Projective bases of division algebras and groups of central type

Letk be a field. For each finite groupG and two-cocylef inZ2(G, kx) (with trivial action), one can form the twisted group algebra\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$k^f G = \oplus _{\sigma \in G} kx_\sigma $$ \end{document} wherexσxτ=f(σ,τ)xστ for all σ, τ∃G. Our main result is a short list ofp-groups containing all thep-groupsG for which there is a fieldk and a cocycle such that the resulting twisted group algebra is ak-central division algebra. We also complete the proof (presented in all but one case in a previous paper by Aljadeff and Haile) that everyk-central division algebra that is a twisted group algebra is isomorphic to a tensor product of cyclic algebras.