Cayley's Theorem on symmetric groups

by Patrick Stevens Jun 14 2016 updated Jun 15 2016

The "fundamental theorem of symmetry", forging the connection between symmetry and group theory.

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)$~$.


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:

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)$~$.


Patrick Stevens

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?