Homogeneous products of characters

Abstract

I. M. Isaacs has conjectured (see \cite{isa00}) that if the product of two faithful irreducible characters of a solvable group is irreducible, then the group is cyclic. In this paper we prove a special case of the following conjecture, which generalizes Isaacs conjecture. Suppose that $G$ is solvable and that $ψ,ϕ\in\Irr(G)$ are faithful. If $ψϕ=mχ$ where $m$ is a positive integer and $χ\in \Irr(G)$ then $ψ$ and $ϕ$ vanish on $G- Z(G)$. In particular we prove that the above conjecture holds for $p$-groups.