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