On the number of conjugacy classes of $π$-elements in finite groups

Attila Maroti, Hung Ngoc Nguyen

Published 2013-06-04, updated 2014-01-18Version 4

Let $G$ be a finite group and $\pi$ be a set of primes. We show that if the number of conjugacy classes of $\pi$-elements in $G$ is larger than $5/8$ times the $\pi$-part of $|G|$ then $G$ possesses an abelian Hall $\pi$-subgroup which meets every conjugacy class of $\pi$-elements in $G$. This extends and generalizes a result of W. H. Gustafson.