Profinite completion of the integers

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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.

Arithmetic functions

Function Value Explanation
order of a profinite group \prod p^\infty for all primes p. This is a supernatural number as is the convention for orders of profinite groups.

Group properties

Abstract group properties

Property Satisfied? Explanation
abelian group
aperiodic group

Topological group properties

Here, the topology is from the profinite group structure.

Property Satisfied? Explanation
profinite group Yes By definition
compact group Yes profinite groups are compact
T0 topological group Yes profinite groups are Hausdorff, hence T_0
totally disconnected group Yes profinite groups are totally disconnected
topologically cyclic group Yes \mathbb{Z}, the subgroup generated by the element that projects to 1 mod p^n for all n, is a dense subgroup. In fact, this group can be thought of as a compactification of \mathbb{Z}.
topologically finitely generated group Yes Follows from being topologically cyclic.
connected topological group No It is nontrivial and totally disconnected.