A cohomological transfer map for profinite groups (Q1369262)

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.