Groups with finitely many normalizers of subnormal subgroups.
The authors study groups with finitely many normalizers of subnormal (respectively infinite subnormal) subgroups. Of course, if every subnormal subgroup is normal in a group \(G\) then \(G\) is a T-group (or a group in which normality is a transitive relation) and if every infinite subnormal subgroup is normal then \(G\) is called an IT-group [see \textit{S. Franciosi} and \textit{F. de Giovanni}, J. Algebra 96, 566-580 (1985; Zbl 0572.20016)]. Thus, generalizations of T- and IT-groups are considered in this article. The main results include: (1) Let \(G\) be a periodic soluble group with finitely many normalizers of infinite subnormal subgroups. Then \(G/\omega(G)\) is finite. (2) Let \(G\) be a soluble group with finitely many normalizers of subnormal subgroups. If \(G\) locally satisfies the maximal condition then \(G/\omega(G)\) is finite. Here \(\omega(G)\) denotes the Wielandt subgroup of \(G\).
scientific article; zbMATH DE number 5078832
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