On The Largest Character Degree And Solvable Subgroups Of Finite Groups

TL;DR

This paper improves a bound relating the size of solvable $ ext{pi}$-subgroups to the largest irreducible character degree in finite groups, refining previous exponential bounds to a smaller exponent.

Contribution

It provides a tighter bound on the size of solvable $ ext{pi}$-subgroups in terms of the largest character degree, reducing the exponent from 3 to approximately 2.471.

Findings

Bound on solvable $ ext{pi}$-subgroups improved

Exponent reduced from 3 to about 2.471

Enhances understanding of subgroup structure in finite groups

Abstract

Let $G$ be a finite group, and $\pi$ be a set of primes. The $\pi$-core $\mathbf{O}_\pi(G)$ is the unique maximal normal $\pi$-subgroup of $G$, and $b(G)$ is the largest irreducible character degree of $G$. In 2017, Qian and Yang proved that if $H$ is a solvable $\pi$-subgroup of $G$, then $|H\mathbf{O}_\pi(G)/\mathbf{O}_\pi(G)|\le b(G)^3$. In this paper, we improve the exponent of $3$ to $3\log_{504}(168)<2.471$.