Cayley's theorem

Statement

Let $G$ be a group. Then $G$ is isomorphic to a subgroup of the symmetric group $\operatorname{Sym}(G)$. Explicitly, the map $g \mapsto \ell_g$, where $\ell_g(x) = gx$, is an injective group homomorphism from $G$ to $\operatorname{Sym}(G)$.

Remarks

The map comes from the action by left multiplication, and it is injective because $\ell_g(e) = g$. If $|G| = n$ is finite, then $G$ is isomorphic to a subgroup of $S_n$. Often a smaller symmetric group is enough, for example $S_3$ is isomorphic to a subgroup of $S_3$ and not only of $S_6$.

History

Arthur Cayley introduced abstract groups in 1854. At that time, groups were usually groups of permutations, and the theorem shows that the abstract notion gives nothing more, up to isomorphism.

Explanations

L1 human

Each element of a group shuffles the elements of the group when you multiply on the left. Different elements give different shuffles, and the product of elements gives the composition of shuffles. So each group is a group of permutations, up to isomorphism.