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

TL;DR

This paper investigates the relationship between the number of conjugacy classes of -elements in finite groups and the structure of the group, establishing conditions under which the group has an abelian Hall -subgroup that intersects all -conjugacy classes.

Contribution

It generalizes Gustafson's result by providing a new threshold condition linking conjugacy class count and subgroup structure in finite groups.

Findings

If conjugacy classes of -elements exceed 5/8 of the -part of |G|, then G has an abelian Hall -subgroup.

The abelian Hall -subgroup intersects every -conjugacy class in G.

The result extends previous work by Gustafson on conjugacy class counts.

Abstract

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.