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Math Help - usng a conjugate

  1. #1
    ux0
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    usng a conjugate

    If G is a group and a,b \in G, prove that ab and ba have the same order. I think i need to use the conjugate to prove this, because conjugate elements have the same order.
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    Quote Originally Posted by ux0 View Post
    If G is a group and a,b \in G, prove that ab and ba have the same order. I think i need to use the conjugate to prove this, because conjugate elements have the same order.

    ba=a^{-1}(ab)a .

    Tonio
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  3. #3
    ux0
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    its just that simple?

    ba=ba

    ba = a^{-1}a(ba) ---------Multiplication of the identity

    ba=a^{-1}(ab)a ---------- Associativity

    <ba>=<ab> --------------- Conjugate elements have the same order
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  4. #4
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    Quote Originally Posted by ux0 View Post
    its just that simple?

    ba=ba

    ba = a^{-1}a(ba) ---------Multiplication of the identity

    ba=a^{-1}(ab)a ---------- Associativity

    <ba>=<ab> --------------- Conjugate elements have the same order

    As painfully and embarrasingly simple as that. Why, did you expect something very difficult?
    And what did you mean in your last line <ba>=<ab>? This is not true, not even close, in general. What you sure meant is that either |ab|=|ba|\,\,\, or \,\,\,ord (ab)=ord (ba)

    Tonio
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  5. #5
    ux0
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    Ya I kind of did, and thanks for the notation correction!
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