A generalization of some results of the theory of permutable subgroups (Q6601241)

Let \(G\) be a finite group, \(A \leq G\) and \(\mathcal{X}=\{X_{1}, \ldots , X_{t} \}\) a set of subgroups of \(G\) such that \((|X_{i}|,|X_{j}|)=1\) if \(i \not =j\). The set \(\mathcal{X}\) is a covering subgroup system for \(A\) if \(A=\langle A \cap X_{1}, \ldots A \cap X_{t} \rangle\).\N\NIn the paper under review, the authors generalize some results of the theory of permutable subgroups. In particular they prove the following Theorem: Let \(A\) be a subgroup of a finite group \(G\) and \(E=X_{1}^{G} \ldots X_{t}^{G}\), where \(\mathcal{X}=\{X_{1}, \ldots, X_{t}\}\) is a covering subgroup system for \(A\) in \(G\). Suppose that \(AX_{i}^{x}=X_{i}^{x}A\) for all \(i\) and all \(x \in G\). If every member of \(\mathcal{X}\) is a soluble (respectively primary) group, then the section \(A^{E}/A_{E}\) is soluble (respectively nilpotent).