Groups in which every proper subgroup is ?ernikov-by-nilpotent or nilpotent-by-?ernikov

Groups With Finitely Many Conjugacy Classes of Subgroups That Are Not Nilpotent-by-Chernikov

“…One consequence of Theorem 6 of that paper is that a locally finite p-group with all proper subgroups (nil-n)-by-Chernikov is soluble and nilpotent-by-Chernikov.) We remark that the case n = 1 of the above theorem from [2] was successfully dealt with in [7], subsequent to the proof for G periodic (and locally graded) in [9]. We shall denote by N n C the class of groups under discussion (and by NC the class of nilpotent-by-Chernikov groups).…”

Nilpotent-by-Černikov CC-groups

“…The main result in this direction is [6,Theorem 2], which asserts that a periodic locally graded group G is abelian-by-Cernikov if and only if every proper subgroup of G is abelian-by-Cernikov. Here G is said to be locally graded if every non-trivial finitely generated subgroup of G contains a proper subgroup of finite index.…”

“…In [6], the authors have studied groups in which every proper subgroup is abelian-by-Cernikov and characterized them in the periodic case. The main result in this direction is [6,Theorem 2], which asserts that a periodic locally graded group G is abelian-by-Cernikov if and only if every proper subgroup of G is abelian-by-Cernikov.…”

“…This condition is necessary to exclude Tarski groups and assures that the groups in consideration are locally finite ([6, 2.1]). We left in [6] of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S144678870003545X…”

“…Section 3 deals with a characterization of central-by-Cernikov groups in terms of the structure of the factors G/C G (x G ), x e G, of the CC-group G. What we shall do there is to bound the index of the radicable part of these Cernikov groups and to show that this will characterize central-by-Cernikov groups (Theorem 3.2). Finally, in Section 4, we shall make use of the results shown in Section 2 in order to obtain a result (Theorem 4.1), which is a contribution to the answer of an open question concerning groups with certain minimal conditions as those considered by the authors in [6].…”