A profinite group is an inverse limit of finite discrete groups. Equivalently, it is a compact Hausdorff totally disconnected topological group.
The profinite topology on a group takes the cosets of finite-index normal subgroups as a neighborhood basis. A profinite group carries the inverse-limit topology from its finite quotients.
An inverse limit carries the coarsest topology making every coordinate projection map continuous. It is the subspace topology inherited from the product of the component spaces.
The profinite completion of a group is the inverse limit of its finite quotient groups:The canonical map has dense image and is injective exactly when is residually finite.
Every homomorphism from a group to a profinite group extends uniquely to a continuous homomorphism . Construct the extension on every finite quotient of and invoke the universal property of an inverse limit; uniqueness follows from the density of in .
A topological group is topologically finitely generated when it has a finite topological generating set.
For a profinite group , a subset is a topological generating set exactly when in every finite quotient. Indeed, a subgroup of a profinite group is dense exactly when its image in every finite continuous quotient is surjective.
Elements of a profinite group are conjugate exactly when their images are conjugate in every . For the nontrivial direction, the finite nonempty sets of conjugators in the form an inverse system, whose inverse limit supplies a conjugator in .
A topological group has the topological Hopf property when every continuous surjective endomorphism is a topological automorphism.
Every topologically finitely generated group that is profinite has the topological Hopf property. There are only finitely many open subgroups of each given index, and inverse image under a surjective endomorphism permutes them. The endomorphism kernel consequently lies in every open normal subgroup and is trivial.
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A **profinite group** is a type of topological group that has a very specific structure. These groups are characterized by several key features: 1. **Definition**: A profinite group is a compact, totally disconnected, Hausdorff topological group that is isomorphic to the inverse limit of a system of finite groups. In more intuitive terms, you can think of profinite groups as "limits" of finite groups that retain a group structure.