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Math Help - Infinite group

  1. #1
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    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.
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  2. #2
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    Re: Infinite group

    Quote Originally Posted by dwsmith View Post
    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.
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  3. #3
    Senior Member Tinyboss's Avatar
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    Re: Infinite group

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