On second maximal subgroups of Sylow subgroups of finite groups. (Q616348)

Given a group \(G\), a subgroup \(K\) is called a second maximal subgroup if there exists a maximal subgroup \(M\) of \(G\) such that \(K\) is a maximal subgroup of \(M\). Several authors have studied the influence of the embedding of second maximal subgroups on the structure of a group. In the present paper attention is focused on the so-called partial cover and avoidance property. 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. The authors classified [in Acta Math. Sin., Engl. Ser. 25, No. 6, 869-882 (2009; Zbl 1190.20014)] the finite groups such that all maximal subgroups of the Sylow \(p\)-subgroups, \(p\) a fixed prime, are partial CAP-subgroups. In the paper under review, the authors give a complete classification of finite groups in which the second maximal subgroups of the Sylow \(p\)-subgroups are partial CAP-subgroups, for a fixed prime \(p\).