Cayley’s The­o­rem on sym­met­ric groups

WikiLast edit: 15 Jun 2016 9:29 UTC by Patrick Stevens

Cayley’s Theorem states that every group appears as a certain subgroup of the symmetric group on the underlying set of .

Formal statement

Let be a group. Then is isomorphic to a subgroup of .

Proof

Consider the left regular action of on : that is, the function given by . This induces a homomorphism given by currying: .

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 is isomorphic to a subgroup of .