Groups in which every proper subgroup is ?ernikov-by-nilpotent or nilpotent-by-?ernikov
Copyright policy )Groups in which every proper subgroup is ?ernikov-by-nilpotent or nilpotent-by-?ernikov
\textit{B. Bruno} and \textit{R. E. Phillips} [Rend. Semin. Mat. Univ. Padova 69, 153-168 (1983; Zbl 0522.20022)] have classified infinite groups in which every proper subgroup is finite-by-nilpotent of class \(c\) whereas \textit{B. Bruno} [Boll. Unione Mat. Ital., VI. Ser. B 3, 797-807 (1984; Zbl 0563.20035) and ibid. D 3, 179-188 (1984; Zbl 0578.20027)] has considered the ``dual'' situation studying the cases in which proper subgroups are Abelian-by-finite and nilpotent-by-finite. The results in the present paper extend the above problems by replacing finite group by Chernikov group and they have a rather different nature than those of Bruno and Phillips because the main theorems give subgroup characterizations of the properties under consideration. These theorems are the following 1) A locally graded group \(G\) is Chernikov-by-nilpotent of class \(c\) if and only if every proper subgroup of \(G\) is Chernikov-by-nilpotent of class \(c\). 2) A periodic locally graded group \(G\) is Abelian-by-Chernikov if and only if every proper subgroup of \(G\) is Abelian-by-Chernikov.
- University of Zaragoza Spain
subgroup characterizations, Chains and lattices of subgroups, subnormal subgroups, Nilpotent groups, Periodic groups; locally finite groups, Subgroup theorems; subgroup growth, Chernikov groups, Chernikov-by-nilpotent groups, Abelian-by-Chernikov groups, locally graded groups, Local properties of groups
subgroup characterizations, Chains and lattices of subgroups, subnormal subgroups, Nilpotent groups, Periodic groups; locally finite groups, Subgroup theorems; subgroup growth, Chernikov groups, Chernikov-by-nilpotent groups, Abelian-by-Chernikov groups, locally graded groups, Local properties of groups
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