a subgroup x of a group g is said to be an h -subgroup if n_g(x) ∩ x^g ≤ x for each element g belonging to g. in [m. bianchi e. a., on finite soluble groups in which normality is a transitive relation, j. group theory, 3 (2000), 147–156] the authors showed that finite groups in which every subgroup has the h -property are exactly soluble groups in which normality is a transitive relation. here ...

According to P. Hall, a subgroup $$H$$ of finite group $$G$$ is called pronormal in if, for any element $$g$$ , the subgroups and $$H^{g}$$ are conjugate $$\langle H,H^{g}\rangle$$ . The simplest examples groups normal subgroups, maximal Sylow subgroups. Pronormal were studied by number authors. For example, Legovini (1981) which every subnormal or pronormal. Later, Li Zhang (2013) described st...

A group is called HNN-free if it has no subgroups that are nontrivial HNN-extensions. 23 We prove that finitely generated HNN-free implies virtually polycyclic for a large class of groups. We also consider finitely generated groups with no free subsemigroups of rank 2 25 and show that in many situations such groups are virtually nilpotent. Finally, as an application of our results, we determine...