Profinite group

A profinite group is an inverse limit of finite discrete groups. For a directed compatible system, its elements are tuples $(g_i)$ whose coordinates agree under all transition homomorphisms. Multiplication and inversion operate coordinatewise. The inverse-limit topology is inherited from the product of the finite discrete spaces.

The product is compact and Hausdorff, and the compatibility equations define a closed subset. Conditions on finitely many coordinates give clopen neighbourhoods; in particular kernels of finite quotient projections form an identity-neighbourhood basis. Conversely, every compact Hausdorff totally disconnected topological group is recovered from its finite quotients by open normal subgroups.

For example, the additive group of p-adic integers is ${\mathbb{Z}}_p={\underleftarrow{\lim }}_n\mathbb{Z}/p^n\mathbb{Z}$. Compatibility says that a residue modulo $p^{n+1}$ reduces to the preceding residue modulo $p^n$. The ordinary integers map densely into this group. Compactness belongs to the completed inverse limit; the profinite topology on the integers is not compact. ^[ref]