Profinite group

In mathematics, a profinite or prof Destinite group is a topological group G, which is the inverse ( projective ) limit of a system of finite groups. This limit is formed in the category of topological groups; Here one considers every finite group as a topological group with the discrete topology. A topological group if and only proendlich if it is compact and totally disconnected.

Examples

• Every finite group is also evident proendlich ( selected as a system of finite groups, only the group itself).
• The p- adic numbers are a simple example of an infinite profinite group.
• Each Galois group of a Galois extension L | K ( provided with the Krull topology) is proendlich.
• If G is any group, then one obtains a profinite group G ^, the profinite completion or profinite completion of G by the inverse limit of G / H increases, where H runs through all normal subgroups of G of finite index.