Def/Order (permutation)

 =Existence= The order of a permutation $\sigma$ exists, by the following argument: define a subset of $\NN$ by: $$S = \{ n \in \NN \mbox{ such that } n > 0 \mbox{ and } \sigma^n = Id_X \}.$$ Using a result of Lagrange, now known as Lagrange's Theorem (Lagrange's original work was closer to what we use here), it is known that $S$ is nonempty, and in particular, the factorial $\vert X \vert !$ is an element of $S$.

Since $S$ is a nonempty set of natural numbers, it has a least element. This least element is the order of $\sigma$.