Editing Profinite group (section)

==Ind-finite groups==

There is a notion of {{em|{{visible anchor||text=ind-finite group}}}}, which is the conceptual [[dual (category theory)|dual]] to profinite groups; i.e. a group <math>G</math> is ind-finite if it is the [[direct limit]] of an [[Direct limit#Formal definition|inductive system]] of finite groups. (In particular, it is an [[ind-group]].) The usual terminology is different: a group <math>G</math> is called [[locally finite group|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 group|abelian]] profinite groups are in duality with locally finite discrete abelian groups. The latter are just the abelian [[torsion group]]s.