The splitting data of cohomology classes (Q798777)

Let G be a finite group, M a G module and assume given \(\alpha \in\hat H^*(G,M)\) (Tate cohomology). The splitting data of \(\alpha\) is defined to be the set of subgroups of G to which the restriction of \(\alpha\) is zero. In this paper the main properties of the splitting data are pointed out, and it is shown that given a set of subgroups of G, satisfying the mentioned properties, it is possible for any \(k\in {\mathbb{Z}}\), to construct a G-module M and an element \(\alpha\) of \(\hat H^ k(G,M)\), such that the given set of subgroups is the splitting data of \(\alpha\). Some restrictions can be imposed as to the order of \(\alpha\). For \(k=2\), we obtain group extensions which furnish interesting examples related to the ''ranks'' of crossed products.