(1) Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel;(2) Department of Mathematics, Indiana University, 47405 Bloomington, IN, USA;(3) Department of Mathematics, Technion-Israel Institute of Technology, 32000 Haifa, Israel
Abstract:
Letk be a field. For each finite groupG and two-cocylef inZ2(G, kx) (with trivial action), one can form the twisted group algebra 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.
wherex σ x τ =f(σ,τ)x στ 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.