Groups with a Fixed Character Degree

TL;DR

This paper investigates the structure of finite solvable groups with a fixed irreducible character degree, establishing a sequence of congruences relating prime factors of specific group order components.

Contribution

It provides a novel characterization of groups with a fixed character degree through prime factor congruences, extending understanding of their algebraic structure.

Findings

Established a sequence of congruences linking prime factors of group order components.

Proved the equivalence of the congruence condition with the presence of a fixed character degree.

Extended the result to both directions, characterizing the group structure.

Abstract

Let $G$ be a finite group, and let $d$ be the degree of an irreducible character of $G$ such that $|G|=d(d+e)$ for some $e>1$. Consider the case when $G$ is solvable, $d$ is square-free, and $(d,d+e)=1$. We wish to explore an equivalent condition on $G$ when $d\in\text{cd}(G)$. We show that if $d\in\text{cd}(G)$ then there is a sequence of congruences relating the prime power factors of $d+e$ to the product of prime factors of $d$ such that the product of the moduli in this sequence of congruences is $d$. Moreover, the argument will hold in both directions.