Note. Cayley's Theorem (next) is an example of a representation theorem. It tells us that any group can be represented as (is isomorphic to) something reasonably concrete. In proving Cayley's Theorem, we associate with each element of a group $G$ a permutation of the set $G.$ What we do is associate with each element $a$ in $G$ the permutation whose first row (in two-row form) is the first row of the Cayley table and whose second row is labeled by $a.$ If the elements in the first row are $a_1,\ldots,a_n$ (in that order), then the elements in the row labeled by $a$ will be $a a_1,\ldots,a a_n$ (in that order).

Example 20.1. The permutation associated with $[3]$ by the idea just described is \[ \left( \begin{matrix} [0]&[1]&[2]&[3]&[4]&[5]\\ [3]&[4]&[5]&[0]&[1]&[2] \end{matrix} \right). \]

Cayley's Theorem. Every group is isomorphic to a permutation group on its set of elements.

Corollary. Every group of finite order $n$ is isomorphic to a subgroup of $S_n.$

Example 20.2. Let \begin{align*} a_1&=[0]\\ a_2&=[1]\\ a_3&=[2]. \end{align*} Then, the construction in the proof of Cayley's theorem yields \[ \theta(a_3)=\theta([2])\\ = \left( \begin{matrix} [0]&[1]&[2]\\ [2]&[0]&[1] \end{matrix} \right) = \left( \begin{matrix} a_1&a_2&a_3\\ a_3&a_1&a_2\\ \end{matrix} \right). \] The idea in the proof of the corollary is simply to delete the $a$'s and keep the subscripts, so that \[ \left( \begin{matrix} 1&2&3\\ 3&1&2\\ \end{matrix} \right) \] is assigned to $a_3.$ Notice that this is an element of $S_3$ because $\abs{\Z_3}=3.$

Corollary. For each positive integer $n,$ there are only finitely many isomorphism classes of groups of order $n.$