natural homomorphisms between the quotients). There is a natural homomorphism and the image of under this homomorphism is dense in The homomorphism is injective if and only if the group is residually finite (i.e., where the intersection runs through all normal subgroups of finite index). The homomorphism is characterized by the following universal property: given any profinite group and any continuous group homomorphism where 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 with Equivalence Any group constructed by the first definition satisfies the axioms in the second definition. Conversely, any group satisfying the axioms in the second definition can be constructed as an inverse limit according to the first definition using the inverse limit where ranges through the open normal subgroups of ordered by (reverse) inclusion. If is topologically finitely generated then it is in addition equal to its own profinite completion Surjective systems In practice, the inverse system of finite groups is almost always , meaning that all its maps are surjective. 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 and then it as its own profinite completion. Examples Finite groups are profinite, if given the discrete topology. The group of -adic integers under addition is profinite (in fact procyclic). It is the inverse limit of the finite groups where ranges over all natural numbers and the natural maps for The topology on this profinite group is the same as the topology arising from the -adic valuation on The group of profinite integers is the profinite completion of In detail, it is the inverse limit of the finite groups where with the modulo maps for This group is the product of all the groups and it is the absolute Galois group of any finite field. The Galois theory of field extensions of infinite degree gives rise naturally to Galois groups that are profinite. Specifically, if is a Galois extension, consider the group consisting of all field automorphisms of that keep all elements of fixed. This group is the inverse limit of the finite groups where ranges over all intermediate fields such that is a Galois extension. For the limit process, the restriction homomorphisms are used, where The topology obtained on is known as the