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On extensions of the Baer-Suzuki theorem. (English) Zbl 0794.20029
Using the classification of the finite simple groups the authors prove the following theorem which is the main result of the paper: Let $p$ be a prime and $G$ be a finite group containing an element $x\in G$ of order $p$ such that $\left[x,g\right]$ is a $p$-element for every $g\in G$. Then $x\in {O}_{p}\left(G\right)$. The authors point out that this theorem was obtained also by W. Xiao [Sci. China, Ser. A 34, No. 9, 1025-1031 (1991; Zbl 0743.20015)] for a $p$-element of arbitrary order.
##### MSC:
 20D20 Sylow subgroups of finite groups, Sylow properties, $\pi$-groups, $\pi$-structure 20E07 Subgroup theorems; subgroup growth 20F45 Engel conditions on groups 20F12 Commutator calculus (group theory)
##### References:
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