  In this paper, we show that if $p$ is a prime and $G$ is a $p$-solvable
group, then $| G:O_p (G) |_p \le (b(G)^p/p)^{1/(p-1)}$ where $b(G)$ is the
largest character degree of $G$. If $p$ is an odd prime that is not a Mersenne
prime or if the nilpotence class of a Sylow $p$-subgroup of $G$ is at most $p$,
then $| G:O_p (G) |_p \le b(G)$.
