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Groups with maximal subgroups of Sylow subgroups $$\sigma$$-permutably embedded. (English) Zbl 1367.20014
Let $$\sigma = \{\sigma_{i} \mid i \in I\}$$ be some partition of the primes $$\mathbb{P}$$ and, for a finite group $$G$$, let $$\sigma(G) = \{\sigma_{i} \cap \pi(G) \mid i \in I \, \text{and} \, \sigma_{i} \cap \pi(G) \neq \emptyset\}$$. A set $$\mathcal H$$ of subgroups of $$G$$ is said to be a complete Hall $$\sigma$$-set of $$G$$ if every non-identity group in $$\mathcal H$$ is a Hall $$\sigma_i$$-subgroup of $$G$$ for some $$\sigma_i\in \sigma(G)$$ and $$\mathcal H$$ contains exactly one Hall $$\sigma_i$$-subgroup for every $$\sigma_i\in \sigma(G)$$. A subgroup $$H$$ of $$G$$ is called $$\sigma$$-permutable ($$\sigma$$-permutably embedded) in $$G$$ if $$G$$ possesses a complete Hall $$\sigma$$-set $$\mathcal H$$ such that $$AH^x=H^xA$$ for any $$H\in \mathcal H$$ and all $$x \in G$$ (if $$H$$ has a complete Hall $$\sigma$$-set and every Hall $$\sigma_i$$-subgroup of $$H$$ is also a Hall $$\sigma_i$$-subgroup of some $$\sigma$$-permutable subgroup of $$G$$, respectively). The main goal of the paper is to prove that every maximal subgroup of every Sylow subgroup of $$G$$ is $$\sigma$$-permutably embedded in $$G$$ if and only if $$G= D \rtimes M$$, where $$D$$ and $$M$$ are $$\sigma$$-Hall subgroups of $$G$$, $$D$$ coincides with the $$\sigma$$-nilpotent residual of $$G$$ and is nilpotent of odd order and every element of $$M$$ induces a power automorphism on $$D/\text{Frat}D$$. As an application of the previous result, the author obtains the classification of the finite groups $$G$$ such that every maximal subgroup of every Sylow subgroup of $$G$$ is $$\sigma$$-permutable in $$G$$.

##### MSC:
 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 20D15 Finite nilpotent groups, $$p$$-groups 20D30 Series and lattices of subgroups 20E28 Maximal subgroups
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