Cohomology and profinite topologies for solvable groups of finite rank

Authors

  • Karl Lorensen

Keywords:

solvable groups, good groups, profinite completions, pro-p completions

Abstract

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 $\mathbb Z/p^\infty$. We show that if $G$ is nilpotent, then the pro-$p$ completion map $G\to \hat{G}_p$ induces an isomorphism $H^\ast(\hat{G}_p,A)\to H^\ast(G,A)$ for any discrete $\hat{G}_p$-module $A$ 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 $H^\ast(\hat{N}_p,A)\to H^\ast(N,A)$ is an isomorphism for any discrete $\hat{N}_p$-module $A$ of finite $p$-power order. Moreover, if $G$ lacks any $\mathbb Z/p^\infty$-sections, the subgroup $N$ enjoys some additional special properties with respect to its pro-$p$ topology. DOI: 10.1017/S0004972711003340

Published

2012-08-27

Issue

Vol. 86 No. 2 (2012)

Section

Articles
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