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  1. #1
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    order

    If G is an abelian group of order pq, where p and q are relatively prime integers. Suppose G has an element v of order p and an element w of order q.
    1)Prove that no powers of v can be equal to any power of w (except for e)
    2)Prove the element vw has order pq.
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  2. #2
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    Quote Originally Posted by apple2009 View Post
    If G is an abelian group of order pq, where p and q are relatively prime integers. Suppose G has an element v of order p and an element w of order q.
    1)Prove that no powers of v can be equal to any power of w (except for e)
    2)Prove the element vw has order pq.
    For 1
    Let v^x = w^y
    v^(xp) = e = w^(yp)
    => q|yp => q|y (as gcd(p,q)=1)

    Can you complete? 2) is a direct application of 1)
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by aman_cc View Post
    For 1
    Let v^x = w^y
    v^(xp) = e = w^(yp)
    => q|yp => q|y (as gcd(p,q)=1)

    Can you complete? 2) is a direct application of 1)
    Aren't you missing the conclusion here?
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  4. #4
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    Quote Originally Posted by Drexel28 View Post
    Aren't you missing the conclusion here?
    Hi Drexel28 - I haven't exactly followed where I went wrong please. Would you mind elaborating a bit please?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by aman_cc View Post
    Hi Drexel28 - I haven't exactly followed where I went wrong please. Would you mind elaborating a bit please?
    No. I don't think that you are wrong, it just lacks the conclusion.

    Since if v^x=w^y\implies q|y we may conclude that w^y=e. A similar argument shows that v^x=e so they can only be equal when they are trivial. I agree with your solution it just didnt have the conclusion.
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    Quote Originally Posted by Drexel28 View Post
    No. I don't think that you are wrong, it just lacks the conclusion.

    Since if v^x=w^y\implies q|y we may conclude that w^y=e. A similar argument shows that v^x=e so they can only be equal when they are trivial. I agree with your solution it just didnt have the conclusion.
    o ok. Thanks. I intentionally left that for the OP to take it fwd.
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  7. #7
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by aman_cc View Post
    o ok. Thanks. I intentionally left that for the OP to take it fwd.
    Then, my complete mistake. I apologize greatly for ruining your intended teaching method.
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