# Cayley's Theorem on symmetric groups

Cayley’s Theorem states that every group $$G$$ appears as a certain subgroup of the symmetric group $$\mathrm{Sym}(G)$$ on the underlying set of $$G$$.

# Formal statement

Let $$G$$ be a group. Then $$G$$ is isomorphic to a subgroup of $$\mathrm{Sym}(G)$$.

# Proof

Consider the left regular action of $$G$$ on $$G$$: that is, the function $$G \times G \to G$$ given by $$(g, h) \mapsto gh$$. This induces a homomorphism $$\Phi: G \to \mathrm{Sym}(G)$$ given by currying: $$g \mapsto (h \mapsto gh)$$.

Now the following are equivalent:

• $$g \in \mathrm{ker}(\Phi)$$ the kernel of $$\Phi$$

• $$(h \mapsto gh)$$ is the identity map

• $$gh = h$$ for all $$h$$

• $$g$$ is the identity of $$G$$

Therefore the kernel of the homomorphism is trivial, so it is injective. It is therefore bijective onto its image, and hence an isomorphism onto its image.

Since the image of a group under a homomorphism is a subgroup of the codomain of the homomorphism, we have shown that $$G$$ is isomorphic to a subgroup of $$\mathrm{Sym}(G)$$.

Parents:

• Symmetric group

The symmetric groups form the fundamental link between group theory and the notion of symmetry.

• I feel like symmetricgroup should be a requisite for this page. However, this page is linked in the body of symmetricgroup, so it seems a bit circular to link it as a requisite. I think this situation probably comes up for most child pages; what’s good practice in such cases?

• I think having it as a requisite is best? I see the issue, but some people may arrive from other pages or search.