Coprimeness among irreducible character degrees of finite solvable groups

Abstract:

Given a finite solvable group $G$, we say that $G$ has property $P_{k}$ if every set of $k$ distinct irreducible character degrees of $G$ is (setwise) relatively prime. Let $k(G)$ be the smallest positive integer such that $G$ satisfies property $P_{k}$. We derive a bound, which is quadratic in $k(G)$, for the total number of irreducible character degrees of $G$. Three exceptional cases occur; examples are constructed which verify the sharpness of the bound in each of these special cases.