Finite groups whose subnormal subgroups permute with all Sylow subgroups

Abstract:

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