If we want to prove the following statement:
"Prove that if $\Omega=[1,2,3,...]$ then $S_{\Omega}$ is an infinite group."
Since the number of permutations of $S_{n}$ is $n!$, is it enough to say that since $n$ is infinite, so is $n!$.
I ask this because the book I am reading explicitly states that this is not an acceptable proof, and I do not see why.
Your "proof" is unacceptable since $n!$ is only defined for $n$ finite.
To prove that $S_\Omega$ is infinite, you need to exhibit infinitely many elements of $S_\Omega$ - say, one for each natural number $n$. Can you see how to do this? (HINT: don't make it more complicated than necessary . . .)