On groups with nilpotent by Černikov proper subgroups.

*(English)*Zbl 0897.20021A 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.

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.

Reviewer: D.J.S.Robinson (Urbana)

##### 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 |