TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-03

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Cayley's theorem: every group $G$ is isomorphic to a subgroup of $Sym (G)$

Statement

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

Facts & Assumptions

Given: A group $G$ with identity $e$ .

[L1]

A left action of $G$ on a set gives a homomorphism into its symmetric group (Actions of $G$ on $X$ correspond exactly to homomorphisms $G \to Sym (X)$ , Left group actions, transitive actions, and faithful actions).

[L2]

The image of a group homomorphism is a subgroup, and a homomorphism is injective exactly when its kernel is trivial (The image of a group homomorphism is a subgroup and its kernel is a normal subgroup, A group homomorphism is injective if and only if its kernel is trivial).

[L3]

A bijective group homomorphism is an isomorphism (Group isomorphisms, automorphisms and the set $Aut (G)$ ).

Proof

technique · direct

1.1

Define $g \cdot x = g x$ on the set underlying $G$ . Then $e \cdot x = x$ and $(g h) \cdot x = g \cdot (h \cdot x)$ , so this is a left action.

L1givenalgebra

2.1

By [L1], the action yields a homomorphism $λ : G \to Sym (G)$ with $λ (g) (x) = g x$ .

step 1.1L1

3.1

If $λ (g)$ is the identity permutation, then evaluating it at $e$ gives $g = λ (g) (e) = e$ . Hence $\ker λ = {e}$ and $λ$ is injective.

step 2.1L2given

4.1

The image $λ [G]$ is a subgroup of $Sym (G)$ , and the injective homomorphism $λ : G \to λ [G]$ is bijective.

step 2.1step 3.1L2

5.1

Thus $λ$ is an isomorphism from $G$ to a subgroup of $Sym (G)$ .

step 4.1L3∎

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

17 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

Brosnan, Cayley's theorem (standard reference, not scraped)

Cayley's theorem: every group GG is isomorphic to a subgroup of Sym⁡(G)Sym(G)