On group theoretical Hopf algebras and exact factorizations of finite groups

We show that a semisimple Hopf algebra A is group theoretical if and only if its Drinfeld double is a twisting of the Dijkgraaf-Pasquier-Roche quasi-Hopf algebra D-omega(Sigma), for some finite group Sigma and some omega is an element of Z(3) (Sigma, k(x)). We show that semisimple Hopf algebras obtained as bicrossed products from an exact factorization of a finite group Sigma are group theoretical. We also describe their Drinfeld double as a twisting of D-omega(Sigma), for an appropriate 3-cocycle omega coming from the Kac exact sequence. (C) 2003 Elsevier Inc. All rights reserved.