On the partial \(\Pi\)-property of some subgroups of prime power order of finite groups (Q6537098)

All groups appearing in this review are finite.\N\NMany authors have examined the structure of a finite group \(G\) under the assumption that certain subgroups of \(G\) of prime power order are well situated in \(G\).\N\NA subgroup \(H\) of a group \(G\) satisfies the partial \(\Pi\)-property in \(G\) if there exists a chief series \(\Gamma_{G}: 1=G_{0} < G_{1} < \cdots < G_{n}=G\) of \(G\) such that for every \(G\)-chief factor \(G_{i}/G_{i-1}\) (\(1 \leq i \leq n\)) of \(\Gamma_{G}\), \(|G/G_{i-1}:N_{G/G_{i-1}}(HG_{i-1}/G_{i-1} \cap G_{i}/G_{i-1})|\) is a \(\pi(HG_{i-1}/G_{i-1} \cap G_{i}/G_{i-1})\)-number. This concept introduced by \textit{X. Chen} and \textit{W. Guo} [J. Group Theory 16, No. 5, 745--766 (2013; Zbl 1311.20015)] generalizes a large number of known embedding properties. There they proved some results assuming some maximal or minimal subgroups of a Sylow subgroup satisfy the partial \(\Pi\)-property.\N\NThe aim of the paper is the study of the influence of the fact that certain subgroups of prime power order of a group \(G\) satisfy the partial \(\Pi\)-property, has on the structure of \(G\). It is known that the \(p\)-length of a \(p\)-supersoluble group is at most \(1\). One of the main results in the paper is the following: ``Let \(P\) be a Sylow \(p\)-subgroup of \(G\), and let \(d\) be a power of \(p\) such that \(1 < d < |P|\). Assume that every subgroup of \(P\) of order \(d\) satisfies the partial \(\Pi\)-property in \(G\), and every cyclic subgroup of \(P\) of order \(4\) (when \(d=2\) and \(P\) is not quaternion-free) satisfies the partial \(\Pi\)-property in \(G\). Then \(G\) is \(p\)-soluble with \(p\)-length at most \(1\).'' \N\NThey give an example that \(p\)-supersolubility can not be obtained in this result.