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

Guo, Xiu Yun

doi:10.1090/S0002-9939-1989-0999757-6

On the intersection of a class of maximal subgroups of a finite group
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Proc. Amer. Math. Soc. 106 (1989), 329-332 Request permission

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 ${S_\pi }(G) = G$.

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