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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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