On Tensor Induction of Group Representations

Let G be a (not necessarily finite) group and p a finite dimensional faithful irreducible representation of G over an arbitrary field; write ~p for p viewed as a projective representation. Suppose that p is not induced (from any proper subgroup) and that ~p~ is not a tensor product (of projective representations of dimension greater than 1 ). Let AT be a noncentral subgroup which centralizes all its conjugates in G except perhaps itself, write H for the normalizer of K in G, and suppose that some irreducible constituent, a say, of the restriction p[K is absolutely irreducible. It is proved that then ( p is absolutely irreducible and) ~p is tensor induced from a projective representation of H, namely from a tensor factor n of ~plH such that n[K = CT and ker it is the centralizer of K in G . 1980 Mathematics subject classification (Amer. Math. Soc.) (1985 Revision): Primary 20 C 15, 20 C 20.