DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

The quotient graph of an action without inversions

Definition

Let a group $G$ act without inversions on an oriented graph $X = (V, E)$ . The quotient graph $G \ X$ has

$V (G \ X) = G \ V, E (G \ X) = G \ E,$

with origin, terminus, and reversal defined on orbits by

$o (G \cdot e) = G \cdot o (e), t (G \cdot e) = G \cdot t (e), \overline{G \cdot e} = G \cdot \overset{ˉ}{e} .$

Because the action is without inversions, $G \cdot e \neq G \cdot \overset{ˉ}{e}$ , so the reversal map is still fixed-point-free and the incidence data is independent of the chosen representatives.

Depends on

Used by

A quotient of a tree can have cycles Counterexample
FALSE: a quotient of a tree by a group action is always a tree False statement
Vertex and edge stabilizers determine the quotient incidences 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

Yuriy Tumarkin, Groups Acting on Trees (standard reference, not scraped)