Before starting the proof of the theorem, we need a preliminary lemma.

Lemma 4.4. Let P be as in the theorem and G be a finite group having P as Sylow p-subgroup. Consider another Sylow p-subgroup P′ of G such that Φ(P) = Φ(P′). If the intersection P ∩ P′ contains a p-centric subgroup Q then P C_{G}(Q) = P′C_{G}(Q).

Proof : As Q is a p-centric subgroup, Lemma 3.6 gives us a generating system Q = < e_{i_{1}}, …, e_{i_{m}}, f_{1}, …, f_{n}, Z > for 0 ≤ m < n of Q, with relations as before. Let Φ := Φ(P) = Φ(P′).

We will prove now that Z(P) = Z(P′). Indeed, let K be the subgroup of G generated by P and P′. As Φ is normal in P and P′, it is normal in K. Moreover P acts trivialy on P/Φ and P′ acts trivialy on P′/Φ so both act trivialy on Q/Φ. This means that the action of K on Q/Φ is also trivial. Now P and P′ are Sylow p-subgroups of K so they are conjugate in K. As Z(P) and Z(P′) are subgroups of Q, Z(P)/Φ = Z(P′)/Φ which is equivalent to Z(P) = Z(P′).

By Lemma 3.6 we can complete the generating system of Q, to symplectic generating systems of P and P′. So there are {e_{i_{m+1}}, …, e_{i_{n}}} and {e′_{i_{m+1}}, …, e′_{i_{n}}} such that {e_{1}, …, e_{n}, f_{1}, …, f_{n}, Z} and {e_{i_{1}}, …, e_{i_{m}}, e′_{m+1}, …, e′_{i_{n}}, f_{1}, …, f_{n}, Z} are symplectic generating systems for P and P′ respectively and such that we have [e_{i}, f_{i}] = [e_{i}, f′_{i}] for i ∈ {i_{m+1}, …, i_{n}}. According to the relations in P and P′ we have [h, e_{i}] = [h, e′_{i}], ∀h ∈ Q and i ∈ {i_{m+1}, …, i_{n}}. This implies that, for the same indices as before, [h, e_{i}^{−1}e′_{i}] = 1 and so e_{i}^{−1}e′_{i} ∈ C_{G}(Q). We conclude that P C_{G}(Q) = P′C_{G}(Q).

Remark 4.5. The fact that Φ(P) = Φ(P′) for any two Sylow p-subgroups of G such that P ∩ P′ contains a p-centric subgroup Q is equivalent to the fact that N_{G}(Φ(P)) controls p-fusion in G. So the lemma above affirms that, if N_{G}(Φ(P)) controls p-fusion in G, then there are no essential subgroups of G in P, which means that N_{G}(P) controls p-fusion in G.

But this is not surprising as Gilotti and Serena [GS] proved that the normalizer of a subgroup Q of a Sylow p-subgroup P of G controls p-fusion in G if and only if Q is strongly closed in P with respect to G and Q admits a central chain of subgroups all weakly closed in P with respect to G. Here a subgroup V of P is said to be strongly closed in P with respect to G if ^{g}V ∩ P ≤ V, ∀g ∈ G and weakly closed in P with respect to G if ^{g}V ≤ P implies ^{g}V = V, g ∈ G.

In our case {1} ◁ Φ(P) ◁ P is a central chain and, obviously P is strongly closed in P with respect to G. So, for proving that N_{G}(P) controls p-fusion in G, it would be sufficient to prove that Φ(P) is weakly closed in P with respect to G. As Φ(P) is a cyclic p-group of order p this is equivalent to proving that Φ(P) is strongly closed in P with respect to G and, therefore, that N_{G}(Φ(P)) controls p-fusion.

In fact the most difficult is to prove that Φ(P) is weakly closed in P with respect to G and for this we need the reduction to the p-centric subgroups of P given by Alperin’s theorem. In what follows we do not consider this approach. We will prove that there is no essential subgroup of G in P.

We first prove the theorem in the case |P/Z(P)| > p^{2}. The remaining case, |P/Z(P)| = p^{2}, needs a more detailed discussion and we treat it in a separate section.

Proposition 4.6. Let P be a generalized extraspecial p-group with |P/Z(P)| > p^{2}. Then P is resistant.

Proof : Let G be a finite group having P as a Sylow p-subgroup. Let Q be a subgroup of P candidate for being essential, so, as before, we can assume that Q is a proper p-centric subgroup of P. By Lemma 3.6, Q has to contain a maximal isotropic subgroup of P. So Q contains Φ(P) and it is a normal subgroup of P as any subgroup of P which contains Φ(P) is normal in P. There are two cases:

1. Q is not maximal isotropic. This forces Q to be non-abelian and so |Φ(Q)| ≥ p. But Φ(Q) ≤ Φ(P) ≃ C_{p} which gives us that Φ(Q) = Φ(P). This fact is true for all Sylow p-subgroups containing Q. So Φ(Q) = Φ(P) = Φ(P′) and by Lemma 4.4, P C_{G}(Q) = P′C_{G}(Q).

2. Q is maximal isotropic. Let C := C_{N_{G}(Q)}(Φ(P)). As P normalises Q and centralizes Φ(P), we have P < C. Let P′ be another Sylow p-subgroup of N_{G}(Q) and let T := P′ ∩ C. The centralizer of a subgroup of P′ of order p is of index at most p in P′ so |P′ : T | ≤ p. The index of a maximal isotropic subgroup in P′ is \sqrt{|P′ : Z|} > p, so |P′ : Q| > p. We have |P′ : T | ≤ p and |P′ : Q| > p so Q is a proper subgroup of T. As Q is maximal isotropic, T is a non-abelian p-group. Let P′′ be a Sylow p-subgroup of C containing T. Now, using the same arguments as in case 1 we have Φ(T) = Φ(P′) = Φ(P′′). At the same time, as P and P′′ are Sylow p-subgroups of C, they are conjugate by an element which centralizes Φ(P) and so Φ(P) = Φ(P′′). Once again we obtain Φ(P) = Φ(P′) and, by Lemma 4.4, P C_{G}(Q) = P′C_{G}(Q).

Cases 1 and 2 show that for all Sylow p-subgroups P and P′ of N_{G}(Q), we have P C_{G}(Q) = P′C_{G}(Q). This implies that W_{G}(Q) has a unique Sylow p-subgroup, so Q cannot be essential. We conclude that there are no essential subgroups in G, for any G having P as a Sylow p-subgroup. By Proposition 2.8, P is resistant.

5. Case |P : Z(P)| = p^{2}

This case needs to be treated separately because we use in the Case 2 of the proof of Proposition 4.2 that the index of a maximal isotropic subgroup in P is at least p^{2}. This is not true if |P : Z(P)| = p^{2}. Let P = < e, f, Z > where Z := Z(P) and [e, f] = z ∈ Z. By Lemma 3.4 we have two possibilities, up to isomorphism, for Z, either Z = < z > ×A or Z = < t > ×A where t^{p} = z.

Proposition 5.1. Let P = < e, f, Z > be a generalized extraspecial p-group, such that Z = < t > ×A with < t^{p} > = Φ(P) as before. Then P is resistant.

Proof : Let G be a group having P as a Sylow p-subgroup. As before the candidates for being essential subgroups of G are p-centric proper subgroups Q of P. Such groups Q are abelian and contain Z. As z = t^{p} ∈ Φ(Q) and Φ(Q) ≤ Φ(P) we have Φ(Q) = Φ(P). Let σ be an automorphism of Q given by conjugation by an element of N_{G}(Q). An automorphism of Q preserves Φ(Q), so σ preserves Φ(P). As Φ(P) = σ(Φ(P)) = Φ(σ(P)), by Lemma 4.4, P C_{G}(Q) = σ(P)C_{G}(Q). Any Sylow p-subgroup of N_{G}(Q) is obtained from P via such an automorphism of Q. So W_{G}(Q) has a unique Sylow p-subgroup and Q cannot be essential. So P is resistant.

We will study now the case where the centre Z = Φ(P) × A with A elementary abelian. For an element x ∈ P we denote by o(x) its order. When P contains elements of order p^{2} we have to distinguish the cases p = 2 and p odd. The results are given in the following two propositions:

Proposition 5.2. Let P = < e, f, Z > be a generalized extraspecial 2-group with < e, f > isomorphic to the quaternion group of order 8 and Z elementary abelian. Then P is resistant.

Proof : Any p-centric subgroups of P is isomorphic to < e, Z >. As this group is of type C_{4} ×A where A is an elementary abelian 2-group, using the same arguments as in Proposition 5.1 we obtain that P is resistant.

The case where p is odd and o(e) = p^{2} is a little bit more complicated. For simplicity we make some notation. Let O_{p′}(G) be the maximal normal p-subgroup of G and Out(Q) := Aut(Q)/Int(Q) be the group of outer automorphisms of Q.

Theorem 5.3. Let p be an odd prime number and P = E×A where A is elementary abelian and E is extraspecial of order p^{3} and exponent p^{2}. Then P is resistant.

Proof

Case 1. Let P = E = < e, f >. The result in this case is due to Glauberman [MP] but the proof we give, which is more elegant, is due to Jacques Thévenaz.

The p-centric subgroups of E are isomorphic to C_{p^{2}} or C_{p} × C_{p}. Let G be a finite group having E as Sylow p-sugroup. First, consider Q < E isomorphic to C_{p^{2}}. As Out(Q) ≃ C_{p} × C_{p−1} which is abelian, W_{G}(Q) ≤ Out(Q) has a unique Sylow p-sugroup and so Q is not essential in G.

Secondly, take Q isomorphic to C_{p} × C_{p}. Suppose that Q is essential in G. As W_{G}(Q) can be embeded in Out(Q) ≃ GL(2, F_{p}) it has 1 or p+1 Sylow p-subgroups. We supposed that Q is essential so W_{G}(Q) needs to have p + 1 Sylow p-subgroups. These p + 1 Sylow p-subgroups generate all SL(2, F_{p}) so W_{G}(Q) ≥ SL(2, F_{p}).

We have C_{G}(Q) = Q × M where M = O_{p′}(C_{G}(Q)). As N_{G}(Q) is an extension of W̃_{G}(Q) by Q, the group H := N_{G}(Q)/M is an extension of W_{G}(Q) by Q. A

Sylow p-subgroup of H is isomorphic of one of N_{G}(Q). Let U := \{( \begin{matrix}1 & ∗ \\ 0 & 1 \end{matrix} )\}

be a Sylow p-subgroup of SL(2, F_{p}). Our aim is to prove that a Sylow p-subgroup of H is a semidirect product of U by Q which is a contradiction because E is not a split extension of U by Q.

We want to prove that H^{2}(SL(2, F_{p}), Q) = 0. Write S := SL(2, F_{p}), Q) and N(U) := N_{SL(2,F_{p})}(U). The restriction in cohomology to U gives a monomorphism res_{U}^{S} : H^{2}(S, Q) → H^{2}(U, Q)^{N(U)} where H^{2}(U, Q)^{N(U)} are the fixed points under the natural action of N(U). Now U = < u > is a cyclic group so [Be, p. 60] its cohomology is

H^{2}(U, Q) = Q^{U}/{(∑_{i=0}^{p−1} u^{i})v|v ∈ Q} .

By a simple computation we obtain Q^{U} = < z >, where z is a generator of Φ(P) and {(∑_{i=0}^{p−1} u^{i})v|v ∈ Q} = 0 so H^{2}(U, Q) = < z >. As z is not fixed by N(U) we have H^{2}(U, Q)^{N(U)} = 0 and therefore H^{2}(S, Q) = 0. This means that the only possible extension of SL(2, F_{p}) by Q is the semidirect product.

Case 2. Now let P = E × A with A elementary abelian. Let G be a finite group having P as Sylow p-subgroup. The p-centric subgroups of P are isomorphic to Q̃ := Q × A with Q as in Case 1.

If Q is isomorphic to C_{p^{2}}, the Frattini subgroup Φ(Q̃) is not trivial and so Φ(P) = Φ(Q̃). Any automorphism of Q̃ preserves Φ(Q̃) and, consequently, Φ(P). So, by Lemma 4.4, Q̃ is not an essential p-sugroup of G.

We will now prove the same for Q̃ when Q is isomorphic to C_{p} × C_{p}. Let z be a generator of Φ(P) and let σ be an automorphism of G given by conjugation by an element of N_{G}(Q̃). Let P′ := σ(P), z′ := σ(z) and H be the group generated by P and P′.

If z′ ∈ Z(P) it means that z′ is a central element in H. So Φ(P′) = < z′ > is invariant by conjugation in H. But Φ(P′) is conjugate to Φ(P) by an element of H. So we have Φ(P) = Φ(P′) and by Lemma 4.4 P C_{H}(Q̃) = P′C_{H}(Q̃). We can do the same if z ∈ Z(P′).

Suppose now that z′ ∉ Z(P) and z ∉ Z(P′). We will see that this is not possible. Indeed, let R := Z(P) ∩ Z(P′). As Z(P) and Z(P′) are distinct subgroups of index p in Q̃, R is of index p in Z(P) and in Z(P′). Moreover R is centralised by H. Let \overline{H} := H/R and \overline{Q} := Q̃/R. As z, z′ ∉ R, \overline{P} := P/R and \overline{P′} := P′/R are extraspecial groups of order p^{3} and exponent p^{2}. Moreover they are Sylow psubgroups of \overline{H}. By Case 1 above, W\overline{H}(\overline{Q}) has a unique Sylow p-subgroup. So \overline{P} C\overline{H}(\overline{Q}) = \overline{P′}C\overline{H}(\overline{Q}) which means that \overline{P} and \overline{P′} are conjugated by an element of C\overline{H}(\overline{Q}). As a conjugation sends Φ(\overline{P}) on Φ(\overline{P′}) which are both subgroups of \overline{Q}, we have Φ(\overline{P}) = Φ(\overline{P′}). This is a contradiction with the fact that Φ(\overline{P}) ≠ Φ(\overline{P′}) as the subgroups generated by \overline{z} and \overline{z′} in H/R are different.

This permits us to conclude that W_{G}(Q̃) has a unique Sylow p-subgroup, and so there are no essential p-sugroups in G. As this is true for any finite group G having P as Sylow p-subgroup, P is resistant.

We prove now that the remaning case is indeed an exception to Theorem 4.2. Let us start with a proprety of resistant groups:

Proposition 5.4. Let P be a p-group and A a finite abelian p-group. If P is not resistant then the direct product P × A is not resistant.

Proof : Let G be a finite group with P as Sylow p-subgroup and let Q be an essential p-subgroup of G embedded in P. Such a G exists because we suppose that P is not resistant. In this case P̃ := P × A is a Sylow p-subgroup of G̃ := G × A. As Q is p-centric in P so is Q̃ := Q × A in P̃. Moreover WG̃(Q̃) ≃ W_{G}(Q). This means that, as S_{p}(W_{G}(Q)) is disconnected, so is S_{p}(WG̃(Q̃)). Then Q̃ is an essential p-subgroup of G̃. This proves that P̃ is not resistant.

Proposition 5.5. Let P = E × A where A is elementary abelian and E is dihedral of order 8 (when p = 2) or is of order p^{3} and exponent p (when p is odd). Then P is not resistant.

Proof : Consider E := < e, f >. We can realise E as a Sylow p-subgroup of

GL(3, F_{p}). One can verify that Q_{1} = \{( \begin{matrix}1 & 0 & ∗ \\ 0 & 1 & ∗ \\ 0 & 0 & 1 \end{matrix} )\} and Q_{2} = \{( \begin{matrix}1 & ∗ & ∗ \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} )\}

are essential subgroups of G. So E is not resistant. As P is isomorphic to E × A where A is elementary abelian, by Proposition 5.4, P is not resistant.

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

Section de Mathématiques, Université de Genève, CP 240, 1211 Genève 24, Switzerland