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Groups with finitely many isomorphic classes of non-abelian subgroups. (English) Zbl 1429.20025

Summary: We study groups in which the non-abelian subgroups fall into finitely many isomorphic classes. We prove that a locally generalized radical group with this property is abelian-by-finite and reduced minimax. The same conclusion holds for locally generalized coradical groups. Here a generalized radical group is a group with an ascending series whose factors are either locally nilpotent or locally finite, and a generalized coradical group is a group with a descending series whose factors are either locally nilpotent or locally finite.

MSC:

20F14 Derived series, central series, and generalizations for groups
20E07 Subgroup theorems; subgroup growth
20E25 Local properties of groups
20F19 Generalizations of solvable and nilpotent groups
20F22 Other classes of groups defined by subgroup chains
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