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