Cohomology and profinite topologies for solvable groups of finite rank

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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 languageEnglish (US)
Pages (from-to)254-265
Number of pages12
JournalBulletin of the Australian Mathematical Society
Volume86
Issue number2
DOIs
StatePublished - Oct 1 2012

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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