Fixed point ratios, Sylow numbers and coverings of $p$-elements in finite groups

TL;DR

This paper explores the use of fixed point ratios in finite groups to analyze Sylow p-subgroups and coverings of p-elements, building on recent research to provide new insights into group structure.

Contribution

It introduces novel applications of fixed point ratios to study Sylow p-subgroups and coverings in finite groups, extending prior work by Burness and Guralnick.

Findings

Derived bounds on the number of Sylow p-subgroups

Established minimal covering sizes for p-elements

Extended fixed point ratio techniques to new group classes

Abstract

Fixed point ratios for primitive permutation groups have been extensively studied. Relying on a recent work of Burness and Guralnick, we obtain further results in the area. For a prime $p$ and a finite group $G$, we use fixed point ratios to study the number of Sylow $p$-subgroups of $G$ and the minimal size of a covering by proper subgroups of the set of $p$-elements of $G$.