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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 xG of order p such that [x,g] is a p-element for every gG. Then xO p (G). 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:
20D20Sylow subgroups of finite groups, Sylow properties, π-groups, π-structure
20E07Subgroup theorems; subgroup growth
20F45Engel conditions on groups
20F12Commutator calculus (group theory)
References:
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