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arXiv · 0710.5468

Projective bases of division algebras and groups of central type II

Michael Natapov
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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^αG for some class αin 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 Λ. 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 Λbut one satisfy certain rigidity property.

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Michael Natapov. 2007-10-29. Projective bases of division algebras and groups of central type II. https://doi.org/10.1007/s11856-008-0020-7

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