Tensor products of primitive modules

Let F be a field and, for i = 1, 2, let Gi be a group and Vi an irreducible, primitive, finite dimensional FGi-module. Set G = G1 × G2 and V = V1 ⊗F V2. The main aim of this paper is to determine sufficient conditions for V to be primitive as a G-module. In particular this turns out to be the case if V1 and V2 are absolutely irreducible and V1 is absolutely quasi-primitive. Thus we extend a result of N.S. Heckster, who has shown that V is primitive whenever G is finite and F is the complex field. We also give a characterization of absolutely quasi-primitive modules. Ultimately, our results rely on the classification of finite simple groups.