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Math Help - Show x^n is discrete in {x^k}

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    Forum Admin topsquark's Avatar
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    Show x^n is discrete in {x^k}

    Hmmm...No listing for Group Theory. I'll post it here for now.

    If x is an element of infinite order in a group G, prove that the elements x^n \text{, }n \in \mathbb{Z}are all distinct.
    I don't even have an idea where to start. Can anyone give me a push in the right direction?

    Thanks!
    -Dan
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    Re: Show x^n is discrete in {x^k}

    Quote Originally Posted by topsquark View Post
    Hmmm...No listing for Group Theory. I'll post it here for now.
    I don't even have an idea where to start. Can anyone give me a push in the right direction
    Suppose that j\ne k but x^j=x^k can x be of infinite order?
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    Forum Admin topsquark's Avatar
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    Re: Show x^n is discrete in {x^k}

    Quote Originally Posted by Plato View Post
    Suppose that j\ne k but x^j=x^k can x be of infinite order?
    I have a thought. Is this where you were going with it?
    Let k > j. Suppose that we have

    x^j = x^k

    Then
    x^{-j} x^j = x^{-j}x^k

    1 = x^{k - j}

    So |x| is of finite order k - j, a contradiction.

    -Dan
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