topologically finitely generated. Indeed, any open subgroup of is of finite index, so its preimage in is also of finite index, and hence it must be open. Suppose and are topologically finitely generated profinite groups that are isomorphic as discrete groups by an isomorphism Then is bijective and continuous by the above result. Furthermore, is also continuous, so is a homeomorphism. Therefore the topology on a topologically finitely generated profinite group is uniquely determined by its structure. Ind-finite groups There is a notion of , which is the conceptual dual to profinite groups; i.e. a group is ind-finite if it is the direct limit of an inductive system of finite groups. (In particular, it is an ind-group.) The usual terminology is different: a group 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. Projective profinite groups A profinite group is if it has the lifting property for every extension. This is equivalent to saying that is projective if for every surjective morphism from a profinite there is a section Projectivity for a profinite group is equivalent to either of the two properties: the cohomological dimension for every prime the Sylow -subgroups of are free pro--groups. Every projective profinite group can be realized as an absolute Galois group of a pseudo algebraically closed field. This result is due to Alexander Lubotzky and Lou van den Dries. Procyclic group A profinite group is if it is topologically generated by a single element that is, if the closure of the subgroup A topological group is procyclic if and only if where ranges over all prime numbers and is isomorphic to either or See also References . . . . Review of several books about profinite groups. . . Infinite group theory Topological groups