In the mathematical branch of group theory , the pro- finite completion is a construction with which the information about all finite factor groups of a group can be summarized.
For a (discrete) group one considers the inverse system , where runs over all normal divisors of finite index and then defines the pro- finite completion of as the inverse limit of this system
Pro-finite completion is a per-finite group . Natural homomorphism has the following universal property : for every homomorphism in a per- finite group there is a continuous homomorphism with .
is injective if and only if residual is finite . Residual finite groups are important in many parts of mathematics.
literature
Ribes, Luis; Zalesskii, Pavel: Profinite groups. Second edition. Results of mathematics and its border areas. 3rd episode. A Series of Modern Surveys in Mathematics, 40. Springer-Verlag, Berlin, 2010. ISBN 978-3-642-01641-7
↑ Nikolov, Nikolay; Segal, Dan: On finitely generated profinite groups. I. Strong completeness and uniform bounds. Ann. of Math. (2) 165 (2007), no. 1, 171-238.