profinite

Profinite refers to a class of topological groups that can be described as inverse limits of finite groups. A profinite group is isomorphic to the inverse limit of an inverse system of finite groups with surjective bonding maps. The associated profinite topology is the coarsest topology making all projection maps continuous. Concretely, a base of open neighborhoods of the identity is given by the open normal subgroups of finite index. Thus profinite groups are compact, Hausdorff, and totally disconnected. Every open subgroup has finite index, and every finite index subgroup is open.

For any group G, its profinite completion Ĝ is the inverse limit of all finite quotients G/N, with the canonical projections. There is a natural homomorphism G → Ĝ; this map is injective exactly when G is residually finite, and its image is dense in Ĝ. The completion is universal for maps from G to finite groups: any homomorphism from G to a finite group factors uniquely through Ĝ. In particular, Ĝ captures all finite quotient information of G.

Examples and applications include the profinite completion of the integers, which is ĤZ ≅ ∏_p Z_p, the product of the p-adic integers. The absolute Galois group of a field is a profinite group, being the inverse limit of the finite Galois groups of finite Galois extensions. Pro-p groups are a common specialization, consisting of profinite groups that arise as inverse limits of finite p-groups. Profinite groups play a central role in number theory and algebraic geometry, where they encode symmetry and Galois-theoretic information in a topologically compact setting.