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

A finite group G is of central type (in the non-classical sense) if it admits a non-degenerate cohomology class [c] a H (2)(G, a",*) (G acts trivially on a",*). 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 a Z (1)(Q,Ci), where Ci = Hom(A, a",*) 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, a",*), where G:= A x 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] a H (1)(Q,Ci) as above, we construct non-degenerate classes [c pi] a H (2)(G,a",*) for certain extensions 1 -> A -> G -> Q -> 1 which are not necessarily split. We thus strictly extend the above family of central type groups.