A generalization of a Hall theorem. (Q2801830)

All groups in this paper are finite. Let \(\sigma=\{\sigma_i \mid i\in I\}\) be a partition of the set \(\mathbb P\) of all primes. If \(G\) is any finite group, denote by \(\pi(G)\) the set of all primes dividing the order of \(G\). A group \(G\) is said to be \textit{\(\sigma\)-primary} if either \(G=1\) or \(|\{\sigma_i\mid\sigma_i\cap\pi(G)\neq\emptyset\}|=1\). And \(G\) is said to be \textit{\(\sigma\)-soluble} if every chief factor of \(G\) is \(\sigma\)-primary. Clearly, this notion extends the one of soluble group, by considering the partition \(\sigma\) of \(\mathbb P\) such that \(\sigma_i\) consists of a single prime, for any \(i\in I\).NEWLINENEWLINE The notions of \textit{Hall \(\Pi\)-subgroup} (for any subset \(\Pi\) of \(\sigma\)) and \textit{\(\sigma\)-Hall subgroup} of a group \(G\) are defined in the natural way in the paper under review. Also the concepts of \textit{complete Hall set of type \(\sigma\)} and \textit{\(\sigma\)-basis} of a group \(G\) are introduced, as extensions of the classical ones. Moreover, if \(A\), \(B\) and \(R\) are subgroups of a group \(G\), it is said that \(A\) \(R\)-permutes with \(B\) if \(AB^x=B^xA\), for some \(x\in R\).NEWLINENEWLINE With this terminology, the following generalizations of Hall's classical characterizations of soluble groups are obtained in the paper:NEWLINENEWLINE Theorem A. Let \(R=R_\sigma(G)\) be the \(\sigma\)-radical of \(G\) (that is, the largest normal \(\sigma\)-soluble subgroup of \(G\)). Then the following conditions are pairwise equivalent: (i) \(G\) is \(\sigma\)-soluble. (ii) For any \(\Pi\), \(G\) has a Hall \(\Pi\)-subgroup and every \(\sigma\)-Hall subgroup of \(G\) \(R\)-permutes with every Sylow subgroup of \(G\). (iii) \(G\) has a \(\sigma\)-basis \(\{H_1,\ldots,H_t\}\) such that for each \(i\neq j\) every Sylow subgroup of \(H_i\) \(R\)-permutes with every Sylow subgroup of \(H_j\).NEWLINENEWLINE Theorem B. Let \(R=R_\sigma(G)\) be the \(\sigma\)-radical of \(G\). Then \(G\) is \(\sigma\)-soluble if and only if for any \(\Pi\) the following hold: \(G\) has a Hall \(\Pi\)-subgroup \(E\), every \(\Pi\)-subgroup of \(G\) is contained in some conjugate of \(E\), and \(E\) \(R\)-permutes with every Sylow subgroup.NEWLINENEWLINE Other characterizations of \(\sigma\)-soluble groups are also obtained in the paper (see Theorem C).