False statementConstruction: AI-adaptedVerification: 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.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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FALSE: the canonical map to the profinite completion is always injective

Statement

The canonical map to the profinite completion is always injective.

Facts & Assumptions

Given: A group $G$ that is not residually finite.

[L1]

The canonical map is injective exactly when the group is residually finite (The canonical map to the profinite completion has kernel equal to the finite residual and has dense image, The finite residual is the intersection of the finite-index normal subgroups, and a group is residually finite when that intersection is trivial).

Refutation

technique · direct

1.1

Because $G$ is not residually finite, the finite residual $R_{f} (G)$ contains some nonidentity element.

given

2.1

By [L1], the finite residual is exactly the kernel of the canonical map. Hence that nonidentity element lies in the kernel, so the canonical map is not injective.

L1step 1.1

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

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)