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:
- the kernel of
- is the identity map
- for all
- is the identity of
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 .
Comments
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?