On strongly \(\Pi \)-permutable subgroups of a finite group
All groups in this report are finite. Consider a partition \(\sigma=\{\sigma_i\mid i\in I\}\) of the set \(\mathbb{P}\) of all primes and a subset \(\emptyset\ne \Pi\subseteq \sigma\) and a group~\(G\). A set \(\mathcal{H}\) of subgroups of \(G\) is a complete Hall \(\Pi\)-set of \(G\) when every non-trivial member of \(\mathcal{H}\) is a Hall \(\sigma_i\)-subgroup of \(G\) for some \(\sigma_i\in \Pi\) and \(\mathcal{H}\) has exactly one Hall \(\sigma_i\)-subgroup of \(G\) for every \(\sigma_i\in \pi\) such that \(\sigma_i\cap \pi(G)\ne \emptyset\). A group \(G\) with a complete Hall \(\Pi\)-set is called \(\Pi\)-full. A subgroup \(A\) of \(G\) is called \(\Pi\)-permutable in \(G\) if \(G\) has a complete Hall \(\Pi\)-set \(\mathcal{H}\) such that \(AH^x=H^xA\) for all \(H\in\mathcal{H}\) and all \(x\in G\); \(\sigma\)-subnormal in \(G\) if \(G\) hs a chain of subgroups \(A=A_0\le A_1\le\dots\le A_t=G\) such that either \(A_{i-1}\trianglelefteq A_i\) or \(A_i/(A_{i-1})_{A_i}\) is a \(\sigma_k\)-group for some \(k=k(i)\) for each \(i\in\{1,\dots, t\}\); and strongly \(\Pi\)-permutable in \(G\) if \(A\) is \(\Pi\)-permutable and \(\sigma\)-subnormal in~\(G\). A group \(G\) is said to be \(\sigma\)-nilpotent if \(G\) is the direct product of the members of a complete Hall \(\sigma\)-set of~\(G\). A subgroup \(A\) of \(G\) is said to be \(\Pi\)-modular when \(G\) is \(\Pi\)-full and \(\langle A, H\cap C\rangle=\langle A, H\rangle \cap C\) for every Hall \(\sigma_i\)-subgroup \(H\) of \(G\) and all \(\sigma_i\in \Pi\) and \(A\le C\le G\). The group \(G\) is said to be a \(\Pi\)-full group of Sylow type if every subgroup of \(G\) has the dominance property Hall \(\sigma_i\)-subgroups for all \(\sigma_i\in \Pi\). The first main theorem of this paper (Theorem~A) shows, for a \(\Pi\)-full group~\(G\) and a \(\sigma\)-nilpotent subgroup \(A\) of \(G\), the equivalence between the following facts: (1) \(A\) is strongly \(\Pi\)-permutable in~\(G\); (2) every characteristic subgroup of \(A\) is strongly \(\Pi\)-permutable in~\(G\); (3) every Hall \(\sigma_i\)-subgroup of \(A\) is strongly \(\Pi\)-permutable in~\(G\) for all \(i\). For the partition \(\sigma=\{\{2\}, \{3\},\dots\}\) we recover classical characterisations of Sylow-permutability for nilpotent subgroups. The second main theorem (Theorem~B) states that for a \(\Pi\)-full group \(G\) of Sylow type and a \(\sigma\)-subnormal subgroup of \(G\), the following statements are equivalent: (1) \(A\) is \(\Pi\)-permutable in~\(G\); (2) \(A\) is \(\Pi\)-permutable in \(\langle A, x\rangle\) for all \(x\in G\); (3) \(A\) is \(\Pi\)-modular in every subgroup \(E\) of \(G\) containing~\(A\). If \(\sigma^*=\{\sigma_i^*\mid j\in J\}\) is a partition of \(\mathbb{P}\) such that \(I\subseteq J\) and \(\sigma_j^*\subseteq \sigma_j\) for all \(j\in J\), we write \(\sigma^*\le \sigma\). Theorem~C states that if \(G\) is \(\Pi\)-full and \(\sigma^*\le \sigma\), then the set of all \(\Pi\)-permutable subgroups \(A\) of \(G\) with \(\sigma^*\)-nilpotent \(A^G/A_G\) is contained in the set of all strongly \(\Pi\)-permutable subgroups of \(G\), and both are sublattices of the lattice of all subgroups of~\(G\). Theorem~D shows that if \(G\) is \(\Pi\)-full, then the set \(\mathcal{L}\) of all \(\Pi\)-subgroups of \(G\) belonging to the set of all \(\Pi\)-permutable subgroups \(A\) of \(G\) with nilpotent \(A^G/A_G\) is a sublattice of the lattice of all subgroups of \(G\), and \(\mathcal{L}\) is modular if, and only if, every two members of \(\mathcal{L}\) permute. This generalises the known fact that the lattice of all S-permutable subgroups of \(G\) is modular if and only if every two members of the lattice permute.
scientific article; zbMATH DE number 7137344
scientific article