Tensor products of primitive modules

Let F be a field and, for i = 1,2, let G i be a group and V i an irreducible, primitive, finite dimensional FG i -module. Set G = G 1 \times G 2 and $V=V_1\otimes _F V_2$ . 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 V 1 and V 2 are absolutely irreducible and V 1 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.