Finite groups with an irreducible character of large degree

Abstract

Let $G$ be a finite group and $d$ the degree of a complex irreducible character of $G$, then write $|G|=d(d+e)$ where $e$ is a nonnegative integer. We prove that $|G|\le e^4-e^3$ whenever $e>1$. This bound is best possible and improves on several earlier related results.