Conditions for \(p\)-supersolubility and \(p\)-nilpotency of finite soluble groups. (Q2866749)

All groups considered in this paper are finite. Let \(\mathfrak Z\) be a complete set of Sylow subgroups of a group \(G\). A subgroup \(H\) of \(G\) is said to be \(\mathfrak Z\)-permutable embedded in \(G\) if every Sylow subgroup of \(H\) is also a Sylow subgroup of some \(\mathfrak Z\)-permutable subgroup of \(G\). A subgroup \(H\) of \(G\) is said to be \(\mathfrak Z\)-permutable in \(G\) if \(H\) permutes with every member of \(\mathfrak Z\).NEWLINENEWLINE In the main results of the paper the authors give criteria to obtain the \(p\)-supersolvability or the \(p\)-nilpotence of a \(p\)-soluble group \(G\), depending on properties of \(\mathfrak Z\)-permutability embedding of the \(p\)-nilpotent radical of a normal subgroup \(N\) of \(G\) containing the \(p\)-supersoluble or \(p\)-nilpotent residual of \(G\).