Modular projective representations of direct products of finite groups

Let G be a finite group, S a field of characteristic p or a complete discrete valuation ring of characteristic p. We denote by S(lambda)G a twisted group ring of the group G and the ring S with an S-factor system A lambda Z(2) (G, S-*) (see [17], pp. 2-4). Let p\\G\ and G = G(p) x B be the direct product of a p-subgroup G. and p'-subgroup B. In this paper we establish necessary and sufficient conditions that every indecomposable S(lambda)G-module is the outer tensor product of an indecomposable S(lambda)G(p)-module and an irreducible (SB)-B-lambda-module.