  Let X be an irreducible, primitive complex character of the finite solvable
group G, and let X* denote the complex conjugate character. If the degree X(1)
is odd, then we show how to associate to X in a unique way, a conjugacy class
of subgroups U of G for which X*X = (1_U)^G, the permutation character on the
cosets of U. We investigate this situation and give a number of applications to
properties of primitive characters of solvable and p-solvable groups.
