ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-01

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A finite group is canonically isomorphic to its profinite completion

Example

A finite group is canonically isomorphic to its profinite completion.

Facts & Assumptions

Given: A finite group $G$ .

[L1]

The profinite completion is initial among continuous homomorphisms from $G$ into profinite groups (The profinite completion is initial among continuous homomorphisms from G to profinite groups).

[L2]

The canonical map is injective exactly when the group is residually finite, and its image is dense (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).

Verification

technique · direct

1.1

The finite group $G$ is residually finite because the identity subgroup has finite index. Hence [L2] makes the canonical map $ι_{G} : G \to \hat{G}$ injective.

L2given

2.1

Endow $G$ with the discrete topology, so $G$ itself is profinite. The identity map $G \to G$ is continuous, and [L1] gives a unique continuous homomorphism $\hat{G} \to G$ left-inverse to $ι_{G}$ . Since $ι_{G} [G]$ is dense by [L2] and finite subsets of a Hausdorff space are closed, the image of $ι_{G}$ must be all of $\hat{G}$ . Therefore $ι_{G}$ is an isomorphism.

L1L2step 1.1construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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)