The symmetric group on an infinite set is infinite

Prove that if $\Omega = \{ 1, 2, \ldots \}$ then $S_\Omega$ is an infinite group. (Do not say $\infty! = \infty$.)

Consider the group of permutations of $\Omega = \{ 1, 2, \ldots \}$; for every $k \in \Omega$, there is a permutation $\sigma_k$ such that $\sigma_k(1) = k$. There are infinitely many of these, all distinct. Thus $S_\Omega$ is an infinite group.