On \(E\)-\(S\)-supplemented subgroups of finite groups. (Q2839276)

In the paper under review, the authors uses some special type of supplementation to give sufficient conditions for a normal subgroup of a finite group \(G\) to have all the non-Frattini \(G\)-chief factors cyclic. They also determine some sufficient conditions for \(p\)-nilpotency, \(p\) prime.NEWLINENEWLINE They say that a subgroup \(H\) of a finite group \(G\) is \(E\)-\(S\)-supplemented in \(G\) if there is a subnormal subgroup \(T\) of \(G\) such that \(G=HT\) and \(H\cap T\) is contained in the subgroup generated by all \(S\)-permutably embedded subgroups of \(H\).NEWLINENEWLINE If \(E\) is a normal subgroup of \(G\) such that some distinguished subgroups of every Sylow subgroup of \(E\) or \(F^*(E)\) are \(E\)-\(S\)-supplemented in \(G\), then every non-Frattini \(G\)-chief factor below \(E\) is cyclic (see Theorem 1.4). -- This result extends a recent theorem of Shemetkov and Skiba.NEWLINENEWLINE The proof of the above theorem depends on the following result which is of independent interest: If the same condition holds for some distinguished family of the Sylow \(p\)-subgroups of \(G\) which is closely related to the above one, where \(p\) is a prime such that \((|G|,p-1)=1\), then \(G\) is \(p\)-nilpotent (Theorem 1.5).NEWLINENEWLINE Some well-known results appear as consequences of the above theorems.