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Math Help - Show that o(ab) = o(ba)

  1. #1
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    Show that o(ab) = o(ba)

    a, b are in group G. Show o(ab) = o(ba).

    I'm stumped. This would be trivial if G were abelian, but it's not. Please help.

    EDIT: The back of the book has:

    o(ab) = m
    => (ba)^m = (a^-1)(a)(ba)^m
    = (a^-1)(ab)^m(a) (**)

    I have NO clue how they got the step marked by (**).
    Last edited by phys251; July 31st 2013 at 07:08 PM.
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  2. #2
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    Re: Show that o(ab) = o(ba)

    Quote Originally Posted by phys251 View Post
    o(ab) = m
    => (ba)^m = (a^-1)(a)(ba)^m
    = (a^-1)(ab)^m(a) (**)

    I have NO clue how they got the step marked by (**).
    Consider some examples.

    a^{-1}a(ba)^3 =a^{-1}a(ba)(ba)(ba) =a^{-1}(ab)(ab)(ab)a =a^{-1}(ab)^3a

    If you need a precise proof, you can prove that a(ba)^m = (ab)^ma by induction on m.
    Thanks from topsquark and phys251
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  3. #3
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    Re: Show that o(ab) = o(ba)

    Ohhhh, you just associate everything once to the left, so now instead of ba's inside parentheses, we have ab's. Thanks.
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