profinite group
1 Definition
A topological group is profinite if it is isomorphic to the inverse limit of some projective system of finite groups. In other words, is profinite if there exists a directed set , a collection of finite groups , and homomorphisms for each pair with , satisfying
-
1.
for all ,
-
2.
for all with ,
with the property that:
- •
-
•
The isomorphism from to (considered as a subspace of ) is a homeomorphism of topological spaces, where each is given the discrete topology and is given the product topology.
The topology on a profinite group is called the profinite topology.
2 Properties
One can show that a topological group is profinite if and only if it is compact and totally disconnected. Moreover, every profinite group is residually finite.
Title | profinite group |
---|---|
Canonical name | ProfiniteGroup |
Date of creation | 2013-03-22 12:48:50 |
Last modified on | 2013-03-22 12:48:50 |
Owner | djao (24) |
Last modified by | djao (24) |
Numerical id | 9 |
Author | djao (24) |
Entry type | Definition |
Classification | msc 20E18 |
Classification | msc 22C05 |
Synonym | profinite |
Related topic | InverseLimit |
Defines | profinite topology |