Abelian subgroups, nilpotent subgroups, and the largest character degree of a finite group

Abstract

Let $H$ be an abelian subgroup of a finite group $G$ and $\pi$ the set of prime divisors of $|H|$. We prove that $|H{O}_{\pi }(G)/O_{\pi }(G)|$ is bounded above by the largest character degree of $G$. A similar result is obtained when $H$ is nilpotent.

16 pages