Pro-end group

In mathematics , a pro- finite or profinite group is a topological group G, which is the inverse (projective) limit of a system of finite groups. This Limes is formed in the category of topological groups; here we consider every finite group as a topological group with the discrete topology . A topological group is pro finite if and only if it is Hausdorffian , compact and totally disjointed .

Examples

• Every finite group is obviously also pro-finite (choose only the group itself as the system of finite groups).
• The p-adic numbers and the per- finite numbers are examples of infinite per-finite groups. ${\displaystyle \mathbb {Z} _{p}}$ ${\displaystyle {\widehat {\mathbb {Z} }}}$
• Every Galois group of a Galois extension L | K (provided with the Krull topology ) is pro finite.
• If G is an arbitrary group, then one obtains a per-finite group Ĝ, the per-finite completion or per-finite completion of G, by taking the inverse limit of G / H, where H runs through all normal divisors of G with finite index .

literature

• John Cassels , Albrecht Froehlich: Algebraic Number Theory. Proceedings of an instructional conference . Academic Press, London 1993, ISBN 0-12-163251-2 (reprint of the London 1965 edition).
• Jürgen Neukirch : Algebraic number theory . Springer, Berlin 2007, ISBN 978-3-540-37547-0 (reprint of the Berlin 1992 edition).
• Laurent Bartholdi: Profinite Groups , Mathematical Snapshots, Oberwolfach 2016