TheoremStatement: 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.

Left translation acts on a Cayley graph by label-preserving automorphisms, freely on vertices

Statement

Left translation acts on a Cayley graph by label-preserving automorphisms, freely on vertices.

Facts & Assumptions

Given: The hypotheses of the Statement.

[F1]

The directed labelled Cayley graph of a group $G$ and a subset $S$ has vertex set $G$ and an arc from $g$ to $g s$ labelled $s$ for each $g \in G$ and $s \in S$ (The directed labelled Cayley graph of a group with respect to a subset).

[F2]

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]

A labelled directed graph is a vertex set with a set of arcs each carrying a label, and a labelled isomorphism is a vertex bijection preserving arcs and labels (Labelled directed graphs, their underlying simple graphs, and label-preserving isomorphisms).

[L2]

A left action of $G$ on $X$ is a function $G \times X \to X$ , written $(g, x) \mapsto g \cdot x$ , such that (Left group actions, transitive actions, and faithful actions).

[L3]

A left action of a group $G$ on a set $X$ is free when (A free group action has no nonidentity element fixing a point).

Proof

technique · direct

1.1

Left multiplication by $h$ sends the arc from $g$ to $g s$ to the arc from $h g$ to $h g s$ , which carries the same label.

F1F2L1

2.1

It is bijective on vertices with inverse left multiplication by $h^{- 1}$ , and the assignment is a homomorphism into the label-preserving automorphism group.

F1L1L2step 1.1

3.1

It fixes a vertex only when $h g = g$ , that is only when $h$ is the identity, so the action is free.

L3step 2.1∎

Depends on

Used by

The label-preserving automorphism action on a Cayley graph is the left regular representation of Cayley's theorem Theorem

Dependency tree · two levels

11 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)