TY - JOUR

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

AU - Lorensen, Karl

N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2012/10

Y1 - 2012/10

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.

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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 -