The largest size of conjugacy class and the $p$-parts of finite groups

TL;DR

This paper establishes a relationship between the size of the largest conjugacy class in a finite group and the structure of its Sylow p-subgroup, revealing new bounds when the group is nonabelian.

Contribution

It proves that for nonabelian finite groups, the size of the Sylow p-subgroup modulo its p-core is less than the largest conjugacy class size.

Findings

For nonabelian groups, |P/O_p(G)| < bcl(G).

Provides bounds relating conjugacy class size and Sylow p-subgroups.

Enhances understanding of group structure via conjugacy class sizes.

Abstract

Let $p$ be a prime and let $P$ be a Sylow $p$-subgroup of a finite nonabelian group $G$. Let $bcl(G)$ be the size of the largest conjugacy class of the group $G$. We show that $|P/O_p(G)| < bcl(G)$ if $G$ is not abelian.