The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups.
-adic integers and the Galois groups of infinite-degree field extensions.
Every profinite group is compact and totally disconnected.
A non-compact generalization of the concept is that of locally profinite groups.
Even more general are the totally disconnected groups.
Profinite groups can be defined in either of two equivalent ways.
[3] In this context, an inverse system consists of a directed set
an indexed family of finite groups
each having the discrete topology, and a family of homomorphisms
and the collection satisfies the composition property
equipped with the relative product topology.
One can also define the inverse limit in terms of a universal property.
In categorical terms, this is a special case of a cofiltered limit construction.
[4] It is defined as the inverse limit of the groups
of finite index (these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between the quotients).
, where the intersection runs through all normal subgroups
is characterized by the following universal property: given any profinite group
is given the smallest topology compatible with group operations in which its normal subgroups of finite index are open, there exists a unique continuous group homomorphism
satisfying the axioms in the second definition can be constructed as an inverse limit according to the first definition using the inverse limit
ranges through the open normal subgroups of
is topologically finitely generated then it is in addition equal to its own profinite completion.
Without loss of generality, it suffices to consider only surjective systems since given any inverse system, it is possible to first construct its profinite group
is ind-finite if it is the direct limit of an inductive system of finite groups.
By applying Pontryagin duality, one can see that abelian profinite groups are in duality with locally finite discrete abelian groups.
A profinite group is projective if it has the lifting property for every extension.
is projective if for every surjective morphism from a profinite
is equivalent to either of the two properties:[7] 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.
is procyclic if it is topologically generated by a single element
ranges over some set of prime numbers