Groups with a Fixed Character Degree

Abstract

This note is concerned with the question: for which positive integers $d,e$, with $d$ square-free and $\text{gcd}(d,d+e)=1$, does there exist a solvable (we will see, in our case, metabelian) group $G$, of order $|G|=d(d+e)$, such that $d$ is an irreducible character degree of $G$? The main theorem is an arithmetic characterization in terms of the prime divisors of $d(d+e)$. This work is motivated by previous investigations into finite groups $G$ that have an irreducible character of comparatively large degree.