Variations on the Baer--Suzuki Theorem

Abstract

The Baer--Suzuki theorem says that if $p$ is a prime, $x$ is a $p$-element in a finite group $G$ and $⟨x,x^g⟩$ 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.

Cited by in corpus (1)

• On varieties of groups satisfying an Engel type identity