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\times G2 and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $V=V_1\otimes _F V_2$\end{document}. 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.