Profinite group

In mathematics, profinite groups are groups that are in a certain sense assembled from finite groups; they share many properties with the finite groups.
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Definition
Formally, a profinite group is a group that is isomorphic to the inverse limit of an inverse system of finite groups. Profinite groups are naturally regarded as topological groups: each of the finite groups carries the discrete topology, and since the inverse limit is a subset of the product of these discrete spaces, it inherits a topology which turns it into a topological group.
Examples
 Any product of finite groups is profinite.
 The padic integers Z_{p</sup> are profinite (with respect to addition): they are the inverse limit of the finite groups Z/pnZ where n ranges over all natural numbers and the natural maps Z/pnZ → Z/pmZ (n≥m) are used for the limit process. }
 The Galois theory of field extensions of infinite degree gives rise naturally to Galois groups that are profinite. Specifically, if L/K is a Galois extension, we consider the group G = Gal(L/K) consisting of all field automorphisms of L which keep all elements of K fixed. This group is the inverse limit of the finite groups Gal(F/K), where F ranges over all intermediate fields such that F/K is a finite Galois extension. For the limit process, we use the restriction homomorphisms Gal(F_{1}/K) → Gal(F_{2}/K), where F_{2} ⊆ F_{1}. The topology we obtain on Gal(L/K) is known as the Krull topology after Wolfgang Krull (1899  1971). Interestingly, every profinite group is isomorphic to one arising from Galois theory.
 The fundamental groups considered in algebraic geometry are also profinite groups, roughly speaking because the algebra can only 'see' finite coverings of an algebraic variety. (The fundamental groups of algebraic topology are in general not profinite.)
Properties and facts
Every profinite group is a compact Hausdorff space: since all finite discrete spaces are compact Hausdorff spaces, their product will be a compact Hausdorff space by Tychonoff's theorem; the direct limit is a closed subset of this product and is therefore also compact Hausdorff.
Every profinite group is totally disconnected and even more: a topological group is profinite if and only if it is Hausdorff, compact and totally disconnected.
Every product of (arbitrarily many) profinite groups is profinite; the topology arising from the profiniteness agrees with the product topology. Every closed subgroup of a profinite group is itself profinite; the topology arising from the profiniteness agrees with the subspace topology. If N is a closed normal subgroup of a profinite group G, then the factor group G/N is profinite; the topology arising from the profiniteness agrees with the quotient topology.
Given an arbitrary group G, there is a related profinite group G^{^}, the profinite completion of G. It is defined as the inverse limit of the groups G/N, where N runs through the normal subgroups in G of finite index (these normal subgroups are partially ordered by inclusion, which translates into an inverse system of natural homomorphisms between them). There is a natural homomorphism η : G → G^{^}, and the image of G under this homomorphism is dense in G^{^}. The homomorphism η is injective if and only if the group G is residually finite (i.e. iff for every nonidentity element g in G there exists a normal subgroup N in G of finite index that doesn't contain g). The homomorphism η is characterized by the following universal property: given any profinite group H and any group homomorphism f : G → H, there exists a unique continuous group homomorphism g : G^{^} → H with f = gη.
Since every profinite group G is compact Hausdorff, we have a Haar measure on G, which allows us to measure the "size" of subsets of G, compute certain probabilities, and integrate functions on G.
Indfinite groups
There is a notion of indfinite group, which is the concept dual to profinite groups; i.e. a group G is indfinite if it is the direct limit of finite groups. The usual terminology is different: a group G is called locally finite if every finitelygenerated subgroup is finite. This is equivalent, in fact, to being 'indfinite'.
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.
See also: locally cyclic group.
Further reading
 Hendrik Lenstra: Profinite Groups, talk given in Oberwolfach, November 2003. online version (http://math.berkeley.edu/~jvoight/notes/oberwolfach/LenstraProfinite.pdf).
 Alexander Lubotzky: review of several books about profinite groups. Bulletin of the American Mathematical Society, 38 (2001), pages 475479. online version (http://www.ams.org/bull/20013804/S0273097901009144/home.html).es:Grupo profinito