ISRAEL JOURNAL OF MATHEMATICS 121 (2001), 173-198 DIVISION ALGEBRAS WITH A PROJECTIVE BASIS BY ELI ALJADEFF Department of Mathematics, Technion -- Israel Institute of Technology Haifa 32000, Israel e-mail: aljade~@~echunix.technion.ac.il AND DARRELL HALLE Department of Mathematics, Indiana University, Bloomington, IN ~7405, USA e-mail: haile@indiana.edu ABSTRACT Let k be any field and G a finite group. Given a cohomology class a 6 H 2 (G, k*), where G acts trivially on k*, one constructs the twisted group algebra kaG. Unlike the group algebra kG, the twisted group algebra may be a division algebra (e.g. symbol algebras, where G -~ Z, x Z,). This paper has two main results: First we prove that if D = kaG is a division algebra central over k (equivalently, D has a projective k-basis) then G is nilpotent and G', the commutator subgroup of G, is cyclic. Next we show that unless char(k) --- 0 and ~ ~ k, the division algebra D = kaG is a product of cyclic algebras. Furthermore, if D v is a p-primary factor of D, then Dp is a product of cyclic algebras where all but possibly one are symbol algebras. If char(k) = 0 and ~ @ k, the same result holds for Dr, p odd. If p = 2 we show that D2 is a product of quaternion algebras with (possibly) a crossed product algebra (L/k, [3), Gal(L/k) ~- Z2 x Z2~. Received January 6, 1999 173