Group theory MOC

Cayley’s theorem

Cayley’s theorem states that any group of order is isomorphic to a subset of the (its) symmetric group . 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.

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