# Cohomology and profinite topologies for solvable groups of finite rank

Research output: Contribution to journalArticle

1 Scopus citations

### Abstract

Assume that 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 az. 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 az-sections, the subgroup N enjoys some additional special properties with respect to its pro-p topology.

Original language English (US) 254-265 12 Bulletin of the Australian Mathematical Society 86 2 https://doi.org/10.1017/S0004972711003340 Published - Oct 1 2012

### All Science Journal Classification (ASJC) codes

• Mathematics(all)