Bounding group orders by large character degrees: A question of Snyder

Let $G$ be a nonabelian finite group and let $d$ be an irreducible character degree of $G$. Then there is a positive integer $e$ so that $\mathrm{\mid }G\mathrm{\mid }=d(d+e)$. Snyder has shown that if $e>1$, then $\mathrm{\mid }G\mathrm{\mid }$ is bounded by a function of $e$. This bound has been improved by Isaacs and by Durfee and Jensen. In this paper, we will show for groups that have a nontrivial, abelian normal subgroup that $\mathrm{\mid }G\mathrm{\mid }\le e^4-e^3$. We use this to prove that $\mathrm{\mid }G\mathrm{\mid }<e^4+e^3$ for all groups. Given that there are a number of solvable groups that meet the first bound, it is best possible. Our work makes use of results regarding Camina pairs, Gagola characters, and Suzuki 2-groups.