ON A THEOREM OF MISLIN

J. L. ALPERIN

1. Introduction

A celebrated theorem of G. Mislin [2], using deep homotopy results, shows the equivalence between a purely cohomological condition and the embedding of a subgroup. Specifically, let H be a subgroup of a finite group, p a prime and assume that a Sylow p-subgroup P of G lies in H. Let k be a field of characteristic p, e.g. the field of p elements. Mislin shows that the following two conditions are equivalent:

(1) The restriction map on cohomology H^{∗}(G, k) → H^{∗}(H, k) is an isomorphism:

(2) H is weakly p-embedded in G, that is, whenever Q is a psubgroup of H then N_{G}(Q)/C_{G}(Q) is isomorphic with N_{H}(Q)/C_{H}(Q).

We shall prove two variations on this result, one about strongly pembedded subgroups, and one about an elementary version. We shall also give a reduction theorem, using one of our results, towards a more direct and representation-theoretic proof of Mislin’s result.

2. Strongly p-embedded subgroups

Preserving the above notation, recall that H is said to be strongly p-embedded in G if N_{G}(Q) is contained in H for any non-identity psubgroup Q of H. We shall prove the following:

Theorem 1. The following two conditions are equivalent:

(1) H is strongly p-embedded in G;

(2) Whenever M is a kG-module then the restriction map H^{∗}(G, M) → H^{∗}(H, M) is an isomorphism.

The implication of the second condition from the first is standard, an immediate consequence of the stable element calculation. Hence, we shall assume the second condition and derive the first. Let Q be a non-identity p-subgroup of H. Let M be the kG-module which is the permutation module on the cosets of Q, so by the Eckmann-Shapiro lemma and our hypothesis, we have

H^{∗}(Q, k) ≃ H^{∗}(G, M) ≃ H^{∗}(H, M)

but by applying Mackey’s theorem to the last term, since M there is the restriction of an induced module, we get that H^{∗}(Q, k) is a direct sum of terms of the form

H^{∗}(H, M_{t})

where t runs over a set of H, Q double coset representatives, and M_{t} is the permutation module on the cosets of tQt^{−1} ∩ H. If t is in H this is just H^{∗}(H, M) so that it follows that all the remaining terms are zero. However, if t is in N_{G}(Q) and t is not in H, then

H^{∗}(H, M_{t}) = H^{∗}(Q, k)

which is not zero, a contradiction, so that N_{G}(Q) must indeed be contained in H.

3. Weakly elementary p-embedded subgroups

We no longer assume that H contains a Sylow p-subroup of G, and we do assume now that k is algebraically closed. We shall say that H is weakly elementary p-embedded in G if we have the following conditions:

(1) Whenever E is an elementary abelian p-subgroup of H then N_{G}(E)/C_{G}(E) ≃ N_{H}(E)/C_{G}(E):

(2) Every elementary abelian p-subgroup of G is conjugate with a subgroup of H;

(3) Two elementary abelian p-subgroups of H are conjugate in G if, and only if, they are conjugate in H;

We denote by V_{L}(U), for a KL-module U for a group L, the variety corresponding to the cohomology of U [1]. We can now state our next result, one which has also been observed by Jon Carlson.

Theorem 2. The restriction map of varieties V_{H}(k) → V_{G}(k) is a bijection if, and only if, H is weakly elementary p-embedded in G.

Suppose that the map of varieties is a bijection. The three properties just listed then follow directly from the properties of the Quillen filtration of the cohomology varieties [1, 109-117]. Let α be an element of V_{G}(k) with vertex E. Choose β in V_{H}(k) mapping to α so a vertex of α must be one for β so a conjugate of E lies in H and 2 holds. The bijection used with a result of Quillen, Theorem 9.1.1 of [1], gives 3. The bijection again, with the free action of N_{G}(E)/C_{G}(E) on V_{G,E}(k)^{+}, and the similar property for H, establishes property 1.

On the other hand, suppose the three conditions hold. If α is in V_{G}(k) then a vertex E of α has a conjugate in H so the map of varieties is surjective. Suppose that β_{1} and β_{2} are elements of V_{H}(k) with vertices and sources F_{i}, γ_{i}, i = 1, 2. (Strictly speaking, as each γ_{i} is a point of the variety, that is, a homomorphism of the cohomology algebra to k, the kernel of this map, the ideal, is the source.) Assume that β_{1} and β_{2} have the same image in V_{G}(k). Then there will be an element of G simultaneously conjugating F_{1} to F_{2} and β_{1} to β_{2}, so we can assume, by condition 3, that F_{1} = F_{2} and then we get β_{1} and β_{2} conjugate in F_{1} = F_{2} by 1 so that it follows that γ_{1} = γ_{2}.

4. A reduction

We now return to the notation of the introduction.

Lemma 1. Let H be weakly p-embedded in G.

(1) If Q is a non-identity p-subgroup of H then N_{H}(Q) is weakly p-embedded in N_{G}(Q).

(2) If K is a normal subgroup of G then HK/K is weakly p-embedded in G/K.

Proof. To establish 1, first note that N_{H}(Q) does contain a Sylow psubgroup of N_{G}(Q) since our hypothesis implies that H controls fusion and since some conjugate of Q in H has the property that its normalizer in H contains such a Sylow subgroup. Now let R be a non-identity p-subgroup of N_{H}(Q). Every automorphism induced by G on QR is induced by an element of H, and this element therefore leaves Q invariant so 1 holds.

Turning to 2, we have that HK/K contains a Sylow p-subgroup of G/K. Let Q be a non-identity p-subgroup of H, contained in the Sylow p-subgroup P of G contained in H. Let g be an element of G such that gK normalizes QK/K. Let S be a Sylow p-subgroup of KQ containing Q. After replacing g by another element of gK, we can assume that g normalizes S. Hence, by hypothesis, there is an element h of H such that h and g induce the same automorphism of S so that gK and hK agree on QK/K as claimed. □

We keep the same notation and now we shall say that the subgroup H controls cohomology if condition 2 of the introduction holds. The remainder of this section is devoted to seeing that Mislin’s result follows from the properties that control of cohomlogy passes to p-local subgroups and quotients; that is, if the above Lemma holds for this property 2 then we have a different approach.

Assuming these results on cohomology, we prove the embedding property by induction on the order of G. Let Q be a non-identity p-subgroup of H. If N_{G}(Q) is a proper subgroup of G then induction applies and N_{G}(Q) = C_{G}(Q)N_{H}(Q) as desired, so we may assume that Q is normal. Now we proceed by induction on the order of Q as well. If Q is elementary, then the isomorphism of cohomology gives the same for the associated varities, so, by Theorem 2, we have the desired result. Finally, assume that Q is not elementary abelian so the Fratttin subgroup D(Q) is a proper non-identity subgroup of Q. Induction applies to G/D(Q) so, as a consequence, H and G induce the same group of automorphims on Q/D(Q). The group of automorphims induced on Q by G, which are the indentity on Q/D(Q), is a p-group, so since H contains a Sylow p-subgroup of G, we have that G and H induce the same automorphisms of Q which are the idenity on Q/D(Q). Thus, the claim is established.

References

[1] L. Evens, “The cohomology of groups,” Clarendon Press 1991.

[2] G. Mislin, On group homomorphisms inducing mod-p cohomology isomorphisms, Comment. Math.Helv. 65 (1990), no. 3, 454-461.