Prove that if $\displaystyle \Omega=\{1,2,....\}$, then $\displaystyle S_{\Omega}$ is an infinite group.
Let $\displaystyle \sigma:\Omega\to\Omega$. I am guessing I need to define a map but I don't know as what.
for every integer $\displaystyle n \geq 1$, there is an obvious injection $\displaystyle S_n \longrightarrow S_{\Omega}$. so $\displaystyle |S_{\Omega}| > |S_n| = n!$ and thus $\displaystyle |S_{\Omega}|= \infty$.