A cohomological transfer map for profinite groups
type: Item
instance of: scholarly article
title: A cohomological transfer map for profinite groups
zbMATH Open document ID: 0887.20024
author: Oliver Schirokauer
published in: Journal of Algebra
publication date: 1998-01-07T00:00:00Z
review text: Let \(G\) be a profinite group, \(A\) an abelian protorsion group on which \(G\) acts continuously, and \(H\) a closed subgroup of \(G\). A new cohomological transfer map from \(H^*(H,A)\) to \(H^*(G,A)\) (where \(H\) is not necessarilly of finite index in \(G\)) is defined. The composition of the restriction map from \(H^*(G,A)\) to \(H^*(H,A)\) and this new transfer map is equal to multiplication by the index \([G:H]\), where the index is understood as an element of \(\widehat\mathbb{Z}\). As an application the following profinite version of a well-known result for finite groups is proved. Theorem. An extension of profinite groups \(A\to E\to G\) with abelian \(A\), splits topologically if the corresponding extension of every Sylow \(p\)-subgroup splits.
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 20E22
zbMATH DE Number: 1076266
zbMATH Keywords: profinite groups
zbMATH Keywords: cohomology transfer
zbMATH Keywords: restriction maps
zbMATH Keywords: split extensions
zbMATH Keywords: subgroups of finite index
reviewed by: Pavel Zalesskij
MaRDI profile type: Publication
full work available at URL: jabr.1997.7076
OpenAlex ID: W2073599637
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DOI: 10.1006/JABR.1997.7076
description: scientific article; zbMATH DE number 1076266