# Math Help - Infinite group

1. ## Infinite group

Prove that if $\Omega=\{1,2,....\}$, then $S_{\Omega}$ is an infinite group.

Let $\sigma:\Omega\to\Omega$. I am guessing I need to define a map but I don't know as what.

2. ## Re: Infinite group

Originally Posted by dwsmith
Prove that if $\Omega=\{1,2,....\}$, then $S_{\Omega}$ is an infinite group.

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

3. ## Re: Infinite group

Or you could note that it contains the transposition (n, n+1) for every natural number n.