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

Abstract

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 $|G|=d(d+e)$. Snyder has shown that if $e>1$, then $|G|$ 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 $|G|\le e^4-e^3$. We use this to prove that $|G|<e^4+e^3$ 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.

We added a proof in the case when p = 2 and G is nonsolvable. We also have fixed some mistakes, and improved the arguments in a couple of places

References in corpus (1)

• p-Group Camina pairs

Cited by in corpus (3)

• The largest character degrees of the symmetric and alternating groups
• Finite groups with an irreducible character of large degree
• Picard groups for some blocks with TI defect groups