Smith, Howard; Wiegold, James Locally graded groups with all subgroups normal-by-finite. (English) Zbl 0855.20028 J. Aust. Math. Soc., Ser. A 60, No. 2, 222-227 (1996). 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. Reviewer: B.H.Neumann (Canberra) Cited in 4 ReviewsCited in 14 Documents MSC: 20E25 Local properties of groups 20F24 FC-groups and their generalizations 20F50 Periodic groups; locally finite groups Keywords:CF-groups; BFC-groups; subgroups of finite index; boundedly finite index; locally graded BFC-groups; nilpotent CF-groups; locally graded CF-groups; Abelian-by-finite groups; finitely generated periodic locally graded groups Citations:Zbl 0853.20023 PDFBibTeX XMLCite \textit{H. Smith} and \textit{J. Wiegold}, J. Aust. Math. Soc., Ser. A 60, No. 2, 222--227 (1996; Zbl 0855.20028)