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Groups with all subgroups normal-by-finite. (English) Zbl 0853.20023

A group \(G\) is a CF-group (core-finite) if \(H/\text{core}_GH\) is finite, for all \(H\leq G\). If there is an integer \(k\) such that \(|H/\text{core}_GH|\leq k\), for all \(H\leq G\), then \(G\) is a BCF-group (boundedly core-finite). Attempting to dualize Neumann’s result that groups in which \(|H^G:H|\) is finite for all \(H\leq G\) are finite-by-abelian, one might hope that all CF-groups were abelian-by-finite. However, Tarski groups are clearly BCF and need to be excluded from the discussion. The main result here is that a locally finite CF-group is abelian-by-finite and BCF. The final result shows that a periodic CF-group which is not abelian-by-finite has an infinite finitely generated homomorphic image \(G_0\) such that every subgroup of \(G_0\) is either finite or has finite index.

MSC:

20F24 FC-groups and their generalizations
20F50 Periodic groups; locally finite groups
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