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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On the $ p$-elements of a finite group


Author: C. Y. Ho
Journal: Proc. Amer. Math. Soc. 48 (1975), 61-66
DOI: https://doi.org/10.1090/S0002-9939-1975-0357596-3
MathSciNet review: 0357596
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Abstract: Let $ G$ be a finite group, $ p$ a prime, and $ x$ a $ p$-element in $ G$. An element $ g$ in $ G$ is called a witness of $ G$ if the subgroup generated by $ x$ and $ g$ is a $ p$-group. The set of all witnesses of $ x$ in $ G$ is denoted by $ W(x)$. This paper shows that $ x$ belongs to a given Sylow $ p$-subgroup $ P$ of $ G$ if one of the following holds: (1) $ G$ is $ p$-solvable and $ W(x) \supset P \cap \{ {x^g}\vert g \in G\} $; (2) $ G$ is $ p$-solvable, $ P = \langle P\backslash Z(P)\rangle $, and $ W(x) \supset P\backslash Z(P)$; (3) $ {\text{cl}}(P) \leqslant 2$ and $ W(x) \supset P$; (4) $ x$ normalizes a subgroup $ {P_1}$ of $ P$ with $ \vert P:{P_1}\vert \leqslant {p^2}$ and $ W(x) \supset P$; (5) $ \vert P\vert = {p^4}$ and $ W(x) \supset P$.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0357596-3
Keywords: Largest solvable normal subgroup, center, nilpotent, nilpotent class, Fitting subgroup, Frattini subgroup, $ p$-solvable, simple group, socle, Sylow $ p$-subgroup, witness
Article copyright: © Copyright 1975 American Mathematical Society