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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 $$G/O_ p(G)$$ is supersolvable. In particular, $$G$$ is solvable.

MSC:
 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 20E28 Maximal subgroups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20D25 Special subgroups (Frattini, Fitting, etc.) 20D30 Series and lattices of subgroups
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References:
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