Explanation

Of course we must justify that the notion `order' makes sense.

If g is a permutation in S_n, then the permutations g, g², g³, ... can not all be distinct, because there are only finitely many permutations in S_n (n! to be precise). So there must exist positive numbers r < s such that g^r = g^s. Since g is a bijection, we find g^{s - r} = e. So there exist positive numbers m with g^m = e, and in particular a smallest such number. Therefore each permutation g has a well-defined order.