ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-09-01

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The profinite completion of the integers is the inverse limit of the rings Z mod n

Example

The profinite completion of $Z$ is

\hat{Z} = \underset{n \geq 1}{\underset{\leftarrow}{l i m}} Z / n Z,

where the transition maps are reduction modulo divisibility.

Facts & Assumptions

Given: The additive group $Z$ .

[L1]

The profinite completion is the inverse limit over all finite-index normal subgroups (The profinite completion is the inverse limit of the finite quotients G over N).

Verification

technique · direct

1.1

Every subgroup of $Z$ has the form $n Z$ , and it has finite index exactly when $n \geq 1$ . Because $Z$ is abelian, these are all normal.

given

2.1

The quotient by $n Z$ is the cyclic group $Z / n Z$ , and if $m ∣ n$ then the map $Z / n Z \to Z / m Z$ is reduction modulo $m$ . So [L1] identifies the profinite completion with the displayed inverse limit.

L1step 1.1∎

Depends on

Used by

The integers sit densely but not closedly inside their profinite completion Example

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Brian Osserman, Math 6112 notes on inverse limits and profinite groups (standard reference, not scraped)
H. W. Lenstra, Profinite groups and Galois groups (standard reference, not scraped)