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.

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