Finite groups whose abelian subgroups are TI-subgroups

Abstract

A group G is called an ATI-group if for any abelian subgroup A of G, A ∩ A x = 1 or A for all x ∈ G. In this paper the finite ATI-groups are classified.

Finite groups all of whose abelian subgroups are QTI-subgroups

“…Suppose that G is a Frobenius group of type (1) or (2). We also conclude by Lemma 2.2 that G is an AQTI-group.…”

“…It is well known that G is a Frobenius or 2-Frobenius group (see [8]), and then Lemmas 3.1 and 3.2 imply that G is of type (1), (2) or (3).…”

“…(3) For any cyclic subgroup A/ x of C G (x)/ x , A is an abelian subgroup of an AQTI-group C G (x), and so A is normal in C G (x) (see (2)). It follows that all cyclic subgroups (and so all subgroups) of C G (x)/ x are normal in C G (x)/ x .…”

A remark on "A note on TI-subgroups of finite groups"

“…In [1], X. Guo, S. Li and P. Flavell classified finite groups all of whose abelian subgroups are TI-subgroups. As a generalization of [1], J. Lu and L. Pang [2] gave a description of finite groups all of whose non-abelian subgroups are TI-subgroups, they called such groups NATI-groups.…”

The structure of non-nilpotent CTI-groups

“…In [13], G. Walls classified the TI-groups. S. Li and X. Guo in [6] classified the ATI-groups of prime power order; also these authors with P. Flavell in [4] determined the structure of ATI-groups.…”