Finite groups whose subnormal subgroups permute with all Sylow subgroups

Abstract

As a generalization of ( t ) (t) -groups and of ( q ) (q) -groups, a group G G is called a ( π − q ) (\pi - q) -group if every subnormal subgroup of G G permutes with all Sylow subgroups of G G . It is shown that if G G is a finite solvable ( π − q ) (\pi - q) -group, then its hypercommutator subgroup D ( G ) D(G) is a Hall subgroup of odd order and every subgroup of D ( G ) D(G) is normal in G G ; conversely, if a group G G has a normal Hall subgroup N N such that G / N G/N is a ( π − q ) (\pi - q) -group and every subnormal subgroup of N N is normal in G G , then G G is a ( π − q ) (\pi - q) -group.