A note on finite groups with few TI-subgroups

Abstract

In [Comm. Algebra, 43(2015), [2680][2681][2682][2683][2684][2685][2686][2687][2688][2689], finite groups all of whose metacyclic subgroups are TI-subgroups have been classified by S. Li, Z. Shen and N. Du. In this note we investigate a finite group all of whose non-metacyclic subgroups are TI-subgroups. We prove that G is a group all of whose non-metacyclic subgroups are TI-subgroups if and only if all nonmetacyclic subgroups of G are normal. Furthermore, we show that a group all of whose non-cyclic subgroups are TI-subgroups has a Sylow tower.