Mathematics Faculty Research

Title

On a Conjecture of Gluck

Document Type

Article

Publication Date

Spring 4-2015

Abstract

Let F(G) and b(G) respectively denote the Fitting subgroup and the largest degree of an irreducible complex character of a finite group G. A well-known conjecture of D. Gluck claims that if G is solvable then |G:F(G)|≤b(G)2. We confirm this conjecture in the case where |F(G)| is coprime to 6. We also extend the problem to arbitrary finite groups and prove several results showing that the largest irreducible character degree of a finite group strongly controls the group structure.

Publication Title

Mathematische Zeitschrift

Volume

279

Issue

3-4

First Page

1067

Last Page

1080