Groups Having a Character of Maximal Degree

TL;DR

This paper classifies certain finite groups with maximal order related to their character degrees, specifically when the order equals $e^4 - e^3$, revealing new structures including a nonsolvable Camina pair.

Contribution

It provides a classification of groups with order exactly $e^4 - e^3$ for specific prime powers of $e$, including new examples of nonsolvable Camina pairs.

Findings

Classification of groups for $e$ prime, 4, 9, 25

Identification of a new nonsolvable Camina pair

Extension of bounds on group order related to character degrees

Abstract

Let $G$ be a group, let $d$ be a character degree, and let $e$ be the integer so that $|G| = d(d+e)$. It has been shown when $e > 1$ that $|G| \le e^4 - e^3$. In this paper, we consider the groups where $|G| = e^4 - e^3$. It is known that $e$ must be a power of a prime. We classify the groups where $e$ is a prime and where $e$ is $4$, $9$, and $25$. In so doing, we find a new nonsolvable Camina pair.