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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]

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]