Conjugacy classes of $\pi$-elements and nilpotent/abelian Hall $\pi$-subgroups

TL;DR

This paper investigates the relationship between the number of conjugacy classes of $pi$-elements in finite groups and the existence of nilpotent or abelian Hall $pi$-subgroups, providing bounds that guarantee such subgroups.

Contribution

It establishes precise lower bounds on the number of conjugacy classes of $pi$-elements that ensure the presence of nilpotent or abelian Hall $pi$-subgroups in finite groups.

Findings

Derived lower bounds for conjugacy classes of $pi$-elements.

Connected bounds to the existence of specific Hall $pi$-subgroups.

Enhanced understanding of the structure of finite groups based on conjugacy class counts.

Abstract

Let $G$ be a finite group and $\pi$ be a set of primes. We study finite groups with a large number of conjugacy classes of $\pi$-elements. In particular, we obtain precise lower bounds for this number in terms of the $\pi$-part of the order of $G$ to ensure the existence of a nilpotent or abelian Hall $\pi$-subgroup in $G$.