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

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 for p > 2; [G,N ] ⊂ N and [N,N ] ⊂ (N) for p = 2. Where N denotes the closure of the subgroup of N generated by p-th powers of elements of N . Let Ω1N denote the subgroup of N generated by elements of order p. We say that N is Ω-extendable in G if Ω1N 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 ∼= Ω1N and every non-trivial element of Ω1N is the image of an element of G̃ of order p. (So every torsion free group is extendable.) For any closed normal subgroup M ⊂ G we define ΦG(M) = [G,M ]M. 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 is a p-th power and Ω1N 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 , . . . , z (1) d of H (G/ΦG(N)), z1, . . . , zk of H(G) such that (i) H∗(G) ∼= H(G/ΦG(N))/(z 1 , . . . , z (1) d )⊗ Z/p[z1, . . . , zk]; (ii) z 1 , . . . , z (1) d classify the extension (Z/p) ∼= ΦG(N)/ΦGΦG(N)→ G/ΦGΦG(N)→ G/ΦG(N); (iii) z1, . . . , zk restrict to a basis of βH(Ω1(N) ∩N). 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 1991 Mathematics Subject Classification. 20J06, 17B50. *Supported by a grant from the E.P.S.R.C. Typeset by AMS-TEX 1 2 PHAM ANH MINH AND PETER SYMONDS 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 a1, . . . , an such that for each i = 1, . . . , n, ai does not annihilate any non-trivial element of R/(a1, . . . , ai−1). Recall the definition of the Koszul complex over R. Given a sequence of elements a1, . . . , an in the centre of R, K = KR(ai, . . . , an) is the the free R-module on certain symbols, which we can take to be monomials xi1 . . . xir in the symbols x1, . . . , xn such that i1 < i2 < · · · < ir (so no squares occur). It is graded by the degree of the monomial. The differential dr : Kr → Kr−1 is the R-linear map defined by dr(xi1 . . . xir ) = r ∑ j=1 (−1)aijxi1 . . . x̂ij . . . xir . If a1, . . . , an is a regular sequence of central elements of R then it is well known that the homology of the associated complex is just R/(a1, . . . , an) in degree 0 and 0 elsewhere. It is tempting to identify K with the exterior algebra over R on x1, . . . , xn, 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 = JR(a1, . . . , an) be the free R-module on all monomials in x1, . . . , xn , again graded by the degree, and define dr : Jr → Jr−1 by the same formula as before. We identify J with the polynomial ring R[x1, . . . , xn] and it becomes a differential graded algebra. Now we consider J as a module over R[x1, . . . , x 2 n]. We do this because d2(x 2 i ) = 0 and hence d is linear over R[x1, . . . , x 2 n]. But as an R[x1, . . . , x 2 n]-module, J ∼= R[x1, . . . , xn]⊗K (even though this is not compatible with any multiplicative structure) and so has homology R[x1, . . . , x 2 n]⊗R/(a1, . . . , an) ∼= R/(a1, . . . , an)[x1, . . . , xn]. We record this as: Lemma 1.1. Let a1, . . . , an is a regular sequence of central elements in a ring R. Then: (1) H∗(KR(a1, . . . , an)) ∼= R/(a1, . . . , an), (2) If 2R = 0 then H∗(JR(a1, . . . , an)) ∼= R/(a1, . . . , an)[x1, . . . , xn]. We now provide some regular sequences in the mod-p cohomology of a finite p-group. We shall use the following notation. Let R = Cpr1 (e1)×Cpr2 (e2)×· · ·×Cprk (ek) be an abelian p-group, with Cpri (ei) the cyclic group of order pi generated by ei. For each 1 ≤ i ≤ k, let xi ∈ H(Cpri (ei)) ∼= Hom(Cpri (ei),Z/p) correspond to the homomorphism that takes ei to 1 and let yi ∈ H(Cpri (ei)) be a generator. It is well known that H(Cpri ) = { Z/2[xi] for p = 2 and ri = 1, Λ[xi]⊗ Z/p[yi] otherwise. We identify H∗(R) with H(Cpr1 (e1))⊗H(Cpr2 (e2))⊗ · · · ⊗H(Cprk (ek)) under the Künneth isomorphism and identify xi and yi with their images in H∗(R). Let λ∗(R) denote the subalgebra of H∗(R) generated by the xi (this need not be an exterior algebra if p = 2) and set λ(R) = λ(R) for brevity. Let B(R) ⊂ H∗(R) be the subspace generated by the yi. Note that B(R) can be characterized as the image of H(R;Z) in H(R). Also B(R) contains the image of the Bockstein map from H(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(R) =  λ(R) +B(R) always λ(R)⊕B(R) if p > 2 or p = 2 and each ri ≥ 2 λ(R) for p = 2 and r1 = 1. (ii) y1, . . . , yk is a regular sequence in H∗(R). Furthermore, given elements γ1, . . . , γk of λ(R), the sequence y1 + γ1, . . . , yk + γk is regular in H∗(R), provided that p > 2, or p = 2 and min(r1, . . . , rk) > 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), we have: (i) If A is contained in the Frattini subgroup of G, then Im (H(G) Res → H(A)) ⊂ B(A). (ii) If there exist elements z1, . . . , zm of H(G) and γ1, . . . , γm ∈ λ(A) such that either z1|A, . . . , zm|A is linearly independent in B(A), or p > 2 and z1|A− γ1, . . . , zm|A− γm is linearly independent in B(A), then z1, . . . , zm is a regular sequence in H∗(G) . Proof. (i) was given in [10, Proposition 1.5]. We now prove (ii). Let iA : A→ A×G, iG : G→ A×G, f : A×G→ G be defined by iA(a) = (a, 1), iG(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 ◦iA) (resp. (f ◦iG)) is just the restriction (resp. identity) map on cohomology. So, for 1 ≤ i ≤ m, f(zi) = zi|A ⊗ 1 + 1⊗ zi 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(zi) induces for 1 ≤ i ≤ n a homomorphism of algebras f∗ i : H (G)/(z1, . . . , zi−1)→ H(A)/(z1|A, . . . , zi−1|A)⊗H(G)/(z1, . . . , zi−1). Let y be a non-trivial element of H(G)/(z1, . . . , zi−1). We now show that ziy is non-trivial, by claiming that f∗ i (ziy) is non-trivial. Write f∗ i (y) = ∑ s≥0 vs and f ∗ i (ziy) = ∑ s≥0 ws with vs, ws in (H(A)/(z1|A, . . . , zi−1|A)) ⊗H(G)/(z1, . . . , zi−1). Then v0 = 1 ⊗ y is non-trivial. Let s1 be maximal such that vs1 is non-trivial. It follows that, for s2 = s1 + 2, ws2 = (zi|A ⊗ 1) · vs1 . By Lemma 1.2 (ii), z1|A, . . . , zi|A is a regular sequence in H∗(A). hence ws2 6= 0. Thus z1, . . . , zm 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 a1, . . . , am of A. Regard A as a trivial Q-module, so H(Q,A) = ⊕i=1H(Q,Z/p 〈ai〉) = H2(Q)⊕m. Pick an element z ∈ H(Q,A). For every closed subgroup R of Q, denote by 0→ A→ Rz πR,z → R→ 1 the central extension of groups corresponding to the cohomology class z|R = ResQR(z) ∈ H(R,A). Let u1, . . . , um be the basis of H(A) = Hom(A,Z/p), dual to that of A. We then have the projection map (ui)∗ : H(Q,A)→ H(Q), 1 ≤ i ≤ m, and z can be expressed as z = (z1, . . . , zm) with zi = (ui)∗(z). 4 PHAM ANH MINH AND PETER SYMONDS Lemma 2.1. (i) All the InfQP (zi) = 0 if and only if there exists a surjective homomorphism τ : P → Qz satisfying πQ,z ◦ τ = π. (ii) Let R = Cpr1 (e1)× · · · × Cprk (ek) be an abelian subgroup of Q. Then: (iia) Rz is abelian if and only if z`|R ∈ B(R), 1 ≤ ` ≤ m; (iib) let S be the maximal elementary abelian subgroup of R. Then (Sz) = 1 if and only if z`|R ∈ λ(R), 1 ≤ ` ≤ m. Proof. (i) Let 0 → A → Pz → P → 1 be a group extension classified by the element z = Inf(QP (z1), . . . , InfQP (zm)). There exists then a commutative diagram 0 −−−−→ A −−−−→ Pz ρ −−−−→ P −−−−→ 1 ∥∥∥ μy yπ 0 −−−−→ A −−−−→ Qz πQ,z −−−−→ Q −−−−→ 1. If all the InfQP (z) = 0 then ρ is split, by σ say. Hence, by setting τ ◦ σ, we have πQ,z ◦ τ = π. Conversely, given τ , we get a splitting Pz → 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 (Sz) = 1⇔ |〈x〉| = p, for all x ∈ Sz ⇔ z`|〈y〉 = 0, for all y ∈ S, 1 ≤ ` ≤ m ⇔ z`|R ∈ λ(R), 1 ≤ ` ≤ m. We also have Lemma 2.2. Suppose that R is an elementary abelian subgroup of Q. Then (Rz) = A if and only if there exists a basis b1, b2, . . . , bm, . . . of R such that Res Q 〈bi〉(zi) 6= 0, Res Q 〈bj〉(zi) = 0, 1 ≤ i ≤ m, 1 ≤ j 6= i. Proof. If such a basis exists let b̃i be an element of the inverse image of bi. Then b̃i has order p and 〈b̃pi 〉 = 〈ei〉. Conversely, if (Rz) = A then every element of A is a p-th power of an element of Rz (Rz is powerful). Let b̃i be such that b̃ p i = ei, b̃j = 1, j > m, and let bi be the image of b̃i in R. Lemma 2.3. Consider the extension 0→ A→ Qz → Q→ 1 discussed above and suppose that there is no non-trivial relation q1z1 + · · ·+ qmzm = 0 with qi ∈ H(Q). Then the sequence H(Q) Inf → H(Qz) Res → H(A) is exact at the middle term. Proof. The Lyndon-Hochschild-Serre spectral sequence for the extension has E2 term H∗(Q) ⊗H∗(A). Note that d2(ui) = zi, 1 ≤ i ≤ m (see the remark after Proposition 2.4). Let q = q1u1 + · · · + qmum be an element of ker d 2 , with q1, . . . , qm ∈ H(Q). Then 0 = d2(q) = −(q1z1 + · · ·+ qmzm) which implies q1 = · · · = qm = 0, by the hypothesis. Therefore d 2 : E 1,1 2 → E 3,0 2 is injective and so E 1,1 3 = 0 and thus E ∞ = 0. This yields an exact sequence 0→ E ∞ → H(Qz)→ E ∞ → 0 and identifying the edge maps gives the result claimed. COHOMOLOGY OF A PRO-p GROUPS WITH A POWERFULLY EMBEDDED SUBGROUP 5 Proposition 2.4. Suppose that A is contained in the Frattini subgroup of Qz and that z1, . . . , zm is a regular sequence in H∗(Q). Then the inflation and restriction maps induce an isomorphism of rings H(Qz) ∼= H(Q)/(z1, . . . , zm)⊗ Z/p[B(A)] provided that one of the following conditions is satisfied: (a) H(Qz) Res → B(A) is surjective; (b) β(zi) ∈ (z1, . . . , zm), 1 ≤ i ≤ m; (c) ker InfQQz = (z1, . . . , zm). Proof. Let {Er, dr} be the Lyndon-Hochschild-Serre spectral sequence corresponding to the central extension 0→ A→ Qz → Q→ 1. Now E2 = H∗(Q)⊗H∗(A). Assume that p > 2 and that condition (a) holds. By considering the maps B(A) = Im(Res : H(Qz)→ H(A)) ∼= E ∞ ⊂ E 0,2 2 we can find elements ỹ1, . . . , ỹm ∈ E 0,2 2 corresponding to y1, . . . , ym ∈ B(A) such that d2(ỹi) = 0. Under the isomorphism E 0,2 2 ∼= H(A) we find that ỹi = yi(modλ(A)). Let S = Z/p[ỹ1, . . . , ỹm]. Then E 2 = S ⊗ Λ∗(E 0,1 2 ) and so E2 = S ⊗ Λ∗(E 0,1 2 )⊗H∗(Q). But d2 vanishes on S, by construction, and thus d2 is S⊗H ∗ (Q)-linear. So {E2, d2} is just the Koszul complex KS⊗H∗(Q)(z1, . . . , zm) of Lemma 1.1. Thus E3 = S ⊗H(Q)/(z1, . . . , zm) = E 3 ⊗ E ∗,0 3 . Now E3 is generated as a ring by the ỹi and E ∗,0 3 so, by the product structure, d3 = 0 and E4 = E3. Subsequently all the differentials are 0 for degree reasons and so E∞ = E3. Let ŷi ∈ H(Qz) have image ỹi ∈ E ∞ and let Ŝ = Z/p[ŷ1 . . . , ŷm]. Let I∗ = Im(Inf : H∗(Q) → H(Qz) ∼= H(Q)/(z1 . . . , zm). Then we have a natural ring homomorphism φ : Ŝ ⊗ I → H(Qz). If we filter Ŝ ⊗ I by F (Ŝ ⊗ I) = Ŝ ⊗ I≥p and H(Qz) 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 ui is transgressive, βui survives to E3 and d3(βui) = −βzi (see e.g. [9] and the remark below). Since E 3 = H(Q)/(H(Q) ∩ (z1, . . . , zm)), either of the conditions (b) or (c) implies that d3(βui) = 0. We set ỹi = β(ui) and proceed as before. If p = 2 then E2 is just the complex JH∗(Q)(z1, . . . , zm) of Lemma 2.1(2), so E3 = H(Q)/(z1, . . . , zm)[u1, . . . , u 2 m]. Set ỹi = ui (= βui) . Then condition (a) implies that d3(ỹi) = 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 d2(ui) = λizi for some non-zero λi ∈ Z/p. In fact λi = ±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 d3(βui) = ±βd2(ui). 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, 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 and [N,N ] ⊂ (N) (which also implies [N,N ] ⊂ N (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. 6 PHAM ANH MINH AND PETER SYMONDS 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 N[N,G,G] ⊂J ⊂ N[N,G] |N[N,G] : J | = p; in other words, G/J is given by the central extension 1→ Z/p→ G/J → G/N[N,G]→ 1 such that N/J is abelian of exponent ≤ 2p, but N/J 6⊂ Z(G/J). (ii) ( [7, Proposition 2]) Set M = N[N,G],K = M[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[N,G], N̂ = N/N[N,G]. We have Theorem 2.6. The following are equivalent: (i) N is powerfully embedded in G; (ii) Res N̂ (ker InfG) ∩ λ(N̂) = {0}. Proof. Let N be powerfully embedded in G (so N[N,G] = N, as [N,G] ⊂ N) and let 0 6= z ∈ kerG. Consider the central extension 0→ Z/p i → Ĝz → Ĝ→ 1 corresponding to z. By Lemma 2.1 (i), Ĝz is a homomorphic image of G, via τ . This implies i(Z/p) ⊂ τ(N) = (N̂z). So (N̂z) 6= 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→ Ĝw → Ĝ→ 1 corresponding to an element 0 6= w ∈ H(Ĝ) such that Ĝw is a quotient of G, (N̂w) = 1, [N̂w, Ĝw] 6= 1. It follows from Lemma 2.1 that w ∈ Ker InfG 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(Ĝ) ∧H(Ĝ)→ H(G) is injective, or p = 2 and ker(InfG) ∩ λ(Ĝ) = {0}. (iii) the map induced from the inflation λ(Ĝ)→ H(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 InfG) ∩ λ(N̂) = {0}, so (i) ⇔ (iii). Note that, for p 6= 2, λ(Ĝ) = H(Ĝ) ∧H(Ĝ) so (ii) is just a restatement of (iii). If p = 2, set G̃ = G/N[N,G], Ñ = N/N[N,G]. A characterization of powerfully embedded normal subgroup N of a 2-group G via the inflation InfG can also be obtained, as follows. First we prepare COHOMOLOGY OF A PRO-p GROUPS WITH A POWERFULLY EMBEDDED SUBGROUP 7 Lemma 2.8. Let L be a central subgroup of a 2-group H, and let w be a cohomology class of H/L satisfying w|L2/L4 = 0. Then w ∈ Im Inf H/L2 H/L4 . Proof. We may suppose that L 6= 1. Set A1 = L/L, A2 = L/L,H1 = H/L,H2 = H/L,K = H/L. A1, A2 are then vector spaces over Z/2 and we have the central extensions 0 → A1 → H1 → K → 1, 0 → A2 → H2 → H1 → 1 . Let z = (z1, . . . , zi) be the cohomology class classifying the extension 0→ A2 → H2 → H1 → 1. Since L is abelian, by Proposition 1.3 (ii), zj |A1 ∈ B(A1), 1 ≤ j ≤ i. Then ((A1)z) = A2 and z1|A1 , . . . , zi|A1 are linearly independent in B(A1), by Lemma 2.2. It follows from Proposition 1.3 (ii) that z1, . . . , zi is a regular sequence in H(H1). Hence, by Lemma 2.3, w|A2 = 0 implies w ∈ Im Inf H1 H2 . For a ∈ G̃ (resp. b ∈ Ĝ), define Na = 〈Ñ , a〉 (resp. N̂b = 〈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(G̃)→ H(G)), μ|Na ∈ B(Na) for every a ∈ G̃ (resp. μ|Ñ ∈ B(Ñ)). Proof. Let N be powerfully (resp. almost powerfully) embedded in G and let 0 6= z ∈ kerG. Consider the central extension 0→ Z/p→ G̃z → G̃→ 1 corresponding to z. By Lemma 2.1 (i), G̃z is a homomorphic image of G. Since Ñ is elementary abelian, (Ñz) = 1. As [G̃z, Ñz] ⊂ (Ñz) = 1 (resp. [Ñz, Ñz] ⊂ (Ñz) = 1), Ñz is central (resp. abelian). Note that Ñz is central iff Na is abelian for every a ∈ G̃. By Lemma 2.1 (iia), it follows that z|Na ∈ B(Na) 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(Ĝ) such that Ĝw is a quotient of G, (N̂w) = 1 and [N̂w, Ĝw] 6= 1 (resp. [N̂w, N̂w] 6= 1). By Lemma 2.1, [N̂w, Ĝw] 6= 1 implies that there exists b ∈ Ĝ such that w|N̂b / ∈ B(N̂b). So w|N̂2 = 0, and w|N̂b / ∈ B(N̂b) for some b ∈ Ĝ (resp. w|Ñ / ∈ B(Ñ)). Applying Lemma 2.8 with H = Ĝ, L = N̂ yields w = Inf Ĝ (t) for some t ∈ H̃∗(G̃). Let a be the projection of b on G̃, then w|N̂b / ∈ B(N̂b) (resp. w|Ñ / ∈ B(Ñ)) implies t|Na / ∈ B(Na) (resp. t|Ñ / ∈ B(Ñ)). As G̃ is a quotient of G, it follows from Lemma 2.1 (i) that ker InfG contains t. We then have Corollary 2.10. For p = 2, the following are equivalent: (i) G is powerful; (ii) ker (Inf:H(G̃)→ H(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 ΦG(N) = N[N,G]. For i ≥ 1, define recursively a sequence of closed normal subgroups ΦG(N) of G as follows: ΦG(N) = N,Φ i+1 G (N) = ΦG(Φ i G(N)). 8 PHAM ANH MINH AND PETER SYMONDS It is clear that [ΦG(N),Φ j G(N)] ⊂ Φ i+j G (N) and the p-power map induces a map Φ i G(N) p → Φ G (N), i, j ≥ 1; also, [G,N ] ⊂ ΦG(N). For i ≥ 0, set Gi = G/Φ G (N), Ai = Ai(N) = ΦG(N)/Φ i+1 G (N) (with the convention that Φ 0 G(N) = N). By [4, Proposition 1.16], G = lim ←− Gi, hence H∗(G) = lim −→ H (Gi). Each Ai is a vector space over Z/p and is central in Gi. We also have successive central extensions 0 −−−−→ A1 −−−−→ G1 −−−−→ G0 −−−−→ 1 0 −−−−→ A2 −−−−→ G2 −−−−→ G1 −−−−→ 1