Pronormality in infinite groups (Q2732523)

Introduced by P. Hall, the pronormal subgroups have been investigated by many authors, especially in the case of finite groups. More recently, several authors have investigated such subgroups in infinite groups. The main reason for this is their important role in the subgroup arrangements in linear groups [see \textit{Z. I. Borevich}, Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 94, 5-12 (1979; Zbl 0442.20024) and \textit{P. V. Gavron} and \textit{L. Yu. Kolotilina}, ibid. 103, 62-65 (1980; Zbl 0461.20013)]. First significant results in the study of infinite groups saturated with pronormal subgroups are connected with a detailed description of locally soluble (or even locally graded in the periodic case) groups, in which all subgroups are pronormal, groups, in which all infinite subgroups are pronormal, groups with pronormal primary subgroups, groups with pronormal cyclic subgroups that have been obtained by \textit{N. F. Kuzennyj} and \textit{I. Ya. Subbotin} [Ukr. Mat. Zh. 39, No. 3, 325-329 (1987; Zbl 0642.20028); (*) Izv. Vyssh. Uchebn. Zaved., Mat. 1988, No. 11(318), 77-79 (1988; Zbl 0685.20025); (**) Ukr. Mat. Zh. 41, No. 3, 323-327 (1989; Zbl 0693.20025); see also the survey by \textit{I. Ya. Subbotin} and \textit{N. F. Kuzennyj}, Contemp. Math. 131, Pt. 1, 383-388 (1992; Zbl 0767.20010)].NEWLINENEWLINENEWLINEIn the paper under review the authors continue to investigate some properties of pronormal subgroups in infinite groups and the connection between pronormal subgroups and groups, in which normality is a transitive relation (T-groups). More specific, the authors are selecting classes of infinite groups, in which the two above-mentioned classes coincide. These results are the most interesting in the article. Here are these theorems:NEWLINENEWLINENEWLINETheorem 3.9. Let \(G\) be an FC-group. The following statements are equivalent: (i) \(G\) is a soluble T-group; (ii) Every subgroup of \(G\) is pronormal; (iii) Every cyclic subgroup of \(G\) is pronormal.NEWLINENEWLINENEWLINETheorem 3.11. Let \(G\) be a periodic locally graded group satisfying the minimal condition on primary subgroups. Then \(G\) is a \(\overline{\text{T}}\)-group iff all its primary subgroups are pronormal.NEWLINENEWLINENEWLINEIn connection with these results it is very natural to admit that in the general situation the locally soluble (or sometimes locally graded) groups with these conditions are fully described in the above-mentioned papers of \textit{N. F. Kuzennyj} and \textit{I. Ya. Subbotin}. Also, as proven by \textit{I. Ya. Subbotin} and \textit{N. F. Kuzennyj} [(*), op. cit.] the conditions (i) and (iii) of Theorem 3.9 are equivalent in the case of an arbitrary locally soluble (not only FC-)group. However, as shown in the above-mentioned papers, the condition (ii) in the general situation restricts to a proper subclass of T-groups. The same situation applies to Theorem 3.11.9 [see (**), op. cit.].NEWLINENEWLINENEWLINESome other results of the article could be commented as above. In general, the article under review could be more useful and valuable if the authors would make the citations needed and right references in the appropriate cases.