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Locally graded groups with all subgroups normal-by-finite. (English) Zbl 0855.20028
This paper continues the study begun by J. T. Buckley, J. C. Lennox, the reviewer, H. Smith and J. Wiegold [ibid. 59, No. 3, 384-398 (1995; Zbl 0853.20023)] of groups in which the core of every subgroup has finite index in it [“CF-groups”] or even has boundedly finite index in it [“BFC-groups”]. The main result is that locally graded BFC-groups are abelian-by-finite [Theorem 1]. It is also shown that nilpotent CF-groups are BFC and abelian-by-finite [Theorem 3], and there are some more technical results. However, the question whether every locally graded CF-group is abelian-by-finite is left open. The discussion of this open question leads to some more problems, of which one [Question 1] asks whether every finitely generated, periodic, locally graded group in which every subgroup is either finite or of finite index is necessarily finite.

20E25 Local properties of groups
20F24 FC-groups and their generalizations
20F50 Periodic groups; locally finite groups