THE COHOMOLOGY OF PRO-ppp GROUPS WITH

A POWERFULLY EMBEDDED SUBGROUP

Pham Anh Minh* and Peter Symonds

Abstract. We calculate the cohomology of a pro-p group with an extendable and almost powerfully embedded subgroup.

We consider the mod-p cohomology of p-groups and pro-p groups with an almost powerfully embedded subgroup satisfying an extendibility condition. The result has a strikingly simple form.

If G is a pro-p group for some prime p we say that a closed, normal, finitely generated subgroup, N, is almost powerfully embedded if:

- [G, N] ⊂ N^{p} for p > 2;

- [G, N] ⊂ N^{2} and [N, N] ⊂ (N^{2})^{2} for p = 2.

Where N^{p} denotes the closure of the subgroup of N generated by p-th powers of elements of N.

Let Ω_{1}N denote the subgroup of N generated by elements of order p. We say that N is Ω-extendable in G if Ω_{1}N is central in G, so in particular is an elementary abelian subgroup, and if also there is a central extension E → G̃ → G, where E ≅ Ω_{1}N and every non-trivial element of Ω_{1}N is the image of an element of G̃ of order p^{2}. (So every torsion free group is extendable.)

For any closed normal subgroup M ⊂ G we define Φ_{G}(M) = [G, M]M^{p}.

Note: It follows from [4, 3.1 and 3.8] that the subgroups defined above are closed, indeed all but the last are open. Also every element of N^{p} is a p-th power and Ω_{1}N is finite.

Denote by H^{∗}(G) the cohomology of G with coefficients in ℤ/p (the Galois or continuous cohomology if G is infinite, see [11]), and by β : H^{∗}(G) → H^{∗+1}(G) the Bockstein homomorphism. Our main theorem is the following consequence of Theorem 3.13.

Theorem. Let G be a pro-p group and N an almost powerfully embedded subgroup, Ω-extendable in G. Then there exist elements z_{i}^{(1)}, …, z_{d}^{(1)} of H^{2}(G/Φ_{G}(N)), z_{1}, …, z_{k} of H^{2}(G) such that

(i) H^{∗}(G) ≅ H^{∗}(G/Φ_{G}(N))/(z_{1}^{(1)}, …, z_{d}^{(1)}) ⊗ ℤ/p[z_{1}, …, z_{k}]; (ii) z_{1}^{(1)}, …, z_{d}^{(1)} classify the extension

(ℤ/p)^{d} ≅ Φ_{G}(N)/Φ_{G}Φ_{G}(N) → G/Φ_{G}Φ_{G}(N) → G/Φ_{G}(N);

(iii) z_{1}, …, z_{k} restrict to a basis of βH^{1}(Ω_{1}(N) ∩ N^{p}).

We also give several partial converses, which give group theoretic information when the cohomology has the form given above. Our basic computational tool is the spectral sequence argument of Proposition 2.4. Finally, in section 4, we calculate the Bocksteins up to an error term which vanishes if the extension is itself extendable.

This problem was investigated by Weigel [13], who considered the case when p is odd and N = G and also by Browder and Pakianathan [2], who considered the case when N = G is uniform and also required

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 20J06, 17B50.

*Supported by a grant from the E.P.S.R.C.

p ≥ 5. The latter authors also considered the Bocksteins. In all cases the key spectral sequence argument is similar.

1. Regular Sequences

A regular sequence in ring R is a sequence of elements a_{1}, …, a_{n} such that for each i = 1, …, n, a_{i} does not annihilate any non-trivial element of R/(a_{1}, …, a_{i−1}).

Recall the definition of the Koszul complex over R. Given a sequence of elements a_{1}, …, a_{n} in the centre of R, K = K_{R}(a_{i}, …, a_{n}) is the the free R-module on certain symbols, which we can take to be monomials x_{i_{1}} … x_{i_{r}} in the symbols x_{1}, …, x_{n} such that i_{1} < i_{2} < ⋯ < i_{r} (so no squares occur). It is graded by the degree of the monomial. The differential d_{r} : K_{r} → K_{r−1} is the R-linear map defined by

d_{r}(x_{i_{1}} … x_{i_{r}}) = ∑_{j=1}^{r}(−1)^{j+1}a_{i_{j}}x_{i_{1}} … x̂_{i_{j}} … x_{i_{r}}.

If a_{1}, …, a_{n} is a regular sequence of central elements of R then it is well known that the homology of the associated complex is just R/(a_{1}, …, a_{n}) in degree 0 and 0 elsewhere.

It is tempting to identify K with the exterior algebra over R on x_{1}, …, x_{n}, and indeed this makes K into a differential graded algebra, but the algebra structure is not part of the definition.

If 2R = 0 there is a slight variation on this theme, which we will need. Let J = J_{R}(a_{1}, …, a_{n}) be the free R-module on all monomials in x_{1}, …, x_{n} , again graded by the degree, and define d_{r} : J_{r} → J_{r−1} by the same formula as before.

We identify J with the polynomial ring R[x_{1}, …, x_{n}] and it becomes a differential graded algebra. Now we consider J as a module over R[x_{1}^{2}, …, x_{n}^{2}]. We do this because d_{2}(x_{i}^{2}) = 0 and hence d is linear over R[x_{1}^{2}, …, x_{n}^{2}].

But as an R[x_{1}^{2}, …, x_{n}^{2}]-module, J ≅ R[x_{1}^{2}, …, x_{n}^{2}] ⊗ K (even though this is not compatible with any multiplicative structure) and so has homology R[x_{1}^{2}, …, x_{n}^{2}]⊗R/(a_{1}, …, a_{n}) ≅ R/(a_{1}, …, a_{n})[x_{1}^{2}, …, x_{n}^{2}]. We record this as:

Lemma 1.1. Let a_{1}, …, a_{n} is a regular sequence of central elements in a ring R. Then: (1) H_{∗}(K_{R}(a_{1}, …, a_{n})) ≅ R/(a_{1}, …, a_{n}),

(2) If 2R = 0 then H_{∗}(J_{R}(a_{1}, …, a_{n})) ≅ R/(a_{1}, …, a_{n})[x_{1}^{2}, …, x_{n}^{2}]. □

We now provide some regular sequences in the mod-p cohomology of a finite p-group. We shall use the following notation. Let R = C_{p^{r_{1}}} (e_{1}) × C_{p^{r_{2}}} (e_{2}) × ⋯ × C_{p^{r_{k}}} (e_{k}) be an abelian p-group, with C_{p^{r_{i}}} (e_{i}) the cyclic group of order p^{r_{i}} generated by e_{i}.

For each 1 ≤ i ≤ k, let x_{i} ∈ H^{1}(C_{p^{r_{i}}} (e_{i})) ≅ Hom(C_{p^{r_{i}}} (e_{i}), ℤ/p) correspond to the homomorphism that takes e_{i} to 1 and let y_{i} ∈ H^{2}(C_{p^{r_{i}}} (e_{i})) be a generator. It is well known that

H^{∗}(C_{p^{r_{i}}}) = \{ \begin{matrix}ℤ/2[x_{i}] & for p = 2 and r_{i} = 1, \\ Λ[x_{i}] ⊗ ℤ/p[y_{i}] & otherwise. \end{matrix}

We identify H^{∗}(R) with H^{∗}(C_{p^{r_{1}}} (e_{1})) ⊗ H^{∗}(C_{p^{r_{2}}} (e_{2})) ⊗ ⋯ ⊗ H^{∗}(C_{p^{r_{k}}} (e_{k})) under the Künneth isomorphism and identify x_{i} and y_{i} with their images in H^{∗}(R).

Let λ^{∗}(R) denote the subalgebra of H^{∗}(R) generated by the x_{i} (this need not be an exterior algebra if p = 2) and set λ(R) = λ^{2}(R) for brevity. Let B(R) ⊂ H^{∗}(R) be the subspace generated by the y_{i}. Note that B(R) can be characterized as the image of H^{2}(R; ℤ) in H^{2}(R). Also B(R) contains the image of the Bockstein map from H^{1}(R), but is not equal to it unless R is elementary abelian. We record the following elementary consequences:

COHOMOLOGY OF A PRO-p GROUPS WITH A POWERFULLY EMBEDDED SUBGROUP 3

Lemma 1.2. (i) H^{2}(R) = \{ \begin{matrix}λ(R) + B(R) & always \\ λ(R) ⊕ B(R) & if p > 2 or p = 2 and each r_{i} ≥ 2 \\ λ(R) & for p = 2 and r_{1} = 1. \end{matrix}

(ii) y_{1}, …, y_{k} is a regular sequence in H^{∗}(R). Furthermore, given elements γ_{1}, …, γ_{k} of λ(R), the sequence y_{1} + γ_{1}, …, y_{k} + γ_{k} is regular in H^{∗}(R), provided that p > 2, or p = 2 and min(r_{1}, …, r_{k}) > 1. □

Define H^{+}(G) to be the ideal of H^{∗}(G) consisting of elements of positive degrees.

Proposition 1.3. Given a central extension of p-groups

0 →^{θ} A → G → K → 1

with A ≅ (ℤ/p)^{m}, we have:

(i) If A is contained in the Frattini subgroup of G, then Im (H^{2}(G) →^{Res} H^{2}(A)) ⊂ B(A).

(ii) If there exist elements z_{1}, …, z_{m} of H^{2}(G) and γ_{1}, …, γ_{m} ∈ λ(A) such that either z_{1}|_{A}, …, z_{m}|_{A} is linearly independent in B(A), or p > 2 and z_{1}|_{A} − γ_{1}, …, z_{m}|_{A} − γ_{m} is linearly independent in B(A), then z_{1}, …, z_{m} is a regular sequence in H^{∗}(G) .

Proof. (i) was given in [10, Proposition 1.5]. We now prove (ii). Let i_{A} : A → A × G, i_{G} : G → A × G, f : A × G → G be defined by i_{A}(a) = (a, 1), i_{G}(g) = (0, g), f(a, g) = θ(a)g, a ∈ A, g ∈ G. Since A is central, f is a homomorphism of groups. Note that (f ∘i_{A})^{∗} (resp. (f ∘i_{G})^{∗}) is just the restriction (resp. identity) map on cohomology. So, for 1 ≤ i ≤ m,

f^{∗}(z_{i}) = z_{i}|_{A} ⊗ 1 + 1 ⊗ z_{i} mod H^{+}(A) ⊗ H^{+}(G).

We now use the argument similar to that of [1, Proof of Theorem 1.1]. The special form of f^{∗}(z_{i}) induces for 1 ≤ i ≤ n a homomorphism of algebras

f_{i}^{∗} : H^{∗}(G)/(z_{1}, …, z_{i−1}) → H^{∗}(A)/(z_{1}|_{A}, …, z_{i−1}|_{A}) ⊗ H^{∗}(G)/(z_{1}, …, z_{i−1}).

Let y be a non-trivial element of H^{∗}(G)/(z_{1}, …, z_{i−1}). We now show that z_{i}y is non-trivial, by claiming that f_{i}^{∗}(z_{i}y) is non-trivial.

Write f_{i}^{∗}(y) = ∑_{s≥0} v_{s} and f_{i}^{∗}(z_{i}y) = ∑_{s≥0} w_{s} with v_{s}, w_{s} in

(H^{∗}(A)/(z_{1}|_{A}, …, z_{i−1}|_{A}))^{s} ⊗ H^{∗}(G)/(z_{1}, …, z_{i−1}).

Then v_{0} = 1 ⊗ y is non-trivial. Let s_{1} be maximal such that v_{s_{1}} is non-trivial. It follows that, for s_{2} = s_{1} + 2, w_{s_{2}} = (z_{i}|_{A} ⊗ 1) · v_{s_{1}}. By Lemma 1.2 (ii), z_{1}|_{A}, …, z_{i}|_{A} is a regular sequence in H^{∗}(A). hence w_{s_{2}} ≠ 0. Thus z_{1}, …, z_{m} is then a regular sequence in H^{∗}(G). □

2. Powerfully embedded subgroups and a cohomological characterization

Let P be a pro-p group and let N be a closed normal subgroup of P . Set Q = P/N and denote by π : P → Q the projection map. Let A be an elementary abelian p-group of rank m and fix a basis a_{1}, …, a_{m} of A. Regard A as a trivial Q-module, so

H^{2}(Q, A) = ⊕_{i=1}^{m}H^{2}(Q, ℤ/p ⟨a_{i}⟩) = H^{2}(Q)^{⊕m}.

Pick an element z ∈ H^{2}(Q, A). For every closed subgroup R of Q, denote by

0 → A → R_{z} →^{π_{R,z}} R → 1

the central extension of groups corresponding to the cohomology class z|_{R} = Res_{R}^{Q}(z) ∈ H^{2}(R, A).

Let u_{1}, …, u_{m} be the basis of H^{1}(A) = Hom(A, ℤ/p), dual to that of A. We then have the projection map

(u_{i})_{∗} : H^{2}(Q, A) → H^{2}(Q), 1 ≤ i ≤ m,

and z can be expressed as z = (z_{1}, …, z_{m}) with z_{i} = (u_{i})_{∗}(z).

Lemma 2.1. (i) All the Inf_{P}^{Q}(z_{i}) = 0 if and only if there exists a surjective homomorphism τ : P → Q_{z} satisfying π_{Q,z} ∘ τ = π.

(ii) Let R = C_{p^{r_{1}}} (e_{1}) × ⋯ × C_{p^{r_{k}}} (e_{k}) be an abelian subgroup of Q. Then: (iia) R_{z} is abelian if and only if z_{ℓ}|_{R} ∈ B(R), 1 ≤ ℓ ≤ m; (iib) let S be the maximal elementary abelian subgroup of R^{2}. Then (S_{z})^{p} = 1 if and only if

z_{ℓ}|_{R} ∈ λ(R), 1 ≤ ℓ ≤ m.

Proof. (i) Let 0 → A → P_{z} → P → 1 be a group extension classified by the element z = Inf(_{P}^{Q}(z_{1}), …, Inf_{P}^{Q}(z_{m})). There exists then a commutative diagram

0 ⟶ A ⟶ P_{z} ⟶^{ρ} P ⟶ 1
∥  ^{μ}↓  ↓^{π}
0 ⟶ A ⟶ Q_{z} ⟶^{π_{Q,z}} Q ⟶ 1.

If all the Inf_{P}^{Q}(z) = 0 then ρ is split, by σ say. Hence, by setting τ ∘ σ, we have π_{Q,z} ∘ τ = π.

Conversely, given τ, we get a splitting P_{z} → A by sending x to τ(ρx)μ(x)^{−1}.

(iia) This is well known: see [3, IV 3 ex.8] or use the calculations in §4.

(iic) We have

(S_{z})^{p} = 1 ⇔ |⟨x⟩| = p,  for all x ∈ S_{z}
⇔ z_{ℓ}|_{⟨y⟩} = 0, for all y ∈ S, 1 ≤ ℓ ≤ m
⇔ z_{ℓ}|_{R} ∈ λ(R),  1 ≤ ℓ ≤ m.  □

We also have

Lemma 2.2. Suppose that R is an elementary abelian subgroup of Q. Then (R_{z})^{p} = A if and only if there exists a basis b_{1}, b_{2}, …, b_{m}, … of R such that Res_{⟨b_{i}⟩}^{Q}(z_{i}) ≠ 0, Res_{⟨b_{j}⟩}^{Q}(z_{i}) = 0, 1 ≤ i ≤ m, 1 ≤ j ≠ i.

Proof. If such a basis exists let b̃_{i} be an element of the inverse image of b_{i}. Then b̃_{i} has order p^{2} and ⟨b̃_{i}^{p}⟩ = ⟨e_{i}⟩.

Conversely, if (R_{z})^{p} = A then every element of A is a p-th power of an element of R_{z} (R_{z} is powerful). Let b̃_{i} be such that b̃_{i}^{p} = e_{i}, b̃_{j} = 1, j > m, and let b_{i} be the image of b̃_{i} in R. □

Lemma 2.3. Consider the extension

0 → A → Q_{z} → Q → 1

discussed above and suppose that there is no non-trivial relation q_{1}z_{1} + ⋯ + q_{m}z_{m} = 0 with q_{i} ∈ H^{1}(Q). Then the sequence

H^{2}(Q) →^{Inf} H^{2}(Q_{z}) →^{Res} H^{2}(A)

is exact at the middle term.

Proof. The Lyndon-Hochschild-Serre spectral sequence for the extension has E_{2} term H^{∗}(Q) ⊗ H^{∗}(A). Note that d_{2}(u_{i}) = z_{i}, 1 ≤ i ≤ m (see the remark after Proposition 2.4). Let q = q_{1}u_{1} + ⋯ + q_{m}u_{m} be an element of ker d_{2}^{1,1}, with q_{1}, …, q_{m} ∈ H^{1}(Q). Then 0 = d_{2}(q) = −(q_{1}z_{1} + ⋯ + q_{m}z_{m}) which implies q_{1} = ⋯ = q_{m} = 0, by the hypothesis. Therefore d_{2}^{1,1} : E_{2}^{1,1} → E_{2}^{3,0} is injective and so E_{3}^{1,1} = 0 and thus E_{∞}^{1,1} = 0.

This yields an exact sequence

0 → E_{∞}^{2,0} → H^{2}(Q_{z}) → E_{∞}^{0,2} → 0

and identifying the edge maps gives the result claimed. □

Proposition 2.4. Suppose that A is contained in the Frattini subgroup of Q_{z} and that z_{1}, …, z_{m} is a regular sequence in H^{∗}(Q). Then the inflation and restriction maps induce an isomorphism of rings

H^{∗}(Q_{z}) ≅ H^{∗}(Q)/(z_{1}, …, z_{m}) ⊗ ℤ/p[B(A)]

provided that one of the following conditions is satisfied:

(a) H^{2}(Q_{z}) →^{Res} B(A) is surjective;

(b) β(z_{i}) ∈ (z_{1}, …, z_{m}), 1 ≤ i ≤ m;

(c) ker Inf_{Q_{z}}^{Q} = (z_{1}, …, z_{m}).

Proof. Let {E_{r}, d_{r}} be the Lyndon-Hochschild-Serre spectral sequence corresponding to the central extension

0 → A → Q_{z} → Q → 1.

Now E_{2} = H^{∗}(Q) ⊗ H^{∗}(A).

Assume that p > 2 and that condition (a) holds. By considering the maps B(A) = Im(Res : H^{2}(Q_{z}) → H^{2}(A)) ≅ E_{∞}^{0,2} ⊂ E_{2}^{0,2} we can find elements ỹ_{1}, …, ỹ_{m} ∈ E_{2}^{0,2} corresponding to y_{1}, …, y_{m} ∈ B(A) such that d_{2}(ỹ_{i}) = 0. Under the isomorphism E_{2}^{0,2} ≅ H^{2}(A) we find that ỹ_{i} = y_{i}(mod λ(A)). Let S = ℤ/p[ỹ_{1}, …, ỹ_{m}]. Then E_{2}^{0,∗} = S ⊗ Λ^{∗}(E_{2}^{0,1}) and so E_{2} = S ⊗ Λ^{∗}(E_{2}^{0,1}) ⊗ H^{∗}(Q).

But d_{2} vanishes on S, by construction, and thus d_{2} is S ⊗H ∗(Q)-linear. So {E_{2}, d_{2}} is just the Koszul complex K_{S⊗H^{∗}(Q)}(z_{1}, …, z_{m}) of Lemma 1.1. Thus E_{3} = S ⊗ H^{∗}(Q)/(z_{1}, …, z_{m}) = E_{3}^{0,∗} ⊗ E_{3}^{∗,0}.

Now E_{3} is generated as a ring by the ỹ_{i} and E_{3}^{∗,0} so, by the product structure, d_{3} = 0 and E_{4} = E_{3}. Subsequently all the differentials are 0 for degree reasons and so E_{∞} = E_{3}.

Let ŷ_{i} ∈ H^{2}(Q_{z}) have image ỹ_{i} ∈ E_{∞}^{0,2} and let Ŝ = ℤ/p[ŷ_{1} …, ŷ_{m}]. Let I^{∗} = Im(Inf : H^{∗}(Q) → H^{∗}(Q_{z}) ≅ H^{∗}(Q)/(z_{1} …, z_{m}). Then we have a natural ring homomorphism ϕ : Ŝ ⊗ I → H^{∗}(Q_{z}). If we filter Ŝ ⊗ I by F^{p}(Ŝ ⊗ I) = Ŝ ⊗ I^{≥p} and H^{∗}(Q_{z}) in the way that yields E_{∞} then ϕ is a homomorphism of filtered rings and induces and isomorphism of the associated graded modules. The filtration is finite in each degree, so ϕ must be an isomorphism. This concludes the proof in this case.

If condition (a) is not available then note that as u_{i} is transgressive, βu_{i} survives to E_{3} and d_{3}(βu_{i}) = −βz_{i} (see e.g. [9] and the remark below). Since E_{3}^{3,0} = H^{3}(Q)/(H^{3}(Q) ∩ (z_{1}, …, z_{m})), either of the conditions (b) or (c) implies that d_{3}(βu_{i}) = 0. We set ỹ_{i} = β(u_{i}) and proceed as before.

If p = 2 then E_{2} is just the complex J_{H^{∗}(Q)}(z_{1}, …, z_{m}) of Lemma 2.1(2), so

E_{3} = H^{∗}(Q)/(z_{1}, …, z_{m})[u_{1}^{2}, …, u_{m}^{2}].

Set ỹ_{i} = u_{i}^{2} (= βu_{i}) . Then condition (a) implies that d_{3}(ỹ_{i}) = 0 for dimension reasons, and the rest of the proof is just as for p odd. □

Remark. By projecting A onto its cyclic factors and comparing spectral sequences, it is easy to see that d_{2}(u_{i}) = λ_{i}z_{i} for some non-zero λ_{i} ∈ ℤ/p. In fact λ_{i} = ±1, see [6] or [5]. The sign appears to be sensitive to the sign convention used in constructing the double complex for the spectral sequence. Similarly with the formula d_{3}(βu_{i}) = ±βd_{2}(u_{i}).

From now on, assume that N is a closed normal and finitely generated subgroup of a pro-p group G. Recall (see e.g. [4, 4]) that N is powerfully embedded in G if [G, N] ⊂ N^{2p}, that G is powerful if it is powerfully embedded in itself. When p = 2, N is almost powerfully embedded in G if [G, N] ⊂ N^{2} and [N, N] ⊂ (N^{2})^{2} (which also implies [N, N] ⊂ N^{4} (see [7])). For convenience, we define almost powerfully embedded as powerfully embedded for p > 2. As noted in [7], we have the following implications:

N powerfully embedded in G ⇒ N almost powerfully embedded in G ⇒ N powerful.

We quote two characterizations of powerfully embedded and almost powerfully embedded.

Proposition 2.5.

(i) ([4, Lemma 2.2(iv)]) If N is not powerfully embedded in G, then there exists a normal subgroup J of G such that

\begin{matrix}N^{2p}[N, G, G] ⊂ & J ⊂ N^{2p}[N, G] \\ |N^{2p}[N, G] : J| & = p; \end{matrix}

in other words, G/J is given by the central extension

1 → ℤ/p → G/J → G/N^{2p}[N, G] → 1

such that N/J is abelian of exponent ≤ 2p, but N/J ⊄ Z(G/J). (ii) ( [7, Proposition 2]) Set M = N^{p}[N, G], K = M^{p}[M, G], A = N/M, B = M/K. N is almost

powerfully embedded in G if and only if the map induced from the p-power A →^{p} B is linear and surjective. □

We now give some cohomological criteria for N to be (almost) powerfully embedded in G. Set Ĝ = G/N^{2p}[N, G], N̂ = N/N^{2p}[N, G]. We have

Theorem 2.6. The following are equivalent:

(i) N is powerfully embedded in G;

(ii) ResĜ ; N̂(ker InfĜ ; G) ∩ λ(N̂) = {0}.

Proof. Let N be powerfully embedded in G (so N^{2p}[N, G] = N^{2p}, as [N, G] ⊂ N^{2p}) and let 0 ≠ z ∈ kerĜ ; G. Consider the central extension

0 → ℤ/p →^{i} Ĝ_{z} → Ĝ → 1

corresponding to z. By Lemma 2.1 (i), Ĝ_{z} is a homomorphic image of G, via τ. This implies i(ℤ/p) ⊂ τ(N^{2p}) = (N̂_{z})^{2p}. So (N̂_{z})^{2p} ≠ 1. Therefore, by Lemma 2.1 (iic), z|N̂ ∉ λ(N̂).

Suppose now that N is not powerfully embedded in G. By Proposition 2.5, there exists a central extension

0 → ℤ/p → Ĝ_{w} → Ĝ → 1

corresponding to an element 0 ≠ w ∈ H^{2}(Ĝ) such that Ĝ_{w} is a quotient of G, (N̂_{w})^{2p} = 1, [N̂_{w}, Ĝ_{w}] ≠ 1. It follows from Lemma 2.1 that w ∈ Ker InfG̃ ; G and w|N̂ ∈ λ(N̂) . □

We then have the following corollary, of which the case p > 2 was given in [12, Theorem 5.1.6].

Corollary 2.7. The following are equivalent:

(i) G is powerful.

(ii) p is odd and the map induced from the inflation

H^{1}(Ĝ) ∧ H^{1}(Ĝ) → H^{2}(G)

is injective, or p = 2 and

ker(InfĜ ; G) ∩ λ(Ĝ) = {0}.

(iii) the map induced from the inflation

λ(Ĝ) → H^{2}(G)

is injective. □

Proof. Recall that G is powerful if and only if it is powerfully embedded in itself. Thus, by Theorem 2.6, G is powerful if and only if (ker InfĜ ; G) ∩ λ(N̂) = {0}, so (i) ⇔ (iii). Note that, for p ≠ 2, λ(Ĝ) = H^{1}(Ĝ) ∧ H^{1}(Ĝ) so (ii) is just a restatement of (iii). □

If p = 2, set G̃ = G/N^{2}[N, G], Ñ = N/N^{2}[N, G]. A characterization of powerfully embedded normal subgroup N of a 2-group G via the inflation InfG̃ ; G can also be obtained, as follows. First we prepare Lemma 2.8. Let L be a central subgroup of a 2-group H, and let w be a cohomology class of H/L^{4} satisfying w|_{L^{2}/L^{4}} = 0. Then w ∈ Im Inf_{H/L^{4}}^{H/L^{2}}.

Proof. We may suppose that L^{2} ≠ 1. Set A_{1} = L/L^{2}, A_{2} = L^{2}/L^{4}, H_{1} = H/L^{2}, H_{2} = H/L^{4}, K = H/L. A_{1}, A_{2} are then vector spaces over ℤ/2 and we have the central extensions

\begin{aligned}0 & → A_{1} & → H_{1} & → K & → 1, \\ 0 & → A_{2} & → H_{2} & → H_{1} & → 1 \end{aligned} .

Let z = (z_{1}, …, z_{i}) be the cohomology class classifying the extension 0 → A_{2} → H_{2} → H_{1} → 1. Since L is abelian, by Proposition 1.3 (ii), z_{j}|_{A_{1}} ∈ B(A_{1}), 1 ≤ j ≤ i. Then ((A_{1})_{z})^{2} = A_{2} and z_{1}|_{A_{1}}, …, z_{i}|_{A_{1}} are linearly independent in B(A_{1}), by Lemma 2.2. It follows from Proposition 1.3 (ii) that z_{1}, …, z_{i} is a regular sequence in H^{∗}(H_{1}). Hence, by Lemma 2.3, w|_{A_{2}} = 0 implies w ∈ Im Inf_{H_{2}}^{H_{1}}. □

For a ∈ G̃ (resp. b ∈ Ĝ), define N_{a} = ⟨Ñ, a⟩ (resp. N̂_{b} = ⟨N̂, b⟩).

Corollary 2.9. For p = 2, the following are equivalent:

(i) N is powerfully (resp. almost powerfully) embedded in G; (ii) for every non-zero element μ of ker (Inf:H^{2}(G̃) → H^{2}(G)), μ|_{N_{a}} ∈ B(N_{a}) for every a ∈ G̃ (resp.

μ|Ñ ∈ B(Ñ)).

Proof. Let N be powerfully (resp. almost powerfully) embedded in G and let 0 ≠ z ∈ kerG̃ ; G. Consider the central extension

0 → ℤ/p → G̃_{z} → G̃ → 1

corresponding to z. By Lemma 2.1 (i), G̃_{z} is a homomorphic image of G. Since Ñ is elementary abelian, (Ñ_{z})^{4} = 1. As [G̃_{z}, Ñ_{z}] ⊂ (Ñ_{z})^{4} = 1 (resp. [Ñ_{z}, Ñ_{z}] ⊂ (Ñ_{z})^{4} = 1), Ñ_{z} is central (resp. abelian). Note that Ñ_{z} is central iff N_{a} is abelian for every a ∈ G̃. By Lemma 2.1 (iia), it follows that z|_{N_{a}} ∈ B(N_{a}) for every a ∈ G̃ (resp. z|Ñ ∈ B(Ñ)).

Suppose now that N is not powerfully (resp. almost powerfully) embedded in G. From the proof of Theorem 2.6, there exists an element w ∈ H^{2}(Ĝ) such that Ĝ_{w} is a quotient of G, (N̂_{w})^{4} = 1 and [N̂_{w}, Ĝ_{w}] ≠ 1 (resp. [N̂_{w}, N̂_{w}] ≠ 1). By Lemma 2.1, [N̂_{w}, Ĝ_{w}] ≠ 1 implies that there exists b ∈ Ĝ such that w|N̂_{b} ∉ B(N̂_{b}). So w|N̂^{2} = 0, and w|N̂_{b} ∉ B(N̂_{b}) for some b ∈ Ĝ (resp. w|Ñ ∉ B(Ñ)). Applying Lemma 2.8 with H = Ĝ, L = N̂ yields w = InfG̃ ; Ĝ(t) for some t ∈ H̃^{∗}(G̃). Let a be the projection of b on G̃, then w|N̂_{b} ∉ B(N̂_{b}) (resp. w|Ñ ∉ B(Ñ)) implies t|_{N_{a}} ∉ B(N_{a}) (resp. t|Ñ ∉ B(Ñ)).

As G̃ is a quotient of G, it follows from Lemma 2.1 (i) that ker InfG̃ ; G contains t. □

We then have

Corollary 2.10. For p = 2, the following are equivalent:

(i) G is powerful;

(ii) ker (Inf:H^{2}(G̃) → H^{2}(G)) ⊂ B(G̃). □

Remark 2.10. Another proof of Theorem 2.6 (for p odd) and Corollary 2.9 (for N almost powerfully embedded in G) also follows from Proposition 2.5 (ii). Indeed, one can deduce that the statement (ii) in Theorem 2.6, together with μ|Ñ ∈ B(Ñ) for p = 2, is equivalent to the condition that the map A →^{p} B be linear and surjective.

3. Cohomology of pro-p groups with powerfully embedded subgroups

Let N be a closed normal and finitely generated subgroup of a pro-p group G. Set Φ_{G}(N) = N^{p}[N, G]. For i ≥ 1, define recursively a sequence of closed normal subgroups Φ_{G}^{i}(N) of G as follows:

Φ_{G}^{1}(N) = N, Φ_{G}^{i+1}(N) = Φ_{G}(Φ_{G}^{i}(N)).

It is clear that [Φ_{G}^{i}(N), Φ_{G}^{j}(N)] ⊂ Φ_{G}^{i+j}(N) and the p-power map induces a map Φ_{G}^{i}(N) →^{p} Φ_{G}^{i+1}(N), i, j ≥ 1; also, [G, N] ⊂ Φ_{G}^{2}(N).

For i ≥ 0, set G_{i} = G/Φ_{G}^{i+1}(N), A_{i} = A_{i}(N) = Φ_{G}^{i}(N)/Φ_{G}^{i+1}(N) (with the convention that Φ_{G}^{0}(N) = N). By [4, Proposition 1.16], G = lim_{⟵} G_{i}, hence H^{∗}(G) = lim_{⟶} H^{∗}(G_{i}). Each A_{i} is a vector space over ℤ/p and is central in G_{i}. We also have successive central extensions

\begin{aligned}0 & ⟶ & A_{1} & ⟶ & G_{1} & ⟶ & G_{0} & ⟶ & 1 \\ 0 & ⟶ & A_{2} & ⟶ & G_{2} & ⟶ & G_{1} & ⟶ & 1 \\ & & & & … \\ 0 & ⟶ & A_{s} & ⟶ & G_{s} & ⟶ & G_{s−1} & ⟶ & 1 \\ & & & & … \end{aligned} .

Define ℓ = ℓ(Φ_{G}^{∗}(N)) to be the largest integer s, if any, satisfying Φ_{G}^{s}(N) ≠ 1, or ℓ(Φ_{G}^{∗}(N)) = ∞ otherwise (so G = G_{ℓ} if ℓ < ∞). Denote by ϕ_{i,j} : G_{j} → G_{i} and ϕ_{i} : G → G_{i} the projection maps, j ≥ i. For 1 ≤ i, let n_{i} be the dimension of A_{i} and let κ_{i−1} = (z_{1}^{(i−1)}, …, z_{n_{i}}^{(i−1)}) ∈ H^{2}(G_{i−1})^{⊕n_{i}} be the cohomology class corresponding to the central extension

0 → A_{i} → G_{i} → G_{i−1} → 1.

We first have

Lemma 3.1. For every 1 ≤ i ≤ ℓ − 1, no non-zero linear combination of the z_{j}^{(i)}’s belongs to Im ϕ_{i−1,i}^{∗}.

Proof. Given 1 ≤ i ≤ ℓ − 1, assume that z is a non-zero linear combination of the z_{j}^{(i)}’s with z ∈ Im ϕ_{i−1,i}^{∗}. Without loss of generality, we may assume that z = ∑_{j=1}^{n_{i+1}} λ_{j}z_{j}^{(i)} with λ_{1} ≠ 0. There exists then a basis (e_{1}, …, e_{n_{i+1}}) of A_{i+1} such that κ_{i} = (z, z_{2}^{(i)}, …, z_{n_{i+1}}^{i)}) ∈ H^{2}(G_{i})^{⊕n_{i+1}}.

Since z ∈ Im ϕ_{i−1,i}^{∗}, its restriction to A_{i} is 0, by Lemma 2.1. Thus ⟨e_{1}⟩ is a factor of Φ_{G}^{i}(N)/Φ_{G}^{i+1}(N), a contradiction. □

Proposition 3.2. The following are equivalent:

(i) N is almost powerfully embedded in G; (ii) ((A_{1})_{κ_{1}})^{p} = A_{2}; also, 0 ≠ z_{j}^{(1)}|_{A_{1}} ∈ B(A_{1}), 1 ≤ j ≤ n_{2}, for p = 2.

Proof. Suppose that N is almost powerfully embedded in G. By Lemma 3.1, z_{j}^{(1)} ≠ 0, 1 ≤ j ≤ n_{2}. It follows from Theorem 2.6 (for p > 2) and Corollary 2.9 (for p = 2) that, for any non-zero linear combination z of the z_{j}^{(1)}’s, z|_{A_{1}} ∉ λ(A_{1}) for p > 2 and z|_{A_{1}} ∈ B(A_{1}) for p = 2. There exists then a basis e_{1}, …, e_{n_{1}} of A_{1} such that z_{j}^{(1)}|_{⟨e_{j}⟩} ≠ 0, z_{j}^{(1)}|_{⟨e_{i}⟩} = 0, 1 ≤ i ≠ j ≤ n_{2}. Hence, by Lemma 2.2, ((A_{1})_{κ_{1}})^{p} = A_{2}.

Conversely, suppose that (ii) holds. If p = 2, (i) also holds, by Corollary 2.9. Assume that p > 2 and N is not powerfully embedded in G. From the proof of Theorem 2.6, there exists a non-zero linear combination z of the z_{j}^{(1)}’s such that z|_{A_{1}} ∈ λ(A_{1}). Arguing as in the proof of Lemma 3.1, there exists an element of A_{2} not belonging to ((A_{1})_{κ_{1}})^{p}, a contradiction. □

For i < j ≤ ℓ and for every element ξ ∈ H^{∗}(G_{i}), consider ξ as an element of H^{∗}(G_{j}) via the inflation map ϕ_{i,j}^{∗}. We then have Theorem 3.3. Let N be almost powerfully embedded in a p-group G. We have:

(i) for 1 ≤ i ≤ ℓ−1, ((A_{i})_{κ_{i}})^{p} = A_{i+1}, n_{1} ≥ n_{2} ⋯ ≥ n_{ℓ}, and the sequence H^{∗}(G_{i}) →^{ϕ_{i,i+1}^{∗}} H^{∗}(G_{i+1}) →^{Res}

B(A_{i+1}) is exact; (ii) for 1 ≤ i ≤ ℓ − 1, z_{1}^{(i)}, …, z_{n_{i+1}}^{(i)} is a regular sequence in H^{∗}(G_{i}); (iii) if n_{1} = ⋯ = n_{r} = d, 2 ≤ r ≤ ℓ, then

(iiia) the map H^{2}(G_{i}) →^{Res} B(A_{i}) is surjective, for 1 ≤ i ≤ r − 1; (iiib) for 2 ≤ i ≤ r − 1,

\begin{aligned}H^{∗}(G_{i}) & = H^{∗}(G_{i−1})/(z_{1}^{(i−1)}, …, z_{d}^{(i−1)}) ⊗ ℤ/p[z_{1}^{(i)}, …, z_{d}^{(i)}] \\ & = H^{∗}(G_{1})/(z_{1}^{(1)}, …, z_{d}^{(1)}) ⊗ ℤ/p[z_{1}^{(i)}, …, z_{d}^{(i)}]. \end{aligned}

Proof. It follows from Proposition 3.2 that ((A_{1})_{κ_{1}})^{p} = A_{2}. As N is powerful, we also have ((A_{i})_{κ_{i}})^{p} = A_{i+1}, i ≥ 2, and n_{1} ≥ n_{2} ⋯ ≥ n_{ℓ} (see e.g. [4, Theorem 2.7]). By Lemma 2.2 (and Proposition 3.2 (ii) for p = 2), z_{1}^{(i)}|_{A_{i}}, …, z_{n_{i+1}}^{(i)}|_{A_{i}} satisfies the assumption of Proposition 1.3 (ii). So z_{1}^{(i)}, …, z_{n_{i+1}}^{(i)} is a regular

sequence in H^{∗}(G_{i}). Therefore H^{∗}(G_{i}) →^{ϕ_{i,i+1}^{∗}} H^{∗}(G_{i+1}) →^{Res} B(A_{i+1}) is exact by Lemma 2.3. (i), (ii) are then proved. (iii) is straightforward from (i), (ii), Lemma 2.3 and Proposition 2.4. □

Remark 3.4. Theorem 3.3 also holds if G is a pro-p group.

Definition. Let N be almost powerfully embedded in G and let d = d(N) be the minimal number of generators of N. Set ℓ = ℓ(Φ_{G}^{∗}(N)), Ω_{1}(N) = \overline{⟨x ∈ N|x^{p} = 1⟩}.

- N is said to be uniform if n_{i} = dim_{ℤ/p} A_{i}(N) = d for every 1 ≤ i ≤ ℓ. In such a case, N is also said to be almost uniformly embedded in G.

- N is called Ω-extendable in G if Ω_{1}(N) is central in G and there exists a central extension 0 → Z → Γ → G → 1 classified by z ∈ H^{2}(G, Z), with Z = (ℤ/p)^{r}, r = rank (Ω_{1}(N)) and Z = (Ω_{1}(N)_{z})^{p} and Γ is called an Ω_{1}(N)-extension of G. G is Ω-extendable if it is Ω-extendable in itself.

Remark 3.5. 1. If N is almost uniformly embedded in G and if Ω = Ω_{1}(N) ⊂ Z(G)∩Φ(N) with d(Ω) = d, the Ω-extendable property of N in G is equivalent to the existence of elements u_{1}, …, u_{d} of H^{2}(G) satisfying B(Ω) = ⟨u_{1}|_{Ω}, …, u_{d}|_{Ω}⟩, by Lemma 2.2. Let (w_{1}, …, w_{d}) ∈ H^{2}(G/Ω)^{⊕d} be any cohomology class classified the extension 0 → (ℤ/p)^{d} → G → G/Ω → 1, and let {E_{r}, d_{r}} be the Lyndon-Hochschild-Serre spectral sequence corresponding to this extension. It follows from the proof of Proposition 2.4 that B(Ω) ⊂ Im Res_{Ω}^{G} if and only if d_{3} = 0, or, equivalently, β(w_{i}) ∈ (w_{1}, …, w_{d}), 1 ≤ i ≤ d. So the Ω-extendibility of N in G is equivalent to the condition that β(w_{i}) ∈ (w_{1}, …, w_{d}), 1 ≤ i ≤ d.

2. If N is almost powerfully embedded in the pro-p group G and T = T (N) is the set of elements of finite order of N, then T is a normal subgroup of G (see [4], Theorem 4.20) and N/T is almost uniformly embedded in G/T. In such a case, Ω_{1}(N) = Ω_{1}(T). Furthermore, as N^{p^{k}} = {g^{p^{k}}|g ∈ N} ([4], Corollary 3.5), it follows that N^{2p} ∩ T = T^{2p} (N^{2} ∩ T = T^{2}, (N^{2})^{2} ∩ T = (T^{2})^{2} for p = 2). So T is also almost powerfully embedded in G.

Examples. 1. If G is an pro-p group of finite rank then, by [4, Corollary 4.3], G contains a characteristic, open, uniform subgroup. If G is powerful then, by [4, Theorem 2.7], there exists a term P_{i}(G) in the lower p-series of G which is uniform and powerfully embedded in G.

2. Let D_{2^{n}} = ⟨a_{n}, b_{n}|a_{n}^{2^{n−1}} = b^{2} = 1, (a_{n}b_{n})^{2} = 1⟩ be the dihedral group of order 2^{n}. We then have a projection map φ_{ij} : D_{2^{j}} → D_{2^{i}} which maps a_{j} (resp. b_{j}) to a_{i} (resp. b_{i}), for j ≥ i. Define 𝔻 = lim_{⟵} D_{2^{n}}.

Then 𝔻 = ⟨a, b|b^{2} = 1, (ab)^{2} = 1⟩ so that the projection map φ_{i} : 𝔻 → D_{2^{i}} maps a (resp. b) to a_{i} (resp. b_{i}), i ≥ 1. It is clear that ⟨a_{n}⟩ is almost powerfully embedded, but not powerfully embedded, and Ω-extendable in D_{2^{n}}. Also, ⟨a⟩ is almost uniformly embedded in 𝔻.

However, any cyclic subgroup of index 2 of a quaternion group Q of order 8 is powerfully embedded in Q, but not Ω-extendable in Q. Also, if M is the extraspecial p-group of order p^{3} and exponent p^{2} with p > 2, then M is powerful, the center of M is powerfully embedded but not Ω-extendable in M.

3. Let n be an integer. For every k ≥ 1, define Γ_{n}(k) = {A ∈ GL_{n}(ℤ/p^{k+1})|A = I_{n} mod (p^{k})}. Then Γ_{n}(k) with k ≥ 1 for p > 2, k ≥ 2 for p = 2 is uniform and Ω-extendable.

Corollary 3.6. If N is almost uniformly embedded in a p-group G with d = d(N) and ℓ = ℓ(Φ_{G}^{∗}(N)) ≥ 3, then, for 2 ≤ j ≤ ℓ − 1,

\begin{aligned}H^{∗}(G_{j}) & = H^{∗}(G_{j−1})/(z_{1}^{(j−1)}, …, z_{d}^{(j−1)}) ⊗ ℤ/p[z_{1}^{(j)}, …, z_{d}^{(j)}] \\ & = H^{∗}(G_{1})/(z_{1}^{(1)}, …, z_{d}^{(1)}) ⊗ ℤ/p[z_{1}^{(j)}, …, z_{d}^{(j)}]. \end{aligned}

Furthermore, if N is also Ω-extendable in G, there exist z_{1}, …, z_{d} in H^{2}(G) such that

H^{∗}(G) = H^{∗}(G_{1})/(z_{1}^{(1)}, …, z_{d}^{(1)}) ⊗ ℤ/p[z_{1}, …, z_{d}].  □

Proof. By the first assumption, n_{1} = ⋯ = n_{ℓ} = d. The first equalities follows then from Theorem 3.3.

Suppose that N is also Ω-extendable in G. By Remark 3.5.1, there exist z_{1}, …, z_{d} in H^{2}(G) such that z_{1}|_{Ω_{1}(N)}, …, z_{d}|_{Ω_{1}(N)} is a basis of B(Ω_{1}(N)). Consider the central extension 1 → (ℤ/p)^{d} → Γ → G → 1 corresponding to z = (z_{1}, …, z_{d}). It follows that N_{z} is, in turn, almost uniformly embedded in Γ. The last equality follows then from the first part of the Corollary. □

The above corollary can be generalized to the case where G is almost powerfully embedded in G (see Theorem 3.13 below).

Lemma 3.7. Suppose that N is almost powerfully embedded in a p-group G and n_{i} > n_{i+1} with a given i.

(i) Let a_{1}, …, a_{j}, …, a_{n_{i}} be elements of Φ_{G}^{i}(N)\Φ_{G}^{i+1}(N) satisfying: (ia) ϕ_{i}^{∗}(a_{1}), …, ϕ_{i}^{∗}(a_{n_{i}}) is a basis of A_{i}; (ib) ord(a_{1}) = ⋯ = ord(a_{j}) = p < ord(a_{k}), for k > j.

Then there exist elements b_{j+1}, …, b_{n_{i}} of Φ_{G}^{i}(N) such that, for k > j, ord(b_{k}) = p, ϕ_{i}^{∗}(b_{k}) = a_{k}. Precisely, for every 1 ≠ x ∈ Φ_{G}^{i}(N) satisfying 1 ≠ x^{p} ∈ Φ_{G}^{j}(N) with j > i + 1, there exists y ∈ Φ_{G}^{j−1}(N) such that (xy)^{p} = 1;

(ii) For every element a of order p of Φ_{G}^{i}(N)\Φ_{G}^{i+1}(N), there exists ξ ∈ H^{+}(G_{i}) satisfying ξ|_{⟨ϕ_{i}(a)⟩} ≠ 0; hence, as an element of H^{+}(G), ξ|_{⟨a⟩} ≠ 0, ξ is nilpotent in H^{+}(G) and ξ ∉ Im ϕ_{i−1}^{∗}.

Proof. (i) follows from [4, Proof of Theorem 4.5]. Set b = ϕ_{i}(a). Then b ≠ 1. Pick an element w of B(A_{i}) satisfying w|_{⟨b⟩} ≠ 0 and set ξ = N_{A_{i}→G_{i}}(w) with N_{A_{i}→G_{i}} the Evens norm map. So ξ|_{⟨b⟩} is not nilpotent. From the commutative diagram

H^{∗}(G_{i}) ⟶^{ϕ_{i}^{∗}} H^{∗}(G) ^{Res}↓  ↓^{Res} H^{∗}(⟨b⟩) ⟶^{Inf} H^{∗}(⟨a⟩) ,

since H^{∗}(⟨b⟩) →^{Inf} H^{∗}(⟨a⟩) is an isomorphism, it follows that ξ|_{⟨a⟩} ≠ 0. So ξ is not nilpotent in H^{∗}(G) and ξ ∉ Im ϕ_{i−1}^{∗}. □

We now have a sufficient and necessary condition for an (almost) powerfully embedded subgroup in G to be also (almost) uniformly embedded, as follows.

Corollary 3.8. Let N be (almost) powerfully embedded in a p-group G with d(N) = d and ℓ(Φ_{G}^{∗}(N)) = ℓ ≥ 2. The following are equivalent:

(i) N is (almost) uniformly embedded in G;

(ii) n_{2} = d and Im ϕ_{1}^{∗} = Im ϕ_{ℓ−1}^{∗}.

Proof. (i) ⇒ (ii) follows from Corollary 3.6. Suppose that (ii) holds. If N is not uniform, then, by Lemma 3.7, there exist j ≥ 2, ξ ∈ H^{∗}(G_{j}) such that ξ|_{A_{j}} ≠ 0 and ξ is not nilpotent in H^{∗}(G). This contradicts the fact that Im ϕ_{1}^{∗} = Im ϕ_{ℓ−1}^{∗} implies 0 = ξ ∈ H^{∗}(G). So N is uniform. □

The next result gives a necessary and sufficient condition for an (almost) powerfully embedded subgroup to be also (almost) uniformly embedded and Ω-extendable in G.

Corollary 3.9. Let N be (almost) powerfully embedded in a p-group G with d(N) = d and ℓ(Φ_{G}^{∗}(N)) = ℓ ≥ 2. The following are equivalent:

(i) N is (almost) uniformly embedded and Ω-extendable in G;

(ii) n_{ℓ} = d and there exist z_{1}, …, z_{d} in H^{2}(G) such that

H^{∗}(G) = H^{∗}(G_{1})/(z_{1}^{(1)}, …, z_{d}^{(1)}) ⊗ ℤ/p[z_{1}, …, z_{d}]

and z_{1}|_{A_{ℓ}}, …, z_{d}|_{A_{ℓ}} is a basis of B(A_{ℓ}); (iii) n_{2} = d, Im ϕ_{1}^{∗} = Im ϕ_{ℓ−1}^{∗} and there exist z_{1}, …, z_{d} ∈ H^{2}(G) such that z_{1}|_{A_{ℓ}}, …, z_{d}|_{A_{ℓ}} is a basis

of B(A_{ℓ}); .

Proof. For convenience, write H^{∗}(G_{1})/(z_{1}^{(1)}, …, z_{d}^{(1)}) = H. The implication (i) ⇒ (ii) follows from Corollary 3.6. Also, by Corollary 3.6, (i) implies n_{2} = d (as N is uniform), ker ϕ_{1}^{∗} = (z_{1}^{(1)}, …, z_{d}^{(1)}), ker ϕ_{ℓ−1}^{∗} = (z_{1}^{(ℓ−1)}, …, z_{d}^{(ℓ−1)}), hence

Im ϕ_{1}^{∗} = Im ϕ_{ℓ−1}^{∗} = H^{∗}(G_{1})/(z_{1}^{(1)}, …, z_{d}^{(1)}).

So (i)⇒ (iii).

Suppose that (ii) holds. By Theorem 3.3 (i), n_{1} ≥ n_{2} ⋯ ≥ n_{ℓ}, so n_{1} = ⋯ = n_{ℓ} = d, hence N is (almost) uniformly embedded in G. As B(A_{ℓ}) ⊂ Im Res_{A_{ℓ}}^{G}, it follows from Remark 3.5.1 that N is Ω-extendable in G. So (i) ⇔ (ii).

Suppose that (iii) holds. By Corollary 3.8, N is (almost) uniformly embedded in G. The existence of the z_{i}’s shows that N is Ω-extendable in G.

□

The following is straightforward from Corollaries 3.6, 3.8 and 3.9.

Theorem 3.10. Let N be a normal subgroup of a p-group G with d(N) = d. The following are equivalent:

(i) N is uniformly (resp. almost uniformly, for p = 2) embedded and Ω-extendable in G with ℓ(Φ_{G}^{∗}(N)) ≥ 2;

(ii) n_{ℓ} = d and there exists a system of linearly independent elements ψ_{1}, …, ψ_{d} of H^{2}(G_{1}) satisfying: (a) the inflation and restriction maps induces an isomorphism

H^{∗}(G) ≅ H^{∗}(G_{1})/(ψ_{1}, …, ψ_{d}) ⊗ ℤ/p[B(A_{ℓ})],

and

(b) for every non-zero linear combination ψ of the ψ_{i}’s, either p > 2 and ψ|_{A_{1}} ∉ λ(A_{1}), or p = 2 and ψ|_{⟨A_{1},a⟩} ∈ B(⟨A_{1}, a⟩) for every a ∈ G_{1} (resp. ψ|_{A_{1}} ∈ B(A_{1})). □

We now have the following corollary, of which the last assertion is a celebrated theorem of Lazard ([8]).

Corollary 3.11. Let G be an infinite pro-p group and let N be a closed normal subgroup of G with d(N) = d < ∞. The following are equivalent:

(i) N is uniformly (resp. almost uniformly, for p = 2) embedded in G;

(ii) There exists a system of linearly independent elements ψ_{1}, …, ψ_{d} of H^{2}(G_{1}) satisfying: (a) the inflation map induces an isomorphism

H^{∗}(G) ≅ H^{∗}(G_{1})/(ψ_{1}, …, ψ_{d}),

and

(b) for every non-zero linear combination ψ of the ψ_{i}’s, either p > 2 and ψ|_{A_{1}} ∉ λ(A_{1}), or p = 2 and ψ|_{⟨A_{1},a⟩} ∈ B(⟨A_{1}, a⟩) for every a ∈ G_{1} (resp. ψ|_{A_{1}} ∈ B(A_{1})).

In particular, if G is any finitely generated pro-p group, the inflation map Inf_{G}^{G/G^{p}[G,G]} induces an isomorphism

Λ_{∗}((G/G^{p}[G, G])^{∗}) ≅ H^{∗}(G)

if and only if G is uniform.

Proof. Suppose that (i) holds. It follows that, for every i ≥ 2, A_{i}(N) is uniformly (resp. almost uniformly, for p = 2) embedded and Ω-extendable in G_{i}. By Theorem 3.10, for i ≥ 2,

H^{∗}(G_{i}) ≅ H^{∗}(G_{1})/(ψ_{1}, …, ψ_{d}) ⊗ ℤ/p[B(A_{i})],

and the condition (ii b) of Theorem 3.10 is satisfied. Since H^{∗}(G) = lim_{⟶} H^{∗}(G_{i}), (ii) also holds.

Conversely, assume that (ii) holds. It follows that ker ϕ_{1}^{∗} = (ψ_{1}, …, ψ_{d}). By Theorem 2.6 and Corollary 2.9, N is powerfully (resp. almost powerfully) embedded in G. If n_{1} > n_{2}, Lemma 3.7 shows the existence of a non-nilpotent element of H^{+}(G) not belonging to Im ϕ_{1}^{∗}, a contradiction. Hence n_{2} = n_{1}. For i ≥ 2, as Im ϕ_{1}^{∗} = Im ϕ_{i}^{∗}, A_{i}(N) is uniformly (resp. almost uniformly, for p = 2) embedded and Ω-extendable in G_{i}, by Corollary 3.9. So N is uniformly (resp. almost uniformly, for p = 2) embedded in G. □

Lemma 3.12. Let N be almost powerfully embedded in a p-group G. For k ≥ 1, a ∈ G, b ∈ Φ_{G}^{i}(N),

(i) [a, b^{p^{k}}] = [a, b]^{p^{k}} mod Φ_{G}^{i+k+2}(N); (ii) if [a, b] = xy with x ∈ Z(G), y ∈ Φ_{G}^{j}(N), then [a, b^{p}] = [a, b]^{p} mod Φ_{G}^{j+2}(N); (iii) if Ω = Ω_{1}(N) is central and ord(b) = p^{2}, then [a, b] ∈ Ω.

Proof. (i) We prove by induction on k. Suppose that k = 1. Applying [4, Chapter 4, Exercise 6] yields [a, b^{p}] = [a, b]^{p} mod Φ_{G}^{j}(N) with j ≥ i + 3 if p is odd or i > 1. If p = 2 and i = 1, as [a, b] ∈ Φ_{G}^{2}(N) and N is powerful, [[a, b], b] ∈ Φ_{G}^{4}(N) (see [4, Chapter 2, Exercise 4]). So [a, b^{2}] = [a, b]^{2}[[a, b], b] = [a, b]^{2} mod Φ_{G}^{4}(N).

Assume that the equality holds for k ≥ 1. So

[a, b]^{p^{k+1}} = ([a, b]^{p^{k}})^{p}
= ([a, b^{p^{k}}] mod Φ_{G}^{i+k+2})^{p}
= [a, b^{p^{k}}]^{p} mod Φ_{G}^{i+k+3}
= [a, b^{p^{k+1}}] mod Φ_{G}^{i+k+3}.

(ii) Without loss of generality, we may assume that Φ_{G}^{j+2}(N) = 1. Therefore, for every g ∈ G, [[a, b], g] ∈ Φ_{G}^{j+1}(N) ⊂ Z(G); in particular, [[a, b], g] = 1 if p = 2 and g ∈ N (see [4, Chapter 2, Exercise 4]). We then have

[a, b^{p}] = [a, b][a, b]^{b} … [a, b]^{b^{p−1}}

= ∏_{m=0}^{p−1}[a, b][[a, b], b^{m}]

= [a, b]^{p} ∏_{m=0}^{p−1}[[a, b], b^{m}] since [[a, b], b^{m}] is central

= [a, b]^{p}[[a, b], b]^{p(p−1)/2}

= [a, b]^{p} mod Φ_{G}^{j+2}(N).

(iii) Let j be the smallest integer with [a, b] ∈ Φ_{G}^{j}(N). As (b^{p})^{p} = 1, b^{p} is central. By (ii), 1 = [a, b^{p}] = [a, b]^{p} mod Φ_{G}^{j+2}(N). By Lemma 3.7 (i), [a, b] is of form

[a, b] = cx

with c ∈ Φ_{G}^{j}(N) ∩ Ω, x ∈ Φ_{G}^{j+1}(N). Applying (ii) yields

x^{p} = (cx)^{p} = [a, b]^{p} = [a, b^{p}] mod Φ_{G}^{j+3}(N)  by (ii)
= 1 mod Φ_{G}^{j+3}(N)  since b^{p} is central.

Therefore x^{p} ∈ Φ_{G}^{j+3}(N). By Lemma 3.7 (i), x is of form x = dz with d ∈ Φ_{G}^{j+1}(N) ∩ Ω, z ∈ Φ_{G}^{j+2}(N), so [a, b] = cdz. Hence, by induction, it follows that [a, b] ∈ Ω. □

Suppose that N is almost powerfully embedded in a pro-p group G and Ω = Ω_{1}(N) is central with Ω ⊄ Φ(G). Write Ω = M × L with M = Ω ∩ Φ(G). L is then a direct factor of G. Therefore, there exist subgroups K, N of G such that

(i) G = K × L, N = N_{1} × L, Ω_{1}(N_{1}) = M; (ii) N_{1} is almost powerfully embedded in K; (iii) N is Ω-extendable in G implies that N_{1} is Ω-extendable in K.

Furthermore, L is elementary abelian and H^{∗}(G) = H^{∗}(K) ⊗ H^{∗}(L). Hence, to consider the cohomology of pro-p groups G having an almost powerfully embedded and Ω-extendable subgroups N in G, we may suppose that Ω = Ω_{1}(N) is central and is contained in the Frattini subgroup of G. We have

Theorem 3.13. Let N be almost powerfully embedded in a pro-p group G with d(N^{p}) = d. Set Ω = Ω_{1}(N), Ω′ = Ω ∩ N^{p}, k = d(Ω), k′ = d(Ω′). The following are equivalent:

(i) N is Ω-extendable in G;

(ii) Ω is abelian and there exist z_{1}, …, z_{k} in H^{2}(G) such that z_{1}|_{Ω}, …, z_{k}|_{Ω} (resp. z_{1}|_{Ω′}, …, z_{k′}|_{Ω′}) is a basis of B(Ω) (resp. B(Ω′)) and

H^{∗}(G) = H^{∗}(G_{1})/(z_{1}^{(1)}, …, z_{d}^{(1)}) ⊗ ℤ/p[z_{1}, …, z_{k′}].

Proof.

Let T be the set of elements of finite order of N. By Remark 3.5, T is a closed normal subgroup which is almost powerfully embedded in G. So Ω = Ω_{1}(T) and, by Lemma 3.7, d(T) = k.

We prove (i) ⇒ (ii). Suppose that N is Ω-extendable in G. Consider the following cases:

- N = T (so d = k′). Set ℓ = ℓ(Φ_{G}^{∗}(N)). We argue by induction on ℓ. If ℓ = 2, then, by Lemma 2.3 and Remark 3.5,

H^{∗}(G) = H^{∗}(G_{1})/(z_{1}^{(1)}, …, z_{d}^{(1)}) ⊗ ℤ/p[z_{1}, …, z_{d}]

with z_{1}|_{Ω′}, …, z_{d}|_{Ω′} a basis of B(Ω′); furthermore, by Remark 3.5, there exist z_{d+1}, …, z_{k} in H^{2}(G) such that z_{1}|_{Ω}, …, z_{k}|_{Ω} is a basis of B(Ω). Assume that (i) holds for ℓ − 1 with ℓ > 2. As discussed above, we also assume that Ω ⊂ Φ(G). Let Ω′′ be a complement of Ω′ in Ω. It follows that Ω′ = Ω/Ω′′ = Ω_{1}(N/Ω′′). Hence N/Ω′′ is Ω-extendable in Q = G/Ω′′ (∗). By Lemma 3.12 (iii) (see also [13, Proposition 5.4 (a)]), N′ = N/Ω is almost powerfully embedded and Ω-extendable in G′ = G/Ω; furthermore, d(N′) = d(Ω_{1}(N′)) = d. Set e = d((N′)^{p}) = d(Ω_{1}(N′) ∩ (N′)^{p}) and R = G′/(N′)^{p}. Since ℓ(Φ_{G′}^{∗}(N′)) = ℓ − 1, it follows from the inductive hypothesis that there exist u_{1}, …, u_{e} in H^{2}(R) and s_{1}, …, s_{e}, u_{e+1}, …, u_{d} in H^{2}(G′) such that, via the restriction maps, the images of s_{1}, …, s_{e}, u_{e+1}, …, u_{d} (resp. s_{1}, …, s_{e}) form a basis of B(Ω_{1}(N′)) (resp. B(Ω_{1}((N′)^{p})) and

H^{∗}(G′) = H^{∗}(R)/(u_{1}, …, u_{e}) ⊗ ℤ/p[s_{1}, …, s_{e}].

By (∗), Lemma 2.3 and Remark 3.5, there exist z_{1}, …, z_{d} in H^{2}(Q) such that z_{1}|_{Ω′}, …, z_{d}|_{Ω′} form a basis of B(Ω′) and

\begin{aligned}H^{∗}(Q) & = H^{∗}(G′)/(s_{1}, …, s_{e}, u_{e+1}, …, u_{d}) ⊗ ℤ/p[z_{1}, …, z_{d}] \\ & = H^{∗}(R)/(u_{1}, …, u_{d}) ⊗ ℤ/p[z_{1}, …, z_{d}]. \end{aligned}

As Ω′′ ∩ N^{p} = 1, ker (H^{2}(Q) →^{Inf} H^{2}(G)) ⊂ Im Inf_{Q}^{R}. There exist then u_{d+1}, …, u_{k} in H^{2}(R) and z_{d+1}, …, z_{k} in H^{2}(G) such that z_{d+1}|_{Ω′′}, …, z_{k}|_{Ω′′} form a basis of B(Ω′′) and

H^{∗}(G) = H^{∗}(R)/(u_{1}, …, u_{k}) ⊗ ℤ/p[z_{d+1}, …, z_{k}] ⊗ ℤ/p[z_{1}, …, z_{d}].

Since H^{∗}(G_{1}) = H^{∗}(R)/(u_{d+1}, …, u_{k})⊗ℤ/p[z_{d+1}, …, z_{k}] and z_{1}^{(i)} = Inf_{G_{1}}^{R}(u_{i}), 1 ≤ i ≤ d, it follows that

H^{∗}(G) = H^{∗}(G_{1})/(z_{1}^{(1)}, …, z_{1}^{(d)}) ⊗ ℤ/p[z_{1}, …, z_{d}];

- T ⫋ N. By Remark 3.5.2, T is almost powerfully embedded in G. It follows from the above case that there exist φ_{1}, …, φ_{ℓ} ∈ H^{2}(G/T^{p}), z_{1}, …, z_{k′}, …, z_{k} ∈ H^{2}(G) such that

H^{∗}(G) = H^{∗}(G/T^{p})/(φ_{1}, …, φ_{ℓ}) ⊗ ℤ/p[z_{1}, …, z_{k′}]

with z_{1}|_{Ω′}, …, z_{k′}|_{Ω′} (resp. z_{1}|_{Ω}, …, z_{k}|_{Ω}) a basis of B(Ω′) (resp. B(Ω)).

Set K = G/T^{p} and M = N/T^{p}. By [4, Theorem 4.20 and its proof], M is almost powerfully embedded in K. Denote by w the element of H^{2}(K_{1}, A_{2}(M)) classifying the extension 0 → A_{2}(M) → K_{2} → K_{1} → 1 and set m = dim_{ℤ/p} A_{2}(M). Note that A_{1}(M) is elementary abelian. There exists then an elementary abelian subgroup B of rank m of A_{1}(M) such that (B_{w})^{p} = A_{2}(M). Let C be the preimage of B via the projection map K → K_{1} = K/C^{p}. It follows that C is almost uniformly embedded in K. Hence, by Corollary 3.9, there exist ψ_{1}, …, ψ_{m} ∈ H^{2}(K/C^{p}) such that

H^{∗}(K) = H^{∗}(K/C^{p})/(ψ_{1}, …, ψ_{m}).

Note that K/C^{p} is nothing but G_{1}. Each φ_{i} is then represented by an element ψ_{m+i} of H^{2}(G_{1}). Therefore

H^{∗}(G) = H^{∗}(G_{1})/(ψ_{1}, …, ψ_{m+ℓ}) ⊗ ℤ/p[z_{1}, …, z_{k′}].

As (ψ_{1}, …, ψ_{m+ℓ}) is nothing but Ker ϕ_{1}^{∗}, it coincides with (z_{1}^{(d)}, …, z_{d}^{(1)}). So (i) ⇒ (ii).

Conversely, suppose that (ii) holds. We need prove that Ω is central. Let G_{z} be given by the central extension

0 → (ℤ/p)^{k} →^{i} G_{z} → G → 1

classified by z = (z_{1}, …, z_{k}) ∈ H^{2}(G)^{⊕k}. It follows that Ω_{1}(G_{z}) = i((ℤ/p)^{k}) is central in G_{z}. Set Ω_{2} = ⟨g ∈ G_{z}|g^{p^{2}} = 1⟩. Then Ω = Ω_{2}/Ω_{1}(G_{z}). By Lemma 3.12 (iii), Ω is central. □

We have the following corollary, of which the case p ≥ 5 was given in [2, Theorem 3.16] and the case p odd in [13, Corollary 4.2], both in the finite case, when n = k = d.

Corollary 3.14. Let G be a powerful pro-p group with d(G) = n and d(Φ(G)) = d. Set Ω = Ω_{1}(G) and k = d(Ω). The following are equivalent:

(i) Ω is abelian and there exist y_{1}, …, y_{k+d−n} in H^{2}(G) and a basis x_{1}, …, x_{n} of H^{1}(G) such that

H^{∗}(G) = \{ \begin{matrix}Λ[x_{1}, …, x_{n}] ⊗ ℤ/p[y_{1}, …, y_{k+d−n}, βx_{d+1}, …, βx_{n}] & for p > 2, \\ Λ[x_{1}, …, x_{d}] ⊗ ℤ/2[y_{1}, …, y_{k+d−n}, x_{d+1}, …, x_{n}] & for p = 2, \end{matrix}

and y_{1}|_{Ω}, …, y_{k+d−n}|_{Ω}, βx_{d+1}|_{Ω}, …, βx_{n}|_{Ω} is a basis of B(Ω); (ii) G is Ω-extendable.

Proof. The implication (i) ⇒ (ii) follows from Theorem 3.13. Suppose that G is Ω-extendable. Let N be a minimal subgroup of G satisfying N^{p} = Φ(G). N is then a closed normal subgroup and powerfully embedded in G. Furthermore, N is Ω-extendable in G and d(N) = d. It follows that Ω/Ω ∩ Φ(G) (resp. Ω ∩ Φ(G)) is of rank n − d (resp. k + d − n). By Theorem 3.13,

H^{∗}(G) = H^{∗}(G/Φ(G))/(z_{1}^{(1)}, …, z_{d}^{(1)}) ⊗ ℤ/p[z_{1}, …, z_{k+d−n}]

with z_{1}|_{Ω∩Φ(G)}, …, z_{k+d−n}|_{Ω∩Φ(G)} a basis of B(Ω ∩ Φ(G)). Since G/Φ(G) is elementary abelian, (i) follows from Lemma 1.2 and Corollary 2.6. □

Theorem 3.15. In the circumstances of Theorem 3.13, let Γ be an Ω-extension of G. Let U be a subgroup of Aut(Γ) which preserves N, and let V be the image of U in Aut(G). Then, as V -modules, H^{∗}(G) ≅ H^{∗}(G_{1})/(z_{1}^{(1)}, …, z_{d}^{(1)}) ⊗ ℤ/p[Ω^{∗}].

Proof. H^{∗}(G) ≅ H^{∗}(G_{1})/(z_{1}^{(1)}, …, z_{d}^{(1)}) is the image of inflation from G_{1}, so is preserved by V. □

The linear span of z_{1}, …, z_{k} is isomorphic to Ω^{∗}. It is the kernel of inflation to Γ, so is preserved by V, hence so is ℤ/p[z_{1}, …, z_{k}]. □

4. The Bockstein

This section is devoted to the study of the Bockstein homomorphism on H^{∗}(G) with G a pro-p group having an almost powerfully embedded subgroup N.

It is sometimes useful to be able to calculate with an explicit construction of group extensions. Let A = C_{p^{r_{1}}} (e_{1})×⋯×C_{p^{rn}} (e_{n}) and let U be an elementary abelian group of rank m, considered as a trivial A-module. Let z = (z_{1}, …, z_{m}) ∈ H^{2}(A, U) and let z̃ be a representative normalized cocycle. Let Az̃ be a group defined as follows. Az̃ = U × A as sets and the multiplication is given by

(u, x) · (v, y) = (u + v + z̃(x, y), xy)

with u, v ∈ U, x, y ∈ A. If z̃ is changed by the codifferential of a normalized cochain c then there is an isomorphism Az̃ → A_{z̃+c} given by (u, x) ↦ (u + c(x), x).

We choose x_{i} ∈ H^{1}(C_{p^{r_{i}}} (e_{i})) as in section 1 and denote by x̃_{i} the representing normalized cocycle. We specify y_{i} ∈ H^{2}(C_{p^{r_{i}}} (e_{i})) by the cocycle

ỹ_{i}(e_{i}^{t}, e_{i}^{s}) = \{ \begin{matrix}1 & for t + s ≥ p^{r_{i}}, \\ 0 & for t + s < p^{r_{i}}. \end{matrix}

As before we also regard these elements as cocycles on A.

Lemma 4.1. If z_{ℓ} = ∑_{i=1}^{n} α_{i}^{(ℓ)}y_{i} + ∑_{1≤i<j≤n} λ_{ij}^{(ℓ)}x_{i}x_{j} then:

(i) If r_{i} = 1 then βx_{i} = y_{i}, (ii) (u, e_{i})^{p^{r_{i}}} = ((α_{i}^{(1)}, …, α_{i}^{(m)}), 1), (iii) [(u, e_{i}), (v, e_{j})] = (u, e_{i})^{−1}(v, e_{j})^{−1}(u, e_{i})(v, e_{j}) = ((λ_{ij}^{(1)}, …, λ_{ij}^{(m)}), 1), where we define λ_{ij}^{(ℓ)} =

−λ_{ji}^{(ℓ)} if i > j and 0 if i = j.

Notice that these sets of equations are invariant under adding a coboundary to z̃.

Proof. Part (i) is an easy calculation with cocycles.

For (ii) one proves, by induction on s, that (u, e_{i})^{s} = (su + ∑_{k=1}^{s−1} z̃(e_{i}, e_{i}^{s}), e_{i}^{s}). Now set s = p^{r_{i}} and evaluate the explicit cocycles.

For (iii), note that (u, e_{i})(v, e_{j}) = (v, e_{j})(u, e_{i})(z̃(e_{i}, e_{j}) − z̃(e_{j}, e_{i}), 1). Now finish by evaluating z̃(e_{i}, e_{j}) − z̃(e_{j}, e_{i}) using explicit cocycles. □

Suppose now that

A = C_{p^{r}}(e_{0}) × C_{p}(e_{1}) × ⋯ × C_{p}(e_{n})

be an abelian group with r ≥ 1, and let B = ⟨e_{1}, …, e_{n}⟩

Let Γ be given by the central extension

0 → (ℤ/p)^{n} →^{i} Γ →^{φ} A → 1

corresponding to a cohomology class (z_{1}, …, z_{n}) ∈ H^{2}(A)^{⊕n}. Suppose that i((ℤ/p)^{n}) ⊂ Φ(Γ), that φ^{−1}(B) is almost powerfully embedded and Ω-extendable in Γ. By Lemma 2.2 and Theorem 3.3, the z_{i} can be chosen such that

z_{i}|_{⟨e_{j}⟩} = \{ \begin{matrix}y_{i} & for i = j, \\ 0 & otherwise. \end{matrix}

Since φ^{−1}(B) is Ω-extendable in Γ, it follows from Theorem 3.13 that each βz_{i} belongs to (z_{1}, …, z_{n}), hence is of the form

βz_{i} = _{∑_{k,ℓ} ; ℓ>0}α_{kℓ}^{(i)}x_{k}z_{ℓ}  (1)

with α_{kℓ}^{(i)} ∈ ℤ/p, 1 ≤ i ≤ n. We have

Lemma 4.2. For 1 ≤ i ≤ n, 0 ≤ k, ℓ ≤ n, we have α_{kk}^{(i)} = 0, α_{kℓ}^{(i)} = −α_{ℓk}^{(i)} and

z_{i} = − ∑_{k<ℓ} α_{kℓ}^{(i)}x_{k}x_{ℓ} mod B(A).

In other words, [(0, e_{k}), (0, e_{ℓ})] = ∏_{i=1}^{n}(0, e_{i})^{−pα_{kℓ}^{(i)}}, 0 ≤ k, ℓ ≤ n.

Proof. Write

z_{i} = y_{i} + ∑_{0<ℓ} γ_{0ℓ}^{(i)}x_{0}x_{ℓ} + ∑_{1≤k<ℓ} γ_{kℓ}^{(i)}x_{k}x_{ℓ} + γ_{i}y_{0},  (2)

with γ_{st}^{(i)}, γ_{i} ∈ ℤ/p, 1 ≤ i ≤ n. (Note that, for p = 2 and 1 ≤ k < ℓ, γ_{kℓ}^{(i)} = 0 by Proposition 3.2, as φ^{−1}(B) is powerful.) So

βz_{i} = ∑_{0<ℓ} γ_{0ℓ}^{(i)}[x_{ℓ}β(x_{0}) − x_{0}y_{ℓ}] + ∑_{1≤k<ℓ} γ_{kℓ}^{(i)}[x_{ℓ}y_{k} − x_{k}y_{ℓ}].  (3)

Also, from (1) and (2), we have

βz_{i} = _{∑_{k,ℓ} ; ℓ>0}α_{kℓ}^{(i)}x_{k}[y_{ℓ} + ∑_{0<s} γ_{0s}^{(ℓ)}x_{0}x_{s} + ∑_{1≤t<s} γ_{ts}^{(ℓ)}x_{t}x_{s} + γ_{ℓ}y_{0}]

= _{∑_{k,ℓ} ; ℓ>0}α_{kℓ}^{(i)}x_{k}y_{ℓ} +_{∑_{k,ℓ} ; ℓ>0} ∑_{0<s} α_{kℓ}^{(i)}γ_{0s}^{(ℓ)}x_{k}x_{0}x_{s} +_{∑_{k,ℓ} ; ℓ>0} ∑_{1≤t<s} α_{kℓ}^{(i)}γ_{ts}^{(ℓ)}x_{k}x_{t}x_{s} +_{∑_{k,ℓ} ; ℓ>0}α_{kℓ}^{(i)}γ_{ℓ}x_{k}y_{0}

= \{ \begin{matrix}_{∑_{k,ℓ} ; ℓ>0}α_{kℓ}^{(i)}x_{k}y_{ℓ} +_{∑_{k,ℓ} ; ℓ>0}α_{kℓ}^{(i)}γ_{ℓ}x_{k}y_{0} mod λ^{3}(A) & for p > 2, \\ _{∑_{k,ℓ} ; ℓ>0}α_{kℓ}^{(i)}x_{k}y_{ℓ} +_{∑_{k,ℓ} ; ℓ>0} ∑_{0<s} α_{kℓ}^{(i)}γ_{0s}^{(ℓ)}x_{k}x_{0}x_{s} +_{∑_{k,ℓ} ; ℓ>0}α_{kℓ}^{(i)}γ_{ℓ}x_{k}y_{0} & for p = 2. \end{matrix}  (4)

Comparing (3) and (4) yields then α_{kk}^{(i)} = 0, α_{kℓ}^{(i)} = −α_{ℓk}^{(i)} and α_{kℓ}^{(i)} = −γ_{kℓ}^{(i)} for k < ℓ. The last equality follows from (2) and Lemma 4.1. □

With the notation as above, let Δ be given by the central extension

0 → (ℤ/p)^{n+1} →^{j} Δ →^{ψ} A → 1

corresponding to a cohomology class (w_{0}, w_{1}, …, w_{n}) ∈ H^{2}(A)^{⊕n+1}. Suppose that j((ℤ/p)^{n+1}) ⊂ Φ(Δ), that r ≥ 2 (so βx_{0} = 0) and ψ^{−1}(Ω_{1}(A)) is almost powerfully embedded and Ω-extendable in Γ. As above, the w_{i} can be chosen such that

w_{i}|_{⟨e_{j}⟩} = \{ \begin{matrix}y_{i} & for i = j, \\ 0 & otherwise, \end{matrix}

and each βw_{i} is of form

βw_{i} = ∑_{k,ℓ} σ_{kℓ}^{(i)}x_{k}w_{ℓ}  (5)

with σ_{kℓ}^{(i)} ∈ ℤ/p, 0 ≤ i ≤ n. We have Lemma 4.3. For 0 ≤ i ≤ n, 0 ≤ k, ℓ ≤ n, we have σ_{kk}^{(i)} = 0, σ_{kℓ}^{(i)} = −σ_{ℓk}^{(i)} and

w_{i} = − ∑_{k<ℓ} σ_{kℓ}^{(i)}x_{k}x_{ℓ} mod B(A).

In other words, [(0, e_{k}), (0, e_{ℓ})] = (0, e_{0})^{−p^{2}σ_{kℓ}^{(0)}} ∏_{i=1}^{n}(0, e_{i})^{−pσ_{kℓ}^{(i)}}, 0 ≤ k, ℓ ≤ n.

Proof. We use a similar argument to the one of the proof of Lemma 4.2. Write

w_{i} = y_{i} + γ_{0ℓ}^{(i)}x_{0}x_{ℓ} + ∑_{1≤k<ℓ} γ_{kℓ}^{(i)}x_{k}x_{ℓ},  (6)

with γ_{st}^{(i)} ∈ ℤ/p, 0 ≤ i ≤ n. (So, for p = 2 and 1 ≤ k < ℓ, γ_{kℓ}^{(i)} = 0.) Therefore

βw_{i} = − ∑_{0<ℓ} γ_{0ℓ}^{(i)}x_{0}y_{ℓ} + ∑_{1≤k<ℓ} γ_{kℓ}^{(i)}[x_{ℓ}y_{k} − x_{k}y_{ℓ}].  (7)

From (5) and (6), we have

βw_{i} = ∑_{k,ℓ} σ_{kℓ}^{(i)}x_{k}[y_{ℓ} + ∑_{0<s} γ_{0s}^{(ℓ)}x_{0}x_{s} + ∑_{1≤t<s} γ_{ts}^{(ℓ)}x_{t}x_{s}]

= ∑_{k,ℓ} σ_{kℓ}^{(i)}x_{k}y_{ℓ} + ∑_{k,ℓ} ∑_{0<s} σ_{kℓ}^{(i)}γ_{0s}^{(ℓ)}x_{k}x_{0}x_{s} + ∑_{k,ℓ} ∑_{1≤t<s} σ_{kℓ}^{(i)}γ_{ts}^{(ℓ)}x_{k}x_{t}x_{s}

= \{ \begin{matrix}∑_{k,ℓ} σ_{kℓ}^{(i)}x_{k}y_{ℓ} mod λ^{3}(A) & for p > 2, \\ ∑_{k,ℓ} σ_{kℓ}^{(i)}x_{k}y_{ℓ} + ∑_{k,ℓ} ∑_{0<s} σ_{kℓ}^{(i)}γ_{0s}^{(ℓ)}x_{k}x_{0}x_{s} & for p = 2. \end{matrix}  (8)

Comparing (7) and (8) yields then σ_{kk}^{(i)} = 0, σ_{kℓ}^{(i)} = −σ_{ℓk}^{(i)} and σ_{kℓ}^{(i)} = −γ_{kℓ}^{(i)} for k < ℓ. The last equality follows from (6) and Lemma 4.1. □

Suppose now that N is almost powerfully embedded in the p-group G with d(N) = d. Let a_{1}, …, a_{r}, a_{r+1}, …, a_{m} be a minimal system of generators of G with a_{1}, …, a_{r} ∈ N (with the convention that r = 0 if N ⊂ Φ(G)). So, for every i ≥ 1, ϕ_{i}(a_{1}), …, ϕ_{i}(a_{m}) also generate G_{i}. Define x_{j}, 1 ≤ j ≤ n in H^{1}(G_{1}) by x_{j}(ϕ_{1}(a_{k})) = δ_{jk}. Set e_{i} = a_{i}, 1 ≤ i ≤ m. There exist then e_{r+1}, …, e_{d} ∈ N such that e_{1}, …, e_{d} generate N. As N is powerful, the e_{i}’s can be renumbered so that e_{1}^{p^{i−1}}, …, e_{n_{i}}^{p^{i−1}} generate Φ_{G}^{i}(N). The x_{j}’s are then considered as elements of H^{∗}(G_{i}), i ≥ 2, via the inflation map.

Theorem 4.4. Suppose that N is almost uniformly embedded in G with ℓ = ℓ(Φ_{G}^{∗}(N)) ≥ 3. For 2 ≤ i ≤ ℓ − 1, the classes z_{1}^{(i)}, …, z_{d}^{(i)} given in Section 2 can be chosen such that:

(i) z_{j}^{(i)}|_{⟨ϕ_{i}(e_{k}^{p^{i−1}} )⟩} = 0 if j ≠ k;

(ii) if

βz_{j}^{(1)} = ∑_{s,t} λ_{st}^{(j)}x_{s}z_{t}^{(1)}  (9)

with λ_{st}^{(j)} ∈ ℤ/p, 1 ≤ j ≤ d, then:

(iia) λ_{ss}^{(j)} = 0; (iib) for 2 ≤ i ≤ ℓ − 2, 1 ≤ j ≤ d,

\begin{aligned}βz_{j}^{(i)} & = ∑_{st} λ_{st}^{(j)}x_{s}z_{t}^{(i)}, \\ βz_{j}^{(ℓ−1)} & = ∑_{st} λ_{st}^{(j)}x_{s}z_{t}^{(ℓ−1)} + η_{j}, \end{aligned}

with η_{j} ∈ Im Inf_{G}^{G_{1}}. If N is Ω-extendable in G, then η_{j} = 0, 1 ≤ j ≤ d.
In particular, if p = 2 or if [N, G] ⊂ N^{p^{2}} for p > 2, we have, for 1 ≤ j ≤ d,

\begin{aligned}βz_{j}^{(i)} & = 0, 1 ≤ i ≤ ℓ − 2, \\ βz_{j}^{(ℓ−1)} & ∈ Im Inf_{G}^{G_{1}}, \end{aligned}

and βz_{j}^{(ℓ−1)} = 0 if N is Ω-extendable in G.

Proof. (i) is clear from Lemma 2.2 and Proposition 3.2. We now prove (ii). Set A = ⟨ϕ_{1}(a_{k}), A_{1}⟩ with a given 1 ≤ k ≤ m. A is then an abelian subgroup of G_{1}. It follows from (9) that

βz_{j}^{(1)}|_{A} = ∑_{s=1}^{m} ∑_{t=1}^{d} λ_{st}^{(j)}x_{s}|_{A}z_{t}^{(1)}|_{A}.

By Lemmas 4.1, 4.2 and 4.3, this implies λ_{ss}^{(j)} = 0 and, for s ≠ t,

[a_{s}, e_{t}] = ∏_{j=1}^{d} e_{j}^{−pλ_{st}^{(j)}} mod Φ_{G}^{3}(N).

Hence, by Lemma 3.12 (i), for s ≠ t

[a_{s}, e_{t}^{p^{i−1}}] = [a_{s}, e_{t}]^{p^{i−1}} mod Φ_{G}^{i+2}(N)
= ∏_{j=1}^{d} e_{j}^{−p^{i}λ_{st}^{(j)}} mod Φ_{G}^{i+2}(N).  (10)

Given i ≤ ℓ − 2, for 1 ≤ k ≤ m, define B_{k} to be the abelian subgroup of G_{i} given by B_{k} = ⟨e_{k}, A_{i}⟩. Note that βz_{j}^{(i)} belongs to (z_{1}^{(i)}, …, z_{d}^{(i)}), for i ≤ ℓ − 2; hence it should be a linear combination of the x_{k}z_{t}^{(i)}’s. By (i), (10), Lemmas 4.2, 4.3 and from the structure of H^{∗}(G_{i}), βz_{j}^{(i)}|_{B_{k}} should be of form

βz_{j}^{(i)}|_{B_{k}} = ∑_{s,t} λ_{st}^{(j)}x_{s}|_{B_{k}}z_{t}^{(i)}|_{B_{k}}.

Therefore

βz_{j}^{(i)} = ∑_{s,t} λ_{st}^{(j)}x_{s}z_{t}^{(i)}.

Analogously, we also get the required decomposition for z_{j}^{(ℓ−1)}, 1 ≤ j ≤ d, by noting that, modulo Im Inf_{G}^{G_{1}}, βz_{j}^{(ℓ−1)} also belongs to (z_{1}^{(ℓ−1)}, …, z_{d}^{(ℓ−1)}), and ξ|_{B_{k}} = 0 for every ξ ∈ Im Inf_{G}^{G_{1}}. Furthermore, if N is Ω-extendable in G, then βz_{j}^{(ℓ−1)} belongs to (z_{1}^{(ℓ−1)}, …, z_{d}^{(ℓ−1)}), hence η_{j} = 0.

Finally, if p = 2 or if [N, G] ⊂ N^{p^{2}} for p > 2, then βz_{j}^{(1)} = 0, for 1 ≤ j ≤ d. So the last equalities follow from what we just proved. The theorem follows. □

Suppose that N is almost powerfully embedded and Ω-extendable in a finite p-group G. As noted above, d(Ω_{1}(N)) = d(N) and we may consider the case where Ω_{1}(N) is contained in the Frattini subgroup of G. The proof of the above theorem can be applied to the subgroup N/Ω_{1}(N) of G/Ω_{1}(N) and, together with Theorem 3.13, yields the following Theorem 4.5. Let N be almost powerfully embedded and Ω-extendable in a finite p-group G. Assume that Ω = Ω_{1}(N) is contained in N^{p}, d(Ω) = d. Pick a basis x_{1}, …, x_{m} of H^{1}(G_{1}) and a cohomology class (w_{1}, …, w_{d}) ∈ H^{2}(G_{1})^{⊕d} corresponding to the central extension 0 → A_{2} → G_{2} → G_{1} → 1. Then there exist z_{1}, …, z_{d} of H^{2}(G), η_{1}, …, η_{d} of Im Inf_{G}^{G_{1}} such that:

(i) H^{∗}(G) = H^{∗}(G_{1})/(w_{1}, …, w_{d}) ⊗ ℤ/p[z_{1}, …, z_{d}]; (ii) z_{1}|_{Ω}, …, z_{d}|_{Ω} is a basis of B(Ω); (iii) if

βw_{j} = ∑_{r,s} λ_{rs}^{(j)}x_{r}w_{s}, 1 ≤ j ≤ k,

then

βz_{j} = ∑_{r,s} λ_{rs}^{(j)}x_{r}z_{s} + η_{j}, 1 ≤ j ≤ k.

The extension Γ of G by (ℤ/p)^{d} corresponding to (z_{1}, …, z_{d}) is also Ω-extendable if and only if all the η_{j}’s vanish.

In particular, if [N, G] ⊂ N^{p^{2}} (for example, if N is powerfully embedded in G for p = 2), then βw_{j} = 0, βz_{j} ∈ Im Inf_{G}^{G_{1}}, 1 ≤ j ≤ k. □

Consider now the case where N is almost powerfully embedded and Ω-extendable in a pro-p group G with Ω = Ω_{1}(N) contained in N^{p}. As noted in Remark 3.5, d(Ω) = d(T) with T the subgroup of N consisting of elements of finite order. It is known that N/T is almost uniformly embedded in G/T. Set d = d(N), k = d(Ω). The cohomology class (z_{1}^{(1)}, …, z_{d}^{(1)}) ∈ H^{2}(G_{1}) can be then chosen such that (z_{k+1}^{(1)}, …, z_{d}^{(1)}) corresponds to the central extension 0 → A_{2}(N/T) → G/T → G_{1} → 1. It follows that, for 1 ≤ j ≤ k, βz_{j}^{(1)} is of form

βz_{j}^{(1)} = _{∑_{r} ; s≤k}λ_{rs}^{(j)}x_{r}z_{s}^{(1)}.

We then have

Corollary 4.6. Let N be almost powerfully embedded and Ω-extendable in a pro-p group G with d(N) = d. Assume that Ω = Ω_{1}(N) is contained in N^{p}, d(Ω) = k. Then there exist a basis x_{1}, …, x_{m} of H^{1}(G_{1}), a cohomology class (w_{1}, …, w_{d}) ∈ H^{2}(G_{1})^{⊕d} corresponding to the central extension 0 → A_{2} → G_{2} → G_{1} → 1, elements z_{1}, …, z_{k} of H^{2}(G), η_{1}, …, η_{k} of Im Inf_{G}^{G_{1}} and λ_{rs}^{(j)} ∈ ℤ/p, 1 ≤ r ≤ m, 1 ≤ s, j ≤ k such that:

(i) H^{∗}(G) = H^{∗}(G_{1})/(w_{1}, …, w_{d}) ⊗ ℤ/p[z_{1}, …, z_{k}]; (ii) z_{1}|_{Ω}, …, z_{k}|_{Ω} is a basis of B(Ω); (iii) for 1 ≤ j ≤ k,

\begin{aligned}βw_{j} & = _{∑_{r} ; s≤k}λ_{rs}^{(j)}x_{r}w_{s}, \\ βz_{j} & = ∑_{r,s} λ_{rs}^{(j)}x_{r}z_{s} + η_{j}. \end{aligned}

The extension Γ of G by (ℤ/p)^{k} corresponding to (z_{1}, …, z_{d}) is also Ω-extendable if and only if all the η_{j}’s vanish.

In particular, if [N, G] ⊂ N^{p^{2}} (for example, if N is powerfully embedded in G for p = 2), then βw_{j} = 0, βz_{j} ∈ Im Inf_{G}^{G_{1}}, 1 ≤ j ≤ k. □

With the notation as above, suppose that G is a powerful pro-p group and βx_{j} = ∑_{s<t}λ_{st}^{(j)}x_{s}x_{t}, 1 ≤ j ≤

n_{1}. Combining the above theorem with Corollary 3.14, we have the following, of which the case p ≥ 5 was given in [2, Theorem 3.16].

Corollary 4.7. Let G be a powerful pro-p group, Ω-extendable with d(G) = n, ℓ = ℓ(Φ_{G}^{∗}(G)) ≥ 2. Set d = d(Φ(G)), Ω = Ω_{1}(G) and k = d(Ω). Then there exist a basis x_{1}, …, x_{n} of H^{1}(G), elements y_{1}, …, y_{k+d−n} of H^{2}(G), η_{1}, …, η_{k−d+n} of Im Inf_{G}^{G/Φ(G)} and λ_{rs}^{(i)} of ℤ/p, 1 ≤ r, s, i ≤ k+d−n, satisfying:

(i) Res_{Ω}^{G} maps {y_{1}, …, y_{k+d−n}, βx_{d+1}, …, βx_{n}} isomorphically onto a basis of B(Ω);

(ii) H^{∗}(G) = \{ \begin{matrix}Λ[x_{1}, …, x_{n}] ⊗ ℤ/p[y_{1}, …, y_{k−d+n}, βx_{d+1}, …, βx_{n}] & for p > 2, \\ Λ[x_{1}, …, x_{d}] ⊗ ℤ/2[y_{1}, …, y_{k−d+n}, x_{d+1}, …, x_{n}] & for p = 2; \end{matrix}

(iii) λ_{rr}^{(i)} = 0, λ_{rs}^{(i)} = −λ_{sr}^{(i)}, βx_{i} = −∑_{r<s}λ_{rs}^{(i)}x_{r}x_{s}, βy_{i} = ∑_{r,s}λ_{rs}^{(i)}x_{r}y_{s} + η_{i}, 1 ≤ i ≤ k − d + n.

The extension Γ of G by (ℤ/p)^{k+d−n} corresponding to (y_{1}, …, y_{k+d−n}) is also Ω-extendable if and only if all the η_{i}’s vanish.

In particular, if p = 2 or if [G, G] ⊂ G^{p^{2}} for p odd, then all the λ_{rs}^{(i)}’s vanish. □

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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