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A subgroup X of a group G is said to be an H-subgroup if NG(X) ∩ Xg ⩽ X for each element g belonging to G. In [M. Bianchi and 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 we extend this characterization to groups without simple sections. [ABSTRACT FROM AUTHOR]

We study groups in which normality is a weakly transitive relation, giving an extension of in Theorem A [On finite T-groups, J. Aust. Math. Soc. 75 (2003) 181-191] due to Ballester-Bolinches and Esteban-Romero pointing out the relations between these groups and those in which all subgroups are almost pronormal. Moreover, we extend a well-known theorem of Peng [Finite groups with pro-normal subgroups, Proc. Amer. Math. Soc. 20 (1969) 232-234] proving that for a large class of generalized FC-groups the weak transitivity of normality is equivalent to having finitely many maximal pronormalizers of subgroups. [ABSTRACT FROM AUTHOR]

A group is called HNN-free if it has no subgroups that are nontrivial HNN-extensions. 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 and show that in many situations such groups are virtually nilpotent. Finally, as an application of our results, we determine the structure of locally graded groups in which every subgroup is pronormal, thus generalizing a theorem of Kuzennyi and Subbotin. [ABSTRACT FROM AUTHOR]

Abstract: We extend to FC⁎, the class of generalized FC-groups introduced in [F. de Giovanni, A. Russo, G. Vincenzi, Groups with restricted conjugacy classes, Serdica Math. J. 28 (2002) 241–254], some results known for FC-groups. The main theorem involves the extended residually finite property (ERF), i.e., all subgroups are closed in the profinite topology. The ERF-groups belonging to several large classes of groups have been determined, for example nilpotent groups [M. Menth, Nilpotent groups with every quotient residually finite, J. Group Theory 5 (2002) 199–217] and FC-groups [D.J.S. Robinson, A. Russo, G. Vincenzi, On groups whose subgroups are closed in the profinite topology, J. Pure Appl. Algebra 213 (2009) 421–429]. Here we generalize these results by classifying completely the ERF-groups belonging to FC⁎. [ABSTRACT FROM AUTHOR]