Groups with finitely many normalizers of non-Abelian subgroups.
Recall that a group \(G\) is said to be `metahamiltonian' if all its non-Abelian subgroups are normal. These groups were introduced and investigated by G. M. Romalis and N. F. Sesekin in a series of articles published about forty years ago. They proved, among others results, that locally soluble metahamiltonian groups have finite commutator subgroup. In the article under review the authors consider the class of groups with finitely many normalizers of non-Abelian subgroups. Many results are known which show that the behaviour of normalizers has a strong influence on the structure of groups. For instance, a famous theorem of B. H. Neumann (1955), improved by I. I. Eremin (1959), states that a group \(G\) is central-by-finite iff the normalizer of every (Abelian) subgroup of \(G\) has finite index. More recently, Ya. D. Polovickiĭ (1980) has shown that a group with finite conjugacy classes is central-by-finite if it has finitely many normalizers of infinite Abelian subgroups. In particular, a group has finitely many normalizers of Abelian subgroups iff it is central-by-finite. The main results obtained in this article are the following (recall that a group \(G\) is said to be `locally graded' if every finitely generated non-trivial subgroup of \(G\) contains a proper subgroup of finite index). Theorem A: Let \(G\) be a locally graded group with finitely many normalizers of non-Abelian subgroups. Then the commutator subgroup of \(G\) is finite. Theorem B: Let \(G\) be a locally graded group with finitely many normalizers of infinite non-Abelian subgroups. Then either \(G\) is a Chernikov group or its commutator subgroup is finite. Finally, an example is given of a nilpotent group with finitely many normalizers of non-Abelian subgroups, which is neither a Chernikov group nor metahamiltonian.
scientific article; zbMATH DE number 5173069