Ind-finite Groups
There is a notion of ind-finite group, which is the concept dual to profinite groups; i.e. a group G is ind-finite if it is the direct limit of an inductive system of finite groups. (In particular, it is a ind-group.) The usual terminology is different: a group G is called locally finite if every finitely-generated subgroup is finite. This is equivalent, in fact, to being 'ind-finite'.
By applying Pontryagin duality, one can see that abelian profinite groups are in duality with locally finite discrete abelian groups. The latter are just the abelian torsion groups.
Read more about this topic: Profinite Group
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