COHOMOLOGY AND FINITE SUBGROUPS OF PROFINITE GROUPS

Pham Anh Minh and Peter Symonds

We prove two theorems linking the cohomology of a pro-p group G with the conjugacy classes of its finite subgroups.

Theorem 3. The number of conjugacy classes of elementary abelian p-subgroups of G is finite if and only if the ring H^{∗}(G, ℤ/p) is finitely generated modulo nilpotent elements.

Theorem 5. If the ring H^{∗}(G, ℤ/p) is finitely generated then the number of conjugacy classes of finite subgroups of G is finite.

1. Pro-p groups and elementary abelian pro-p subgroups

Let G be an infinite pro-p group. Fix a fundamental system U of open neighborhoods U of 1 ∈ G such that ∩_{U∈U}U = 1 and each U is an open normal subgroup of G. For convenience, write V ≤ U for V ⊂ U

and U, V ∈ U. For U ∈ U, set G_{U} = G/U. Denote by φ_{U,V} : G_{V} → G_{U} the projection map with V ≤ U. {G_{U}, φ_{U,V} } is an inverse system and

G = lim_{⟵} G_{U}.

Let φ_{U} : G → G_{U}, U ∈ U, be the projection map. Let E (resp. E_{U}) be the set of elementary abelian pro-p subgroups (resp. p-subgroups) of G (resp. G_{U}). (We use the convention that {1} is elementary abelian of rank 0.) It is clear that φ_{U}(E) and φ_{U,V} (F) are elements of E_{U}, for every E ∈ E, F ∈ E_{V} , V ≤ U. Denote by ψ_{U} : E → E_{U} (resp. ψ_{U,V} : E_{V} → E_{U}, V ≤ U) the map induced from φ_{U} (resp. φ_{U,V}).

Proposition 1. Given V ∈ U, there exists W ∈ U with W ≤ U such that, for every E ∈ E_{W}, ψ_{V,W} (E) ∈ Im ψ_{V}.

Proof. Let F be an element of E_{V}. Suppose that F ∉ Im ψ_{V}. Then there exists S ∈ U with S ≤ V such that F ∉ Im ψ_{V,S}. Since G_{V} is finite, there exists W ∈ U with W ≤ V such that F ∈ Im ψ_{V,W} implies F ∈ Im ψ_{V}, for every F ∈ E_{V}.

Let E be an arbitrary element of E_{W}. Since ψ_{V,W} (E) is an element of E_{V}, it also belongs to Im ψ_{V}. The proposition is proved. □

The following notation will be used. Let K be a (profinite) p-group. H^{∗}(K) will always denote the (continuous) cohomology of K with coefficients ℤ/p. Denote by I(K) the ideal of H^{∗}(K) consisting of elements which restrict trivially to every elementary abelian subgroup of K, and by ℜ_{K} the nilradical of H^{∗}(K). Set

H(K) = \{ \begin{matrix}H^{ev}(K) & for p > 2, \\ H^{∗}(K), & for p = 2 \end{matrix}

and let H^{+}(K) be the ideal of H^{∗}(K) consisting of elements of positive degrees. We have

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20J06, 17B50.

Proposition 2. Given η ∈ I(G), there exist W ∈ U and ξ ∈ I(G_{W}) such that (φ_{W})^{∗}(ξ) = η. In particular, if H^{∗}(G) is finitely generated, there exists U ∈ U such that I(G) ⊂ (φ_{U})^{∗}(I(G_{U})).

Proof. Let η be an element of I(G). Since H^{∗}(G) = lim_{⟶} H^{∗}(G_{U}), there exist V and ζ ∈ H^{∗}(G_{V}) such that φ_{V}^{∗} (ζ) = η. Set ξ = φ_{V,W}^{∗} (ζ) with W given in Proposition 1. Also φ_{W}^{∗} (ξ) = η. We now prove that ξ ∈ I(G_{W}).

Let E be an element of E_{W} and set K = ψ_{V,W} (E). By Proposition 1, there exists M ∈ E such that ψ_{V} (M) = K. Consider the commutative diagram

H^{∗}(G_{W}) ⟵^{φ_{V,W}^{∗}} H^{∗}(G_{V}) ⟶^{φ_{V}^{∗}} H^{∗}(G) ^{Res}↓  ^{Res}↓  ↓^{Res} H^{∗}(E) ⟵_{(φ_{V,W} |_{E})^{∗}} H^{∗}(K) ⟶_{(φ_{V} |_{M})^{∗}} H^{∗}(M)  .

As φ_{V} |_{M} is a monomorphism and η|_{M} = 0, the right square of the diagram tells us that ζ|_{K} = 0. So, by the commutativity of the left square, ξ|_{E} = 0. Hence ξ ∈ I(G_{W}).

If H^{∗}(G) is finitely generated, V can be chosen such that H^{∗}(G) = Im (φ_{V})^{∗}. It follows from what we just proved that I(G) ⊂ (φ_{W})^{∗}(I(G_{W})). □

We can now deduce the profinite case of the following theorem of Quillen from the finite case, where many fairly elementary proofs are now known.

Theorem 1 (Quillen [8]). I(G) ⊂ ℜ_{G}. In other words, every element of I(G) is nilpotent.

Proof. Straightforward from Proposition 2, by noting that I(G_{W}) ⊂ ℜ_{G_{W}}, by the finite case, as G_{W} is finite. □

This result also appears in [9].

Proposition 3. Suppose that there exist U ∈ U and a ∈ G \ U with ord(a) = p. Set b = φ_{U}(a) ∈ G_{U}. Then there exists ξ ∈ H^{+}(G_{U}) satisfying:

(a) ξ|_{⟨b⟩} ≠ 0;

(b) φ_{U}^{∗}(ξ)|_{⟨a⟩} ≠ 0. In particular, φ_{U}^{∗}(ξ) is not nilpotent in H^{+}(G).

Proof. By [7, Theorem 2.7], there exists ξ ∈ H^{+}(G_{U}) such that ξ|_{⟨b⟩} is not nilpotent. From the commutative diagram

\begin{matrix}H^{∗}(G_{U}) & ⟶^{φ_{U}^{∗}} & H^{∗}(G) \\ ^{Res}↓ & & ↓^{Res} \\ H^{∗}(⟨b⟩) & ⟶_{(φ_{U} |_{⟨a⟩})^{∗}} & H^{∗}(⟨a⟩) \end{matrix} ,

since H^{∗}(⟨b⟩) →^{(φ_{U} |_{⟨a⟩})^{∗}} H^{∗}(⟨a⟩) is an isomorphism, it follows that φ_{U}^{∗}(ξ)|_{⟨a⟩} is not nilpotent. The proposition follows. □

Theorem 2. A pro-p group A is torsion-free if and only if H^{+}(A) ⊂ ℜ_{A}.

Proof. If A is not torsion-free, it follows from Proposition 3 that H^{+}(A) contains a non-nilpotent element. If A is torsion-free, then I(G) = H^{+}(A); hence, by Theorem 1, any element of H^{+}(A) is nilpotent. □

Corollary 1. (i) A subgroup A of G is torsion-free if and only Im (H^{+}(G) →^{Res} H^{+}(A)) ⊂ ℜ_{A}; (ii) If H^{∗}(G)/ℜ_{G} is finitely generated, then G contains an open, normal, torsion-free subgroup U.

Proof. (i) If A is torsion-free, it follows from Theorem 2 that H^{+}(A) consists of nilpotent elements; hence any element of Im (H^{+}(G) →^{Res} H^{+}(A)) is nilpotent. Conversely, suppose that A contains an element a of order p. By Proposition 3, there exists ζ ∈ H^{+}(G) such that ζ|_{⟨a⟩}, hence ζ|_{A}, is not nilpotent. (ii) Suppose that H^{∗}(G)/ℜ_{G} is finitely generated. Since H^{∗}(G) = lim_{⟶} H^{∗}(G_{V}), there exists U ∈ U such

that H^{∗}(G_{U})/ℜ_{G_{U}} →^{(φ_{U})^{∗}} H^{∗}(G)/ℜ_{G} is surjective. Hence Im (H^{+}(G) →^{Res} H^{+}(U)) ⊂ ℜ_{U}. By (i), U is torsion-free.

The corollary follows. □

Remark 1. The following example shows that the converse of Corollary 1(ii) does not hold. For every n ∈ ℕ, define 𝔄_{n} to be the procyclic group ℤ_{2} generated by e_{n}. Set 𝔄 = ˆ⊕_{n}𝔄_{n} and let ⟨a⟩ ≅ ℤ/2 act on 𝔄 by ^{a}e_{n} = e_{n}^{−1}. Define G = 𝔄 ⋊ ℤ/2. For every n, ⟨e_{n}a⟩ ≅ ℤ/2 and ⟨e_{n}a⟩ is not conjugate with ⟨e_{m}a⟩ if m ≠ n. Hence there are infinitely many conjugacy classes of elementary abelian subgroups of G. According to Theorem 3 below, H^{∗}(G)/ℜ_{G} is not finitely generated, although 𝔄 is open, normal and torsion-free in G.

We will give a necessary and sufficient condition for H^{∗}(G) to be finitely generated as a ring. First we prepare.

Lemma 1. Suppose that H^{∗}(G) is finitely generated and M is a finite 𝔽_{p}G-module. Then H^{∗}(G, M) is noetherian over H^{∗}(G).

Proof. We prove by induction on n = dim_{𝔽_{p}} M. Suppose that the lemma holds for n − 1. Consider the exact sequence of 𝔽_{p}G-modules

0 → 𝔽_{p} →^{f} M →^{g} N → 0

with N = M/f(𝔽_{p}). We then have the corresponding long exact sequence of cohomology

… H^{∗−1}(G, N) →^{δ_{∗}} H^{∗}(G) →^{f_{∗}} H^{∗}(G, M) →^{g_{∗}} H^{∗}(G, N) → ….

By the inductive assumption, H^{∗}(G, N) is noetherian over H^{∗}(G), hence so is Im g_{∗}. Pick elements ξ_{1}, …, ξ_{m} of H^{∗}(G, M) so that {g_{∗}(ξ_{1}), …, g_{∗}(ξ_{m})} is a set of generators of Im g_{∗}. It follows that H^{∗}(G, M) is generated by ξ_{1}, …, ξ_{m}, as a module over Im f_{∗}. Hence H^{∗}(G, M) is noetherian over H^{∗}(G). □

Corollary 2. The following are equivalent:

(a) H^{∗}(G) is finitely generated;

(b) G contains an open normal, torsion-free subgroup U such that H^{∗}(U) is finite;

(c) there exists an open subgroup K of G such that H^{∗}(K) is finitely generated;

(d) H^{∗}(K) is finitely generated, for any open subgroup K of G.

Proof. The implication (b) ⇒ (a) was proved by Quillen ([7, Proposition 13.5]) using a spectral sequence argument. It is clear that (d) ⇒ (c) and (d) ⇒ (a). Suppose that H^{∗}(G) is finitely generated and K is open in G. By the Eckmann-Shapiro lemma, H^{∗}(K) = H^{∗}(G, M) with M = Hom_{K}(𝔽_{p}G, 𝔽_{p})). Since M is noetherian as a 𝔽_{p}-module, H^{∗}(K) is noetherian over H^{∗}(G), by Lemma 1. As H^{∗}(G) is finitely generated, so is H^{∗}(K). In particular, if U is given as in Corollary 1 (ii), then H^{∗}(U) is finite dimensional, as H^{+}(U) ⊂ ℜ_{U}. We then have (a) ⇒ (b) and (a) ⇒ (d).

Finally, suppose that K is open in G and H^{∗}(K) is finitely generated. It follows that K contains an open, normal, torsion-free subgroup U such that H^{∗}(U) is finite. As U is also open in G, U contains an open, normal subgroup V of G. Since V is torsion-free and open in U, H^{∗}(V) is finite. So H^{∗}(G) is finitely generated. The implication (c) ⇒ (a) is then proved. □

Remark 2. In [7], it was proved that, if H^{∗}(G) is finitely generated, then G has only finitely many conjugacy classes of elementary abelian pro-p subgroups. However the converse does not hold: the group 𝔄 given above has only one elementary abelian subgroup, which is the one of rank 0, while H^{∗}(𝔄) is an exterior algebra with an infinite number of generators of degree 1, hence is not finitely generated.

From now on, fix F a set of representatives of conjugacy classes of elementary abelian pro-p subgroups of G. We now give a cohomological criterion for F to be finite, as follows.

Theorem 3. The following are equivalent:

(a) There exists an open normal subgroup A of G such that EA ≠ F A, for E ≠ F in F;

(b) H^{∗}(G)/ℜ_{G} is finitely generated (as a ring);

(c) F is finite.

Proof. (a) ⇒ (c): Let S be the set of subgroups of G_{A}. The map f : F → S, E ↦ EA/A is then injective. As f maps F injectively into the finite set S, F is finite.

The implication (c) ⇒ (a) is clear. We now prove (b) ⇒ (c). Suppose that H^{∗}(G)/ℜ_{G} is finitely generated. Let U be the open, normal, torsion-free subgroup of G as given in Corollary 1(ii). It follows from the

proof of the corollary that, for every V ≤ U, V is torsion-free and H^{∗}(G_{V})/ℜ_{G_{V}} →^{(φ_{V} )^{∗}} H^{∗}(G)/ℜ_{G} is surjective.

Let M be a set of representatives of conjugacy classes of maximal elementary abelian pro-p subgroups of G and let E, F be two different elements of M. Since U is torsion-free, it follows that φ_{V} maps E (resp. F) isomorphically to φ_{V} (E) (resp. φ_{V} (F)), for every V ≤ U. Furthermore, as E ≠ F, there exists W ≤ U such that each of φ_{W} (E), φ_{W} (F) is not conjugate (in G_{W}) to any subgroup of the other. According to [7, Theorem 2.7], there exist ξ, η ∈ H^{∗}(G_{W}) such that ξ|_{φ_{W} (E)}, η|_{φ_{W} (F)} are not nilpotent, and ξ|_{φ_{W} (F)} = 0, η|_{φ_{W} (E)} = 0. Set ξ′ = (φ_{W})^{∗}(ξ), η′ = (φ_{W})^{∗}(η). It follows that ξ′|_{E}, η′|_{F} are not nilpotent, and ξ′|_{F} = 0, η′|_{E} = 0; in particular, ξ′ and η′ are not nilpotent. Since

H^{∗}(G_{U})/ℜ_{G_{U}} →^{(φ_{U})^{∗}} H^{∗}(G)/ℜ_{G} is surjective, there exist ζ, θ ∈ H^{∗}(G_{U}) such that ζ|_{φ_{U} (E)}, θ|_{φ_{U} (F)} are not

nilpotent, and ζ|_{φ_{U} (F)} = 0, θ|_{φ_{U} (E)} = 0. Also, by [7, Theorem 2.7], it follows that each of φ_{U}(E), φ_{U}(F) is not conjugate to any subgroup of the other; in particular, φ_{U}(E) ≠ φ_{U}(F). So ψ_{U} maps M injectively into E_{U}. Hence M is finite, and so is F.

Finally, let us prove (c) ⇒ (b). Suppose that F is finite. It follows that there exists T ∈ U such that, for every E ∈ E, φ_{T} |_{E} is an isomorphism, and ψ_{T} maps F injectively into E_{T}. For every E ∈ F, the restriction map Res_{φ_{T} (E)}^{G_{T}} induces an action of H(G_{T}) on H^{∗}(φ_{T} (E)), hence on H^{∗}(E); furthermore, by [4, Corollary 7.4.7], H^{∗}(E) is a finitely generated H(G_{T})-module. Besides, the inflation (φ_{T})^{∗} also induces an action of H(G_{T}) on H^{∗}(G) so that Res_{E}^{G} is a H(G_{T})-homomorphism, for every E ∈ F. Set 𝔍 = ∏_{E∈F} H^{∗}(E). Since F is finite, 𝔍 is a finitely generated H(G_{T})-module. As H^{∗}(G)/I(G) is isomorphic to a submodule of 𝔍, it is also a finitely generated H(G_{T})-module. Furthermore, I(G) ⊂ ℜ_{G} implies that H^{∗}(G)/ℜ_{G} is, in turn, a finitely generated H(G_{T})-module. Since H(G_{T}) is a finitely generated algebra, so is H^{∗}(G)/ℜ_{G}. □

It has been pointed out to us by Henn that such a result can also be deduced from material in [6].

Let E be the category with objects the elementary abelian pro-p subgroups of G and with morphisms from A to B defined to be the homomorphisms θ : A → B of the form θa = gag^{−1} for some g ∈ G. Set 𝔏 = lim_{⟵}_{E∈E}H(E). It is clear that the projection map

∏_{E∈E} H^{∗}(E) → 𝔍 = ∏_{E∈F} H^{∗}(E)

induces a monomorphism, between H(G)-modules, from 𝔏 to 𝔍. From the proof of the above theorem, 𝔍 is finitely generated over H(G). We then have

Proposition 4. If F is finite, then 𝔏 is finitely generated over H(G). □

According to [8, Proposition 13.4], if F is finite, the map

H(G) →^{Res^{G}} 𝔏

is an F -isomorphism. In other words, given x ∈ I(G) and y ∈ 𝔏, there exists an integer n = n_{x,y} such that x^{n} = 0 and y^{p^{n}} ∈ Im Res^{G}. We now give a sufficient condition for Res^{G} to be a uniform F -isomorphism (i.e., the integer n can be chosen independently of x and y).

Theorem 4. If H^{∗}(G) is finitely generated, then Res^{G} is a uniform F -isomorphism.

Proof. Suppose that H^{∗}(G) is finitely generated. F is then finite and the map H(G) →^{Res^{G}} 𝔏 is an F - isomorphism. By Proposition 4, 𝔏 is finitely generated over H(G). So there exists an integer r such that y^{p^{r}} ∈ Im Res^{G}, for every y ∈ 𝔏.

By Proposition 2, there exists U ∈ U such that I(G) ⊂ (φ_{U})^{∗}(I(G_{U})). As G_{U} is finite, there exists an integer s such that x^{s} = 0, for every x ∈ I(G). The theorem is then proved, by setting n = n_{x,y} = max(r, s). □

2. Pro-p groups and finite p-subgroups

Let G be an infinite pro-p group. The purpose of this section is to prove the following

Theorem 5. If H^{∗}(G) is finitely generated, then G has only finitely many conjugacy classes of finite p-subgroups.

An analogous result is known for discrete groups of finite virtual cohomological dimension over p (see [1], 13.2) and for analytic pro-p groups (where the hypothesis is vacuous, see [2]).

Our proof depends on some deep results on unstable algebras over the Steenrod algebra.

We will need:

Lemma 2. If H^{∗}(G) is finitely generated and C is a finite central subgroup of G, then H^{∗}(G/C) is finitely generated.

Proof. Without loss of generality, we may suppose that C is of order p. Write K = G/C and let z ∈ H^{2}(K) be the cohomology class classifying the central extension

1 → C → G → K → 1.

There exists then an open, normal subgroup L of K such that Res_{L}^{K}(z) = 0. Therefore the preimage H of L in G is isomorphic to C × L. In other words, L can be considered as an open subgroup of G. By Corollary 2, H^{∗}(L), and so H^{∗}(K), are finitely generated. □

For every subgroup P of G, denote by N_{G}(P) (resp. C_{G}(P)) the normalizer (resp. centralizer) of P in G. We have

Lemma 3. If P is a subgroup of G of order p, then:

(i) N_{G}(P) = C_{G}(P); (ii) H^{∗}(C_{G}(P)) and H^{∗}(C_{G}(P)/P) are finitely generated, provided that so is H^{∗}(G).

Proof. (i) follows from the fact that N_{G}(P)/C_{G}(P) is embedded into Aut(P) which is of order p − 1.

(ii) By Lemma 2, we need only prove that H^{∗}(C_{G}(P)) is finitely generated. By Corollary 2, there exists U open, normal, torsion-free in G. Set K = ⟨U, P ⟩. K is then of p-rank 1 (i.e. every elementary abelian subgroup of K is of rank at most 1), open in G and has finitely generated cohomology, by Corollary 2. Furthermore, C_{K}(P) is open in C_{G}(P). By Corollary 2, it suffices to prove that H^{∗}(C_{K}(P)) is finitely generated. According to the theory in [3] (see also [7, Corollary 1.7]) the unstable algebra T^{V} H^{∗}(G) is noetherian and also T^{V} H^{∗}(G) = ∏_{(ρ)} H^{∗}(C_{K}(imρ)), where the product is taken over conjugacy classes of homomorphisms ρ : ℤ/p → K. It follows that H^{∗}(C_{K}(P)) is finitely generated. □

Proof of Theorem 5. As H^{∗}(G) is finitely generated, it follows from Corollary 2 that there exists an open normal, torsion-free subgroup U of G. Define

n_{G} = min{n||G/U| = p^{n} for some open, normal, torsion-free subgroup U of G}.

We argue by induction on n_{G}. If n_{G} = 1, the conclusion follows, as every finite subgroup of G is elementary abelian of rank 1. Suppose that the theorem holds if n_{G} < m.

Assume that n_{G} = m. It is known that the number of conjugacy classes of elementary abelian psubgroups of rank 1 of G is finite. Let {C_{1}, …, C_{k}} be a set of representatives of such conjugacy classes. It is clear that, for any finite p-subgroup P of G, there exist g ∈ G and i such that P^{g} contains C_{i} as a central subgroup. Let N_{i} be the set of finite subgroups of G containing C_{i} as a central subgroup, 1 ≤ i ≤ k. It is then sufficient to prove that the number of conjugacy classes in N_{i} is finite, 1 ≤ i ≤ k.

Fix such an i. Note that N_{i} coincides with the set of finite subgroups of K = C_{G}(C_{i}) containing C_{i}, hence is in 1 − 1 correspondence with the set of finite subgroups of H = C_{G}(C_{i})/C_{i}. Therefore we need only prove that H has many finitely conjugacy classes of finite subgroups. Let U be open, normal, torsion-free in G with |G/U| = p^{n_{G}} and set V = U ∩ K. V is then open, normal, torsion-free in K and |K/V | ≤ |G/U| = p^{n_{G}}. So n_{K} ≤ n_{G}. As n_{H} = n_{K} − 1 < n_{G}, it follows from the inductive hypothesis that H has many finitely conjugacy classes of finite subgroups. The theorem is proved. □

References

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6. H.-W. Henn, J. Lannes, L. Schwarz, The categories of unstable modules and unstable algebras over the Steenrod algebra modulo nilpotent objects, Amer. Jour. Math. 115 (1993), 1053-1106.

6. D. Quillen, A cohomological criterion for p-nilpotence, J. Pure Appl. Alg. 1 (1971), 361-372.

8. , The spectrum of an equivariant cohomology ring II, Ann. Math. 94 (1971), 573-602.

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Department of Mathematics, College of Science, University of Hue, Dai hoc Khoa hoc, Hue, Vietnam.

  E-mail address: paminh@dng.vnn.vn

Department of Mathematics, U.M.I.S.T., P.O. Box 88, Manchester M60 1QD, England.

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