Groups with many normal or self-normalizing subgroups (Q1275732)

A subgroup \(H\) of a group \(G\) is said to be a \(\xi\)-group if either \(H\) is normal in \(G\) or coincides with its normalizer in \(G\). A group \(G\) is called an \(\mathcal E\)-group if all its subgroups have the property \(\xi\). These groups were studied by \textit{G. Giordano} [Matematiche 26(1971), 291-296 (1972; Zbl 0254.20024)]. \textit{N. S. Chernikov} completely described (infinite and finite) groups in which all abelian subgroups have the property \(\xi\) [see Dopov. Akad. Nauk Ukr. RSR, Ser. A 1974, 977-978 (1974; Zbl 0297.20041); and Group-theoretical studies, Kiev 1978, 117-127 (1978; Zbl 0444.20024)]. \textit{G. Cutolo} studied groups in which every \(X\)-subgroup is a \(\xi\)-subgroup for some relevant classes of groups \(X\) [Boll. Unione Mat. Ital., VII. Ser., A 3, No. 2, 215-223 (1989; Zbl 0679.20023)]. The article under review is dedicated to the investigation of groups with the minimal condition on non-\(\xi\)-subgroups. The author proves that a locally graded group of this type either is a Chernikov group or is an \(\mathcal E\)-group (Theorem A). Theorem B states the same for a locally graded group with finitely many conjugacy classes of subgroups that do not have the property \(\xi\).