CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-14

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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.

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The fundamental group of a connected Lie group is abelian

Statement

For every connected Lie group $G$ with identity $e$ , the fundamental group $π_{1} (G, e)$ is abelian.

Facts & Assumptions

Given: A connected Lie group $G$ with identity $e$ .

[F1]

The fundamental group of any topological group is abelian. The fundamental group of a topological group is abelian.

Proof

technique · direct

1.1

Smooth multiplication and inversion are continuous, so the underlying space of $G$ is a topological group.

givenalgebra

2.1

Apply [F1] to this topological group to conclude that $π_{1} (G, e)$ is abelian. Connectedness is retained because it is the convention needed by the covering-Lie-group applications, although [F1] shows that this conclusion itself holds for the identity component without using global connectedness. No choice axiom or boundary case beyond the trivial group is involved.

F1step 1.1∎

Depends on

The fundamental group of a topological group is abelian

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Pavel Etingof, MIT 18.745 Lie Groups and Lie Algebras I (standard reference, not scraped)