Profinite completion of the integers

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

The profinite completion of the integers is defined in the following equivalent ways. Note that these definitions can be interpreted both as defining the group abstractly and as defining the group as a topological group:

1. The group obtained by taking the profinite completion of the group of integers (viewed as a discrete group).
2. The external direct product, over all prime numbers $p$, of the additive group of p-adic integers for the prime $p$. Note that we take the product topology from the topologies on these groups.