Finite groups with an irreducible character of large degree

TL;DR

This paper establishes a sharp upper bound on the order of finite groups with an irreducible character of large degree, refining previous results and providing a precise mathematical relationship involving the group's order and character degree.

Contribution

The authors prove a new optimal bound on the order of finite groups based on the degree of an irreducible character, improving earlier bounds and demonstrating its sharpness.

Findings

The bound |G| ≤ e^4 - e^3 is proven to be best possible.

The result generalizes and strengthens previous inequalities relating group order and character degrees.

The bound applies for all e > 1, providing a precise measure of the group's size.

Abstract

Let $G$ be a finite group and $d$ the degree of a complex irreducible character of $G$, then write $|G|=d(d+e)$ where $e$ is a nonnegative integer. We prove that $|G|\leq e^4-e^3$ whenever $e>1$. This bound is best possible and improves on several earlier related results.