Proverbial completion

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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.

definition

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

in the category of topological groups .

Universal property

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 .

Other properties

  • If is finitely generated , then every subgroup of finite index is open and .
  • If is finitely generated, then holds for every finite group
.
  • For a group denote the set of all finite factor groups of . Then for finitely generated groups and :
.

Examples

The perfinite completion of the group of integers is
.
It is isomorphic to the product of the p-adic numbers over all prime numbers :
.
.
  • The natural homomorphism
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

Web links

Individual evidence

  1. Ribes-Zalesskii, op.cit., Proposition 3.2.2
  2. 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.
  3. Ribes Zalesskii, op.cit., Corollary 3.2.8