LemmaStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck 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.

Left and right amenability agree by inversion

Statement

Let $G$ be a group. Then $G$ admits a left-invariant mean if and only if it admits a right-invariant mean.

Here a mean $m$ is right invariant when $m (f \cdot g) = m (f)$ for every $g \in G$ , where $(f \cdot g) (x) = f (x g^{- 1})$ .

Facts & Assumptions

Given: A group $G$ .

[L1]

Amenability is defined by existence of a left-invariant mean (Left-invariant means and amenable groups).

Proof

technique · direct

1.1

Suppose $m$ is left invariant. For bounded $f$ , define $I f (x) = f (x^{- 1})$ and $m_{r} (f) = m (I f)$ . Then $m_{r}$ is a mean, and for every $g \in G$ one has $I (f \cdot g) = g^{- 1} \cdot I f$ , so left invariance of $m$ gives $m_{r} (f \cdot g) = m_{r} (f)$ . Thus a left-invariant mean produces a right-invariant mean.

L1givenconstruct

2.1

Replacing $g$ by $g^{- 1}$ in the same computation shows that inversion also carries right-invariant means back to left-invariant means. Hence the two notions are equivalent.

step 1.1∎

Depends on

Left-invariant means and amenable groups

Used by

Equivalent boundary formulations of the Folner condition Lemma

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

Cornelia Drutu and Michael Kapovich, Lectures on Geometric Group Theory (standard reference, not scraped)