Covering the set of $p$-elements in finite groups by Sylow $p$-subgroups

TL;DR

This paper investigates whether Sylow p-subgroups are sufficient to cover all p-elements in finite groups, showing that in many cases they are, despite the general negative answer.

Contribution

The work demonstrates that Sylow p-subgroups often suffice to cover the p-elements set in finite groups, highlighting conditions where this holds.

Findings

Sylow p-subgroups cover G_p in many finite groups

Counterexamples show not all groups are covered by Sylow p-subgroups

Provides conditions under which Sylow p-subgroups cover G_p

Abstract

Let $G_p$ be the set of $p$-elements of a finite group $G$. Do we need all the Sylow $p$-subgroups of $G$ to cover $G_p$? Although this question does not have an affirmative answer in general, our work indicates that the answer is yes more often than one could perhaps expect.