Variations on the Baer--Suzuki Theorem

TL;DR

This paper explores variations of the Baer--Suzuki theorem in finite groups, demonstrating that certain modified conditions still imply a $p$-group structure, and addresses related questions on commutators of $p$-elements.

Contribution

It introduces new variants of the Baer--Suzuki theorem and proves that some of these variants hold, extending understanding of $p$-elements in finite groups.

Findings

Certain variations of the Baer--Suzuki theorem are valid.

The paper provides answers to questions about commutators of $p$-elements.

It clarifies conditions under which $p$-elements generate $p$-groups.

Abstract

The Baer--Suzuki theorem says that if $p$ is a prime, $x$ is a $p$-element in a finite group $G$ and $\langle x, x^g \rangle$ is a $p$-group for all $g \in G$, then the normal closure of $x$ in $G$ is a $p$-group. We consider the case where $x^g$ is replaced by $y^g$ for some other $p$-element $y$. While the analog of Baer--Suzuki is not true, we show that some variation is. We also answer a closely related question of Pavel Shumyatsky on commutators of conjugacy classes of $p$-elements.