Cohomology and profinite topologies for solvable groups of finite rank

Details Cohomology and profinite topologies for solvable groups of finite rank Cohomology and profinite topologies for solvable groups of finite rank

2011-03-08

arxiv.org/abs/1103.1610v3

Mathematics

Group Theory

This paper supersedes arXiv:1009.2645v5: the two theorems in the introduction to the latter paper are both corollaries to Theo

Scientific paper

Assume $G$ is a solvable group whose elementary abelian sections are all finite. Suppose, further, that $p$ is a prime such that $G$ fails to contain any subgroups isomorphic to $C_{p^\infty}$. We show that if $G$ is nilpotent, then the pro-$p$ completion map $G\to \hat{G}_p$ induces an isomorphism $H^\ast(\hat{G}_p,M)\to H^\ast(G,M)$ for any discrete $\hat{G}_p$-module $M$ of finite $p$-power order. For the general case, we prove that $G$ contains a normal subgroup $N$ of finite index such that the map $H^\ast(\hat{N}_p,M)\to H^\ast(N,M)$ is an isomorphism for any discrete $\hat{N}_p$-module $M$ of finite $p$-power order. Moreover, if $G$ lacks any $C_{p^\infty}$-sections, the subgroup $N$ enjoys some additional special properties with respect to its pro-$p$ topology.

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