Profinite group

In mathematics, profinite groups are topological groups that are in a certain sense assembled from finite groups. They share many properties with their finite quotients: for example, both Lagrange's theorem and the Sylow theorems generalise well to profinite groups.
A non-compact generalization of a profinite group is a locally profinite group.
Definition.
Profinite groups can be defined in either of two equivalent ways.
First definition.
A profinite group is a topological group that is isomorphic to the inverse limit of an inverse system of discrete finite groups. In this context, an inverse system consists of a directed set formula_1, a collection of finite groups formula_2, each having the discrete topology, and a collection of homomorphisms formula_3 such that formula_4 is the identity on formula_5 and the collection satisfies the composition property formula_6. The inverse limit is the set:
formula_7
equipped with the relative product topology. In categorical terms, this is a special case of a cofiltered limit construction. One can also define the inverse limit in terms of a universal property.
Second definition.
A profinite group is a Hausdorff, compact, and totally disconnected topological group: that is, a topological group that is also a Stone space. Given this definition, it is possible to recover the first definition using the inverse limit formula_8 where formula_9 ranges through the open normal subgroups of formula_10 ordered by (reverse) inclusion.
Profinite completion.
Given an arbitrary group formula_10, there is a related profinite group formula_23, the profinite completion of formula_10. It is defined as the inverse limit of the groups formula_25, where formula_9 runs through the normal subgroups in formula_10 of finite index (these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients). There is a natural homomorphism formula_28, and the image of formula_10 under this homomorphism is dense in formula_23. The homomorphism formula_31 is injective if and only if the group formula_10 is residually finite (i.e.,
formula_33, where the intersection runs through all normal subgroups of finite index). 
The homomorphism formula_31 is characterized by the following universal property: given any profinite group formula_35 and any group homomorphism formula_36, there exists a unique continuous group homomorphism formula_37 with formula_38.
Ind-finite groups.
There is a notion of ind-finite group, which is the conceptual 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 an 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.
Projective profinite groups.
A profinite group is projective if it has the lifting property for every extension. This is equivalent to saying that "G" is projective if for every surjective morphism from a profinite "H" → "G" there is a section "G" → "H".
Projectivity for a profinite group "G" is equivalent to either of the two properties:
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 formula_10 is "procyclic" if it is topologically generated by a single element formula_40 i.e., formula_41 of the subgroup formula_42.
A topological group formula_10 is procyclic iff formula_44 where formula_45 ranges over all rational primes and formula_46 is isomorphic to either formula_47 or formula_48.