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Influence of normality on maximal subgroups of Sylow subgroups of a finite group. (English) Zbl 0802.20019
This paper deals with the influence of the normality of the maximal subgroups of Sylow $p$-subgroups of a finite group $G$ on the structure of $G$. Here is a sample of the results obtained: 1) If $G$ is a finite solvable group and every maximal subgroup of the Sylow subgroups of the Fitting subgroup $F(G)$ is normal in $G$, then $G$ is supersolvable. 2) Let $G$ be a finite group, let $H \triangleleft G$ and assume that $G/H$ is supersolvable and all maximal subgroups of the Sylow subgroups of $H$ are normal in $G$. Then $G$ is supersolvable. 3) Let $G$ be a finite group, let $p = \max(\pi(G))$ and assume that every maximal subgroup of the Sylow $q$-subgroups of $G$ is normal in $G$ for all $q \in \pi(G) - \{p\}$. Then $G$ has a Sylow tower and $G/O\sb p(G)$ is supersolvable. In particular, $G$ is solvable.

20D20Sylow subgroups of finite groups, Sylow properties, $\pi$-groups, $\pi$-structure
20E28Maximal subgroups of groups
20D10Solvable finite groups, theory of formations etc.
20D25Special subgroups of finite groups
20D30Series and lattices of subgroups of finite groups
Full Text: DOI
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