On the number of $p$-elements in a finite group

TL;DR

This paper investigates the ratio of $p$-elements to Sylow $p$-subgroups in finite groups, proposing a conjecture and proving it for $p$-solvable groups and under certain conditions for almost simple groups.

Contribution

It introduces a conjecture relating $p$-elements and Sylow $p$-subgroups and proves it for specific classes of finite groups.

Findings

Proved the conjecture for $p$-solvable groups.

Established the conjecture's validity under conditions for almost simple groups.

Provided new bounds on the ratio of $p$-elements to Sylow $p$-subgroups.

Abstract

In this paper we study the ratio between the number of $p$-elements and the order of a Sylow $p$-subgroup of a finite group $G$. As well known, this ratio is a positive integer and we conjecture that, for every group $G$, it is at least the $(1-\frac{1}{p})$-th power of the number of Sylow $p$-subgroups of $G$. We prove this conjecture if $G$ is $p$-solvable. Moreover, we prove that the conjecture is true in its generality if a somewhat similar condition holds for every almost simple group.