Groups with a Fixed Character Degree

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.