# Cohomology and profinite topologies for solvable groups of finite rank

Research output: Contribution to journalArticle

1 Citation (Scopus)

### 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

Solvable Group
Finite Rank
Cohomology
Topology
Isomorphism
Subgroup
Module
Normal subgroup
Completion
Isomorphic

### All Science Journal Classification (ASJC) codes

• Mathematics(all)

### Cite this

@article{9efb83fedba648969462462e839b3ab1,
title = "Cohomology and profinite topologies for solvable groups of finite rank",
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.",
author = "Karl Lorensen",
year = "2012",
month = "10",
day = "1",
doi = "10.1017/S0004972711003340",
language = "English (US)",
volume = "86",
pages = "254--265",
journal = "Bulletin of the Australian Mathematical Society",
issn = "0004-9727",
publisher = "Cambridge University Press",
number = "2",

}

In: Bulletin of the Australian Mathematical Society, Vol. 86, No. 2, 01.10.2012, p. 254-265.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Cohomology and profinite topologies for solvable groups of finite rank

AU - Lorensen, Karl

PY - 2012/10/1

Y1 - 2012/10/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84866022815&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84866022815&partnerID=8YFLogxK

U2 - 10.1017/S0004972711003340

DO - 10.1017/S0004972711003340

M3 - Article

AN - SCOPUS:84866022815

VL - 86

SP - 254

EP - 265

JO - Bulletin of the Australian Mathematical Society

JF - Bulletin of the Australian Mathematical Society

SN - 0004-9727

IS - 2

ER -