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Math Help - Group Theory Proof

  1. #1
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    Group Theory Proof

    Can somebody help me with this proof please?

    Suppose that G is a group with elements a and b of orders p and q respectively, where p and q are distinct primes.

    Suppose also that for some integers i and j, that a^i = b^j. Show that p|i and q|j.

    Thanks in advance.
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi
    Quote Originally Posted by Cairo View Post
    Suppose that G is a group with elements a and b of orders p and q respectively...
    This means that a^p=b^q=e where e is the identity element of the group.
    ... where p and q are distinct primes.

    Suppose also that for some integers i and j, that a^i = b^j. Show that p|i and q|j.
    You may try using (a^i)^p=(b^j)^p and (a^i)^q=(b^i)^q.
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  3. #3
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    could be having a dumb moment....but I'm still not following.
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  4. #4
    Super Member flyingsquirrel's Avatar
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    As (a^i)^p=a^{ip}=(a^p)^i=e^i=e and  a^{ip}=b^{jp}, we get that  e=b^{jp} which requires that there exists n_1\in\mathbb{N} such that jp = n_1 q. What can be deduced from this ?
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  5. #5
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    that p|i ?
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  6. #6
    Super Member flyingsquirrel's Avatar
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    Bad luck, it's the other one... (it can be shown acknowledged that if u|vw then u|v or u|w)
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