The influence of \(\pi\)-quasinormality of some subgroups of a finite group.
A subgroup \(H\) of a finite group \(G\) is said to be \(\pi\)-quasinormal in \(G\) if it permutes with every Sylow subgroup of \(G\). Suppose that \(G\) is a group with a normal subgroup \(H\) such that all maximal subgroups of any Sylow subgroup of \(F^*(H)\) are \(\pi\)-quasinomal in \(G\). The main result in the paper states that if \(G/H\) is supersolvable then \(G\) is also supersolvable. The authors also obtain several applications of the above result to saturated formations containing the class of supersolvable groups.
scientific article; zbMATH DE number 2051598