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

TL;DR

This paper establishes bounds on the index of the largest normal p-subgroup in a p-solvable group based on its largest character degree, with specific improvements under certain conditions.

Contribution

It provides new bounds relating the index of the p-core to the largest character degree in p-solvable groups, extending previous results with sharper inequalities.

Findings

Bound |G:O_p(G)|_p ≤ (b(G)^p/p)^{1/(p-1)} for p-solvable groups.

Improved bound |G:O_p(G)|_p ≤ b(G) under certain conditions.

Applicable to groups where p is not a Mersenne prime or Sylow p-subgroup has small nilpotence class.

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