Large orbits of Hall subgroups of solvable linear groups

TL;DR

This paper establishes a bound on the index of a specific subgroup in a finite solvable group based on the largest character degree of a Hall subgroup, linking group structure and character theory.

Contribution

It provides a new inequality connecting the index of a subgroup and the character degree of a Hall subgroup in solvable groups.

Findings

Bound on subgroup index in terms of character degree

Link between group structure and character theory

Inequality involving Hall subgroups and character degrees

Abstract

Suppose that $G$ is a finite solvable group and let $H$ be a Hall $\pi$-subgroup, let $b(H)$ be the largest character degree of $H$, we show that $|G:O_{\pi' \pi}(G)|_{\pi} \leq b(H)^2$.