Communications of the
Korean Mathematical Society CKMS

In \cite{J}, Johnson introduced the primitivity of subgroups and proved that a finite group $G$ is supersolvable if every primitive subgroup of $G$ has a prime power index in $G$. In that paper, he also posed an interesting problem: what a group looks like if all of its primitive subgroups are maximal. In this note, we give the detail structure of such groups in solvable case. Finally, we use the primitivity of some subgroups to characterize $T$-group and the solvable $PST_0$-groups.

Keywords: finite groups, primitive subgroups, maximal subgroups, the solvable $PST_0$-groups

MSC numbers: 20D10, 20D15