The cohomology of pro-\(p\) groups with a powerfully embedded subgroup. (Q1434775)

Let \(G\) be a pro-\(p\) group. A closed normal finitely generated subgroup \(N\) is said to be almost powerfully embedded in \(G\) if \([G,N]\subset N^p\) for \(p>2\) and \([G,N]\subset N^2\), \([N,N]\subset (N^2)^2\) for \(p=2\). The authors of the paper study the mod-\(p\) cohomology of \(p\)-groups and pro-\(p\) groups with almost powerfully embedded subgroups satisfying an extendibility condition. To state the main result we need to explain what is the condition. Let \(\Omega_1N\) denote the subgroup of \(N\) generated by elements of order \(p\). Then \(N\) is said to be \(\Omega\)-extendable in \(G\) if \(\Omega_1N\) is central in \(G\) and there is a central extension \(E\to\widetilde G\to G\), where \(E\cong\Omega_1N\) and every non-trivial element of \(\Omega_1N\) is the image of an element of \(\widetilde G\) of order \(p^2\). Note that every torsion-free pro-\(p\) group is extendable. Put \(\Phi_G(M)=[G,M]M^p\) and denote by \(H^*(G)\) the cohomology of \(G\) and by \(\beta\colon H^*(G)\to H^{*+1}(G)\) the Bockstein homomorphism. The main theorem of the paper is the following Theorem. Let \(G\) be a pro-\(p\) group and \(N\) an almost powerfully embedded subgroup, \(\Omega\)-extendable in \(G\). Then there exist elements \(z_1^{(1)},\dots,z_d^{(1)}\) of \(H^2(G/\Phi_G(N))\), \(z_1,\dots,z_k\) of \(H^2(G)\) such that (i) \(H^*(G)\cong H^*(G/\Phi_G(N))/(z_1^{(1)},\dots,z_d^{(1)})\otimes \mathbb{Z}/p[z_1,\dots,z_k]\); (ii) \(z_1^{(1)},\dots,z_d^{(1)}\) classify the extension \((\mathbb{Z}/p)^d\cong \Phi_G(N)/\Phi_G(\Phi_G(N))\to G/\Phi_G(\Phi_G(N))\to G/\Phi_G(N)\); (iii) \(z_1,\dots,z_k\) restrict to a basis of \(\beta H^1(\Omega_1(N)\cap N^p)\). The authors give also very nice cohomological criteria for a group \(G\) to be powerful.