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COHOMOLOGY AND PROFINITE TOPOLOGIES FOR SOLVABLE GROUPS OF FINITE RANK

Part of: Connections with homological algebra and category theory Structure and classification of infinite or finite groups Special aspects of infinite or finite groups

Published online by Cambridge University Press:  16 February 2012

KARL LORENSEN
KARL LORENSEN*
Affiliation:
Altoona College, Pennsylvania State University, Altoona, PA 16601, USA (email: [email protected])
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Abstract

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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 Cp∞. We show that if G is nilpotent, then the pro-p completion map induces an isomorphism for any discrete -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 is an isomorphism for any discrete -module M of finite p-power order. Moreover, if G lacks any Cp∞-sections, the subgroup N enjoys some additional special properties with respect to its pro-p topology.

Keywords

solvable groupsnilpotent groupsprofinite completionspro-p completionsprofinite topologypro-p topologySerre’s good groups

MSC classification

Secondary: 20F16: Solvable groups, supersolvable groups 20J06: Cohomology of groups 20E18: Limits, profinite groups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 86 , Issue 2 , October 2012 , pp. 254 - 265
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2012

References

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