DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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 directed labelled Cayley graph of a group with respect to a subset

Definition

Let $G$ be a group and let $S \subseteq G$ . The directed labelled Cayley graph of $G$ with respect to $S$ is the labelled directed graph

${Cay}_{d i r} (G, S)$

with vertex set $G$ and labelled arc set

$A : = {(g, s, g s) : g \in G, s \in S} \subseteq G \times S \times G .$

Thus each generator $s \in S$ supplies from every vertex $g$ a directed edge labelled $s$ to the vertex $g s$ . If $e \in S$ , the graph has a labelled loop at every vertex.

Depends on

Used by

Dependency tree · two levels

13 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. Drutu and M. Kapovich, Geometric Group Theory, Section 7.9 (standard reference, not scraped)