On groups of central type, non-degenerate and bijective cohomology classes

Abstract

A finite group $G$ is of central type (in the non-classical sense) if it admits a non-degenerate cohomology class $[c]\in H^2(G,\C^*)$ ($G$ acts trivially on $\C^*$). Groups of central type play a fundamental role in the classification of semisimple triangular complex Hopf algebras and can be determined by their representation theoretical properties. Suppose that a finite group $Q$ acts on an abelian group $A$ so that there exists a bijective 1-cocycle $\pi\in Z^1(Q,\ach)$, where $\ach=\rm{Hom}(A,\C^*)$ is endowed with the diagonal $Q$-action. Under this assumption, Etingof and Gelaki gave an explicit formula for a non-degenerate 2-cocycle in $Z^2(G,\C^*)$, where $G:=A\rtimes Q$. Hence, the semidirect product $G$ is of central type. In this paper we present a more general correspondence between bijective and non-degenerate cohomology classes. In particular, given a bijective class $[\pi]\in H^1(Q,\ach)$ as above, we construct non-degenerate classes $[c_{\pi}]\in H^2(G,\C^*)$ for certain extensions $1\to A\to G\to Q\to 1$ which are not necessarily split. We thus strictly extend the above family of central type groups.