Finite Groups in Which All Nonabelian Subgroups Are Ti-Subgroups

Abstract

In this paper, we show that G is a finite group in which every nonabelian subgroup is a TI-subgroup if and only if every nonabelian subgroup of G is normal in G.

Finite groups in which some particular subgroups are TI-subgroups

“…Arguing as in proof of Theorem 2, we can obtain the following three results, which are generalizations of Theorem 1, [3] and [4] respectively. Here we omit their proofs.…”

“…In [1], Guo, Li and Flavell classified groups in which all abelian subgroups are TI-subgroups. As a generalization of [1], we showed in [3] that G is a group in which all nonabelian subgroups are TI-subgroups if and only if all nonabelian subgroups of G are normal in G, and we showed in [4] that G is a group in which all nonnilpotent subgroups are TI-subgroups if and only if all nonnilpotent subgroups of G are normal in G.…”

“…Since G is an NCTI-group, it follows that all nonabelian subgroups of G are TI-groups. By [3], we have that all nonabelian subgroups of G are normal in G. Then M is abelian. It follows that every Sylow subgroup of M must be cyclic by [2,Theorem 10.5.6].…”