Projective bases of division algebras and groups of central type II

TL;DR

This paper completes the classification of groups that serve as projective bases for division algebras over a field, showing that all groups on a specific list can be realized as such bases and examining their uniqueness.

Contribution

It proves that every group on a known list can be a projective basis for a division algebra and analyzes the uniqueness of these bases.

Findings

All groups on the list mbda can be realized as projective bases.

Most groups on the list satisfy a rigidity property regarding basis uniqueness.

The classification of projective bases of division algebras is now complete.

Abstract

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.