# Infinite group

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• October 4th 2011, 05:38 AM
dwsmith
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.
• October 4th 2011, 03:44 PM
NonCommAlg
Re: Infinite group
Quote:

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 $|S_{\Omega}|= \infty$.
• October 4th 2011, 05:38 PM
Tinyboss
Re: Infinite group
Or you could note that it contains the transposition (n, n+1) for every natural number n.
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