Bounding an index by the largest character degree of a solvable group

Abstract

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 $|G:O_p(G){|}_p\le b(G)$.

Updated the introduction based on referee's report