Sufficient conditions for supersolubility of finite groups (Q1295505)

A subgroup \(H\) of a finite group \(G\) is \(S\)-quasinormal in \(G\) if it permutes with each Sylow subgroup of \(G\); a subgroup \(K\) of \(G\) is \(S\)-quasinormally embedded in \(G\) if each Sylow subgroup of \(K\) is also a Sylow subgroup of some \(S\)-quasinormal subgroup of \(G\). The authors prove: Theorem 1. If each maximal subgroup of the Sylow subgroups of \(G\) is \(S\)-quasinormally embedded in \(G\), then \(G\) is supersolvable. Theorem 2. If \(G\) is a solvable group with normal subgroup \(H\) such that \(G/H\) is supersolvable and all maximal subgroups of the Sylow subgroups of the Fitting subgroup of \(H\) are \(S\)-quasinormally embedded in \(G\), then \(G\) is supersolvable.