On \(s\)-quasinormal and \(c\)-normal subgroups of a finite group. (Q943506)

All groups considered in this paper are finite. A subgroup \(H\) of a group \(G\) is called \(s\)-quasinormal in \(G\) if \(H\) commutes with every Sylow \(p\)-subgroup \(S\) of \(G\) (i.e., \(HS=SH\)). \textit{A. Ballester-Bolinches} and \textit{M. C. Pedraza-Aguilera} [J. Pure Appl. Algebra 127, No. 2, 113--118 (1998; Zbl 0928.20020)] generalized \(s\)-quasinormal subgroups to \(s\)-quasinormally embedded subgroups and proved that if the maximal subgroups of the Sylow subgroups of \(G\) are \(s\)-quasinormally embedded in \(G\), then \(G\) is supersolvable. It was also proved in the same paper that \(G\) is supersolvable if and only if there is a normal subgroup \(H\) in \(G\) such that \(G/H\) is supersolvable and the maximal subgroups of the Sylow subgroups of \(H\) are \(s\)-quasinormally embedded in \(G\). -- A subgroup \(H\) of \(G\) is said to be \(s\)-quasinormally embedded in \(G\) if for all primes \(p\), a Sylow \(p\)-subgroup of \(H\) is a Sylow \(p\)-subgroup of some \(s\)-quasinormal subgroup of \(G\). A subgroup \(H\) of \(G\) is called \(c\)-normal in \(G\) [\textit{Y. Wang}, J. Algebra 180, No. 3, 954--965 (1996; Zbl 0847.20010)] if \(G\) has a normal subgroup \(N\) such that \(G=NH\) and \(N\cap H\leq\text{Core}(H)\), where \(\text{Core}(H)\) is the core of \(H\) in \(G\). Let \(\mathfrak F\) be a saturated formation containing the class of all supersolvable groups and \(G\) a finite group. The main theorems in the paper are the following: (1) \(G\in\mathfrak F\) if and only if there is a normal subgroup \(H\) such that \(G/H\in\mathfrak F\) and every maximal subgroup of all Sylow subgroups of \(H\) is either \(c\)-normal or \(s\)-quasinormally embedded in \(G\); (2) \(G\in\mathfrak F\) if and only if there is a soluble normal subgroup \(H\) such that \(G/H\in\mathfrak F\) and every maximal subgroup of all Sylow subgroups of \(F(H)\) are either \(s\)-quasinormally embedded in \(G\) or \(c\)-normal in \(G\).