Cayley's Theorem

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Cayley's Theorem states that every group is isomorphic to a permutation group, i.e., a subgroup of a symmetric group; in other words, every group acts on some set. Although the result is simple, it is deep, as it characterizes group structure as the structure of a family of bijections.

Proof

We prove that each group $G$ is isomorphic to a group of bijections on itself. Indeed, for all $g\in G$, let $f_g$ be the mapping $f_g : x \mapsto gx$ from $G$ into itself. Then $f_g$ is a bijection, for all $g$; and for all $g,h \in G$, $f_g \circ f_h = f_{gh}$. Thus $G$ is isomorphic to the set of permutations $\{ f_g | g \in G\}$ on $G$. $\blacksquare$

The action of $G$ on itself as described in the proof is called the left action of $G$ on itself. Right action is defined similarly.

See also

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