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

For a finite group the Cayley graph is a finite simple graph in the published sense and the two distances agree

Statement

Let $G$ be a finite group and let $S$ be a finite generating set. Then $Cay (G, S)$ is a connected finite simple graph in the published sense, and its path metric agrees with the published graph distance.

Facts & Assumptions

Given: A finite group $G$ and a finite generating set $S$ .

[F1]

The Cayley graph of a group $G$ with respect to a subset $S$ has vertex set $G$ and edge set ${{g, g s} : g \in G, s \in (S \cup S^{- 1}) ∖ {e}}$ (The Cayley graph of a group with respect to a subset).

[L1]

On a finite vertex set these graph notions are the published ones and the path metric is the published graph distance (On a finite vertex set the graph notions agree, and on connected graphs the two path distances agree).

[L2]

The path metric of a connected simple graph assigns to two vertices the least length of a path joining them (The path metric of a connected simple graph).

[L3]

$E \subseteq [V]^{2} : = {{u, v} \subseteq V : u \neq v} .$ (A finite simple graph is a finite vertex set together with a set of two-element vertex subsets).

[L4]

Let $u$ and $v$ lie in the same connected component of a graph $G$ . Their distance is (Graph distance within a component, eccentricity, diameter and girth, including the acyclic convention).

[L5]

A set $A$ is finite when $A \approx n$ for some $n \in N$ . (The cardinality $∣ A ∣$ of a finite set).

Proof

technique · direct

1.1

The vertex set is the group, which is finite, and the edges are two-element subsets, so the published definition applies verbatim; because $S$ generates $G$ , the Cayley graph is connected.

F1L1L3L5given

2.1

Both distances are the least length of a path in the same sense, so they agree.

F1L1L2L4step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

27 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. Loh, Geometric Group Theory: An Introduction (2015 course version), 264 pp. (standard reference, not scraped)
C. Drutu and M. Kapovich, Geometric Group Theory (with an appendix by B. Nica), 837 pp. (standard reference, not scraped)