Local embeddings of some families of subgroups of finite groups. (Q1034312)

A subgroup \(H\) of a finite group \(G\) is said to have the partial cover and avoidance property in \(G\) (or to be a partial CAP-subgroup of \(G\)) if there exists a chief series of \(G\) such that \(H\) either covers or avoids each chief factor of this series. This concept was first introduced by \textit{Y. Fan}, \textit{X. Guo} and \textit{K. P. Shum} [Chin. Ann. Math., Ser. A 27, No. 2, 169-176 (2006; Zbl 1109.20017)], who called these subgroups semi cover-avoiding subgroups. In the paper under review the authors study the influence of the partial cover and avoidance property of certain families of subgroups on the structure of a finite group. Among other results in that direction, the following are proved: Theorem 3.2: Let \(p\) be a prime dividing the order of a group \(G\). Suppose that all maximal subgroups of every Sylow \(p\)-subgroup of \(G\) are partial CAP-subgroups. Then, either \(G\) is a group whose Sylow \(p\)-subgroups are cyclic groups of order \(p\), or \(G\) is a \(p\)-supersoluble group. Theorem 3.8: Let \(\mathcal F\) be a saturated formation containing the class of all supersoluble groups. Let \(G\) be a group and let \(N\) be a normal subgroup of \(G\) such that \(G/N\in\mathcal F\). If every maximal subgroup of every Sylow subgroup of \(F^*(N)\) (the generalized Fitting subgroup of \(N\)) is a partial CAP-subgroup of \(G\), then \(G\in\mathcal F\). Known results by \textit{X. Guo} and \textit{L. L. Wang} [in Acta Math. Sin., Engl. Ser. 23, No. 9, 1689-1696 (2007; Zbl 1138.20016)] can be derived as corollaries of some results in this paper.