Cayley's theorem

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

Given an arbitrary group we can define an injective monomorphism into the symmetric group of its underlying set . For any , we define the bijection

Then is an monomorphism.

Proof

Let . Clearly

hence is a Group homomorphism. is also injective: iff iff . Hence is a monomorphism.

A generalization is the Yoneda lemma.

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