Projective bases of division algebras and groups of central type II

Let G be a finite group and let k be a field. We say that G is a projective basis of a k-algebra A if it is isomorphic to a twisted group algebra k^\alpha G for some class \alpha in H^2(G,k^\times), where the action of G on k^\times is trivial. In a preceding paper by Aljadeff, Haile and the author (Projective bases of division algebras and groups of central type, Israel J. Math. 146 (2005) 317-335) it was shown that if a group G is a projective basis in a k-central division algebra then G is nilpotent and every Sylow-p subgroup of G is on the short list of families of p-groups, denoted by \Lambda. In this paper we complete the classification of projective bases of division algebras by showing that every group on that list is a projective basis for a suitable division algebra. We also consider the question of uniqueness of a projective basis of a k-central division algebra. We show that basically all groups on the list \Lambda but one satisfy certain rigidity property.