TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-04

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Zp is the pro-p completion of the integers

Statement

The pro- $p$ completion of the additive group $Z$ is $Z_{p}$ , and the defining completion map is the canonical residue-class map $ι : Z \to Z_{p}$ .

Facts & Assumptions

Given: The additive group $Z$ .

[F1]

The pro- $p$ completion is the inverse limit over normal subgroups with finite $p$ -group quotients (The pro-p completion of an abstract group is the inverse limit over its finite p-group quotients).

[L1]

The canonical map into $Z_{p}$ is $m \mapsto (m m o d p^{n})_{n}$ (The canonical map from Z to Zp sends an integer to its coherent residue classes modulo p^n).

[L2]

Compatible tuples satisfy the inverse-limit universal property (The compatible-tuple construction satisfies the inverse-limit universal property in groups).

Proof

technique · direct

1.1

Every subgroup of the additive group $Z$ has the form $d Z$ . The quotient $Z / d Z$ is a finite $p$ -group exactly when $d = p^{n}$ for some $n \geq 0$ , because every finite quotient of a cyclic group is cyclic. Thus the inverse system in [F1] is exactly the system of quotients $Z / p^{n} Z$ with the usual reduction maps.

F1givenalgebra

2.1

The compatible-tuple inverse limit of the system from step 1.1 is precisely $Z_{p}$ , and [L1] is the resulting canonical cone map from $Z$ to that inverse limit. By [L2], this is the universal pro- $p$ completion map. Therefore the pro- $p$ completion of $Z$ is $Z_{p}$ .

L1L2step 1.1∎

Depends on

Used by

Zp is the full profinite completion of the integers False statement
The profinite completion of the integers is the direct product of the p-adic integer groups over all primes Proposition

Dependency tree · two levels

9 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

Gareth Wilkes, Profinite Groups and Group Cohomology lecture notes (standard reference, not scraped)
Jordan Bell, The profinite completion of the integers, the p-adic integers, and Prufer p-groups (standard reference, not scraped)