Bounding group orders by large character degrees: A question of Snyder

TL;DR

This paper investigates bounds on the order of nonabelian finite groups based on irreducible character degrees, establishing new bounds for groups with abelian normal subgroups and improving existing general bounds.

Contribution

It proves a new upper bound of |G| ≤ e^4 - e^3 for groups with a nontrivial, abelian normal subgroup, and refines the general bound to |G| < e^4 + e^3.

Findings

For groups with abelian normal subgroups, |G| ≤ e^4 - e^3.

All groups satisfy |G| < e^4 + e^3.

The bounds are tight for certain solvable groups.

Abstract

Let $G$ be a nonabelian finite group and let $d$ be an irreducible character degree of $G$. Then there is a positive integer $e$ so that $|G| = d(d+e)$. Snyder has shown that if $e > 1$, then $|G|$ is bounded by a function of $e$. This bound has been improved by Isaacs and by Durfee and Jensen. In this paper, we will show for groups that have a nontrivial, abelian normal subgroup that $|G| \le e^4 - e^3$. We use this to prove that $|G| < e^4 + e^3$ for all groups. Given that there are a number of solvable groups that meet the first bound, it is best possible. Our work makes use of results regarding Camina pairs, Gagola characters, and Suzuki 2-groups.