False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Every profinite group is pro-p for some prime

Statement

Every profinite group is pro- $p$ for some prime $p$ .

Facts & Assumptions

Given: The profinite completion $\hat{Z}$ .

[L1]

$\hat{Z} ≅ \prod_{q} Z_{q}$ over all primes $q$ (The profinite completion of the integers is the direct product of the p-adic integer groups over all primes).

[F1]

A pro- $p$ group is an inverse limit of finite $p$ -groups (A pro-p group is a profinite group that is an inverse limit of finite p-groups).

Refutation

technique · direct

1.1

The profinite group $\hat{Z}$ has nontrivial continuous quotients $Z_{q}$ for every prime $q$ by [L1].

L1given

2.1

If $\hat{Z}$ were pro- $p$ for some fixed prime $p$ , then every finite quotient of $\hat{Z}$ would be a $p$ -group by [F1]. For each prime $q \neq p$ , the projection to the $q$ -adic factor followed by reduction modulo $q$ gives a quotient $\hat{Z} ↠ Z_{q} ↠ Z_{q} / q Z_{q} ≅ Z / q Z,$ which is a nontrivial $q$ -group. This contradiction shows that not every profinite group is pro- $p$ .

F1step 1.1algebra

3.1

Therefore the statement is false.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Jordan Bell, The profinite completion of the integers, the p-adic integers, and Prufer p-groups (standard reference, not scraped)
Brian Osserman, Inverse limits and profinite groups (standard reference, not scraped)