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On groups with nilpotent by Černikov proper subgroups. (English) Zbl 0897.20021
A group is called locally graded if every non-trivial finitely generated subgroup has a non-trivial finite image. The authors study locally graded groups in which every proper subgroup is nilpotent-by-Chernikov.
The main results are as follows. Theorem A: Let $$G$$ be a locally graded group in which every proper subgroup is nilpotent-by-Chernikov. Then either $$G$$ is nilpotent-by-Chernikov or else $$G$$ is a perfect, countable, locally finite $$p$$-group with all proper subgroups nilpotent. Theorem B: Let $$G$$ be a locally graded group. (i) If all proper subgroups are nilpotent-by-finite, then $$G$$ is either nilpotent-by-finite or periodic. (ii) If all proper subgroups are abelian-by-finite, then $$G$$ is either abelian-by-finite or periodic.
Finally, the authors answer a question of Otal and Peña by establishing Theorem C: A locally graded group in which every proper subgroup is abelian-by-Chernikov is itself abelian-by-Chernikov.

##### MSC:
 20E25 Local properties of groups 20F19 Generalizations of solvable and nilpotent groups 20F22 Other classes of groups defined by subgroup chains 20F50 Periodic groups; locally finite groups 20E07 Subgroup theorems; subgroup growth
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