TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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

Statement

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

Facts & Assumptions

Given: The hypotheses of the Statement.

[L1]

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

[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).

[L2]

Every group $G$ is isomorphic to the subgroup of $Sym (G)$ formed by its left translations $λ_{g} : x \mapsto g x$ . (Cayley's theorem: every group $G$ is isomorphic to a subgroup of $Sym (G)$ ).

[L3]

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).

[L4]

Group isomorphisms, automorphisms and the set $Aut (G)$ . (Group isomorphisms, automorphisms and the set $Aut (G)$ ).

Proof

technique · direct

1.1

Cayley’s theorem embeds a group in the symmetric group on its underlying set by left multiplication, and the action just constructed has exactly those permutations.

F1L1L2L3

2.1

So the label-preserving automorphism action is that embedding followed by the inclusion of the automorphism group in the symmetric group; the Cayley graph refines the regular action rather than supplying a second embedding theorem.

L1L2L4step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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)