  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 $\psi,\phi\in\Irr(G)$ are faithful. If $\psi \phi=m\chi$ where $m$ is
a positive integer and $\chi \in \Irr(G)$ then $\psi$ and $\phi$ vanish on $G-
Z(G)$. In particular we prove that the above conjecture holds for $p$-groups.

Group Theory