Let k be any field and G a finite group. Given a cohomology class α ∈ H2(G, k*), where G acts trivially on k*, one constructs the twisted group algebra kαG. Unlike the group algebra kG, the twisted group algebra may be a division algebra (e.g. symbol algebras, where G ≅ Zn × Zn). This paper has two main results: First we prove that if D = kαG 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 √-1 ∉ k, the division algebra D = kαG is a product of cyclic algebras. Furthermore, if Dp 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 √-1 ∉ k, the same result holds for Dp, p odd. If p = 2 we show that D2 is a product of quaternion algebras with (possibly) a crossed product algebra (L/k, β), Gal(L/k) ≅ Z2 × Z2n.