PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

Finite groups are amenable

Statement

Every finite group is amenable.

Facts & Assumptions

Given: A finite group $G$ .

[L1]

A group is amenable exactly when it admits a left-invariant mean (Left-invariant means and amenable groups).

Proof

technique · direct

1.1givenconstruct

Define $m (f) = ∣ G ∣^{- 1} \sum_{x \in G} f (x)$ on $ℓ^{\infty} (G)$ . This is linear, positive, and satisfies $m (1_{G}) = 1$ , so it is a mean.

2.1L1step 1.1algebra∎

For every $g \in G$ , the map $x \mapsto g^{- 1} x$ is a permutation of $G$ , so $\sum_{x \in G} f (g^{- 1} x) = \sum_{x \in G} f (x)$ . Therefore $m (g \cdot f) = m (f)$ , and [L1] makes $G$ amenable.

Depends on

Left-invariant means and amenable groups

Used by

Dependency tree · two levels

3 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

C. Löh, Geometric Group Theory: An Introduction (2015 course version) (standard reference, not scraped)
Cornelia Drutu and Michael Kapovich, Lectures on Geometric Group Theory (standard reference, not scraped)