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On the intersection of a class of maximal subgroups of a finite group


Author: Xiu Yun Guo
Journal: Proc. Amer. Math. Soc. 106 (1989), 329-332
MSC: Primary 20D20
DOI: https://doi.org/10.1090/S0002-9939-1989-0999757-6
MathSciNet review: 999757
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Abstract: Let $ G$ be a finite group and $ \pi $ a set of primes. We consider the family of subgroups of $ G:\mathcal{F} = \{ M:M < \cdot G,{[G:M]_\pi } = 1,[G:M]$ is composite} and denote $ {S_\pi }(G) = \bigcap \left\{ M: M \in \mathcal{F} \right\} $ if $ \mathcal{F}$ is non-empty, otherwise $ {S_\pi }(G) = G$. The purpose of this note is to prove

Theorem. Let $ G$ be a $ \pi $-solvable group. Then $ {S_\pi }(G)$ has the following properties: (1) $ {S_\pi }(G)/{O_\pi }(G)$ is supersolvable. (2) $ {S_\pi }({S_\pi }(G)) = {S_\pi }(G)$. (3) $ G/{O_\pi }(G)$ is supersolvable if and only if $ {S_\pi }(G) = G$.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1989-0999757-6
Keywords: Finite group, maximal subgroup
Article copyright: © Copyright 1989 American Mathematical Society

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