Cayley's theorem

Cayley's theorem states that any group $𝐺$ of order $𝑛$ is isomorphic to a subset of the (its) symmetric group $𝐺! ≅ 𝑆 𝑛$ . #m/thm/group

Given an arbitrary group $(𝐺, \cdot)$ we can define an injective monomorphism into the symmetric group of its underlying set $𝐺$ . For any $ℎ \in 𝐺$ , we define the bijection

𝜑 ℎ : 𝐺 \to 𝐺 : 𝑔 \mapsto ℎ \cdot 𝑔

Then $𝑓 : 𝐺 \to 𝐺! : ℎ \mapsto 𝜑 ℎ$ is an monomorphism.

Proof

Let $𝑎, 𝑏, 𝑥 \in 𝐺$ . Clearly

(𝜑 𝑎 \circ 𝜑 𝑏) (𝑥) = 𝜑 𝑎 (𝜑 𝑏 (𝑥)) = 𝜑 𝑎 (𝑏 \cdot 𝑥) = 𝑎 \cdot 𝑏 \cdot 𝑥 = 𝜑 𝑎 \cdot 𝑏 (𝑥)

hence $𝑓$ is a Group homomorphism. $𝑓$ is also injective: $𝜑 𝑎 (𝑥) = 𝜑 𝑏 (𝑥)$ iff $𝑎 𝑥 = 𝑏 𝑥$ iff $𝑎 = 𝑏$ . Hence $𝑓$ is a monomorphism.

A generalization is the Yoneda lemma.

#state/tidy | #lang/en | #SemBr