Finite groups with metacyclic QTI-subgroups

Abstract

Let G be a finite group. A subgroup A of G is called a TI-subgroup of G if A ∩ Ax = 1 or A for all x ∈ G. A subgroup H of G is called a QTI-subgroup if CG(x) ⊆ NG(H) for every 1 ≠ x ∈ H, and a group G is called an MCTI-group if all its metacyclic subgroups are QTI-subgroups. In this paper, we show that every nilpotent MCTI-group is either a Dedekind group or a p-group and we completely classify all the MCTI-p-groups. We show that all MCTI-groups are solvable and that every nonnilpotent MCTI-group must be a Frobenius group having abelian kernel and cyclic complement.