Characterizations of Schunck classes of finite soluble groups
The motivation for this paper is the characterization of Schunck classes and saturated formations by means of certain embedding properties of their associated projectors. A subgroup \(H\) of a group \(G\) is said to be \(S\)-quasinormal in \(G\) if it permutes with every Sylow subgroup of \(G\). \(H\) is said to be \(S\)-quasinormally embedded in \(G\) if every Sylow subgroup of \(H\) is a Sylow subgroup of some \(S\)-quasinormal subgroup of \(G\). A subgroup \(H\) of a group \(G\) is said to be subnormally embedded in \(G\) if every Sylow subgroup of \(H\) is a Sylow subgroup of some subnormal subgroup of \(G\). A Schunck class \(\mathcal H\) is an \(S\)-quasinormally embedded Schunck class (respectively subnormally embedded) if for each group \(G\), every \(\mathcal H\)-projector is an \(S\)-quasinormally embedded (respectively subnormally embedded) subgroup of \(G\). The main results are two theorems obtaining that the Schunck classes \(S\)-quasinormally embedded are normally embedded Schunck classes and that the saturated formations subnormally embedded are the classes \({\mathcal H}={\mathcal S}_\pi\), for \(\pi=\text{char}({\mathcal H})\).
scientific article; zbMATH DE number 1203894