Division algebras with a projective basis

Abstract

Letk be any field andG a finite group. Given a cohomology class α∈H2(G,k*), whereG acts trivially onk*, one constructs the twisted group algebrakαG. Unlike the group algebrakG, the twisted group algebra may be a division algebra (e.g. symbol algebras, whereG⋞Zn×Zn). This paper has two main results: First we prove that ifD=kαG is a division algebra central overk (equivalentyD has a projectivek-basis) thenG is nilpotent andG’ the commutator subgroup ofG, is cyclic. Next we show that unless char(k)=0 and\(\sqrt { - 1} \notin k\), the division algebraD=kαG is a product of cyclic algebras. Furthermore, ifDp is ap-primary factor ofD, thenDp is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k)=0 and\(\sqrt { - 1} \notin k\), the same result holds forDp, p odd. Ifp=2 we show thatD2 is a product of quaternion algebras with (possibly) a crossed product algebra (L/k,β), Gal(L/k)⋞Z2×Z2n.