On the partial Π-property of second minimal or second maximal subgroups of Sylow subgroups of finite groups (Q6636344)

\textit{X. Chen} and \textit{W. Guo} [J. Group Theory 16, No. 5, 745--766 (2013; Zbl 1311.20015)] introduced the concept of the partial \(\Pi\)-property of subgroups of finite groups: a subgroup \(H\) of a finite group \(G\) satisfies the partial \(\Pi\)-property in \(G\) if there exists a chief series \(1 = G_0 < G_1 < \cdots < G_n = G\) of \(G\) such that, for \(1\leq i\leq n,\) \(|G/G_{i-1} : N_{G/G_{i-1}}(HG_{i-1}/G_{i-1} \cap G_i/G_{i-1})|\) is a \(\pi(HG_{i-1}/G_{i-1} \cap G_i/G_{i-1})\)-number. The authors investigate the structure of the groups in which some second minimal or second maximal subgroups of a Sylow subgroup satisfy the partial \(\Pi\)-property. Let \(P\) be a Sylow \(p\)-subgroup of a group \(G\). Then \(G\) is \(p\)-soluble with \(p\)-length at most 1 if one of the following occurs: \(|P|\geq p^2\) and every \(2\)-minimal subgroup of \(P\) satisfies the partial \(\Pi\)-property in \(G\); \(|P| \geq p^3,\) every \(2\)-maximal subgroup of \(P\) satisfies the partial \(\Pi\)-property in \(G\) and every cyclic subgroup of \(P\) of order \(4\) satisfies the partial \(\Pi\)-property in \(G\) when \(P\cong Q_8.\)