On \(SS\)-supplemented subgroups of finite groups and their properties. (Q2902672)

The authors provide the background and the preliminaries necessary for the introduction of \(SS\)-supplementation: A subgroup \(H\) in a finite group \(G\) is \(SS\)-supplemented in \(G\) if there exists a subgroup \(K\) contained in \(G\), the \(SS\)-supplement of \(H\), such that \(G=HK\) and \(H\cap K\) is \(S\)-quasinormal in \(K\). If each subgroup is \(SS\)-supplemented in \(G\), \(G\) is said to be \(SS\)-supplemented. In general, the class of \(SS\)-supplemented subgroups contains properly both the class of all \(C\)-supplemented subgroups as well as the class of all \(SS\)-quasinormal subgroups.NEWLINENEWLINE The significance of \(SS\)-complementation relates to the solvability of a group. For example, a group \(G\) is solvable if and only if each Sylow subgroup of \(G\) is \(SS\)-supplemented. This implies a Hall subgroup is complemented in \(G\) if and only if \(H\) is \(SS\)-supplemented in \(G\). Moreover a group \(G\) is solvable if and only if each maximal subgroup of \(G\) has a subnormal \(SS\)-supplement in \(G\). This can be refined to \(G\) being solvable if \(G\) has a solvable maximal subgroup \(H\) such that \(H\) has a normal \(SS\)-supplement \(K\) in \(G\). The \(SS\)-supplemented groups are supersolvable and are characterized. The remainder of the article carefully relates the role of \(SS\)-supplementation in examining groups with respect to saturated formations that contain the class of supersolvable groups, especially with respect to the Fitting and generalized Fitting subgroups.NEWLINENEWLINE The authors conclude with several results that yield conditions for \(p\)-nilpotence via \(SS\)-supplementation. The results are a significant contribution to the expansion of the concepts related to the structure of a finite group \(G=HK\in\mathfrak F\) whenever \(H,K\in\mathfrak F\), \(\mathfrak F\) a saturated formation containing the class of supersolvable groups.