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Math Help - euler func.

  1. #1
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    euler func.

    Prove that if a and b are elements of a finite group G, then

    ord(ab) = ord(ba).
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  2. #2
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    ...........

    alright. and then?
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  3. #3
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    If ord(ab) = n then (ab)^n  = e\quad  \Rightarrow \quad \underbrace {(ab)(ab) \cdots (ab)}_n = e.

    So a\left[ {\underbrace {(ba)(ba) \cdots (ba)}_{n - 1}} \right]b = e\quad  \Rightarrow \quad \left[ {\underbrace {(ba)(ba) \cdots (ba)}_{n - 1}} \right] = a^{ - 1} b^{ - 1}

    (ba)^n  = \underbrace {(ba)(ba) \cdots (ba)}_{n - 1}(ba) = \left( {a^{ - 1} b^{ - 1} } \right)(ba) = e
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  4. #4
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    Quote Originally Posted by Plato View Post
    If ord(ab) = n then (ab)^n = e\quad \Rightarrow \quad \underbrace {(ab)(ab) \cdots (ab)}_n = e.

    So a\left[ {\underbrace {(ba)(ba) \cdots (ba)}_{n - 1}} \right]b = e\quad \Rightarrow \quad \left[ {\underbrace {(ba)(ba) \cdots (ba)}_{n - 1}} \right] = a^{ - 1} b^{ - 1}

    (ba)^n = \underbrace {(ba)(ba) \cdots (ba)}_{n - 1}(ba) = \left( {a^{ - 1} b^{ - 1} } \right)(ba) = e
    umm, excuse me, how are we sure that n is the least positive integer such that (ba)^n = e?
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  5. #5
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    Quote Originally Posted by kalagota View Post
    umm, excuse me, how are we sure that n is the least positive integer such that (ba)^n = e?
    If (ab)^n=e\; \Rightarrow \;(ba)^n=e then the same argument shows that (ba)^n=e \;  \Rightarrow \; (ab)^n=e.
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  6. #6
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by Opalg View Post
    If (ab)^n=e\; \Rightarrow \;(ba)^n=e then the same argument shows that (ba)^n=e \; \Rightarrow \; (ab)^n=e.
    you don't actually answered the question.. of course i know what that means.. the question was how are we sure that n is the smallest integer that satisfies (ba)^n = e.. Ü anyways, never mind.. i almost forgot the proof (, it was a year ago!)
    Last edited by ThePerfectHacker; December 1st 2007 at 02:06 PM.
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  7. #7
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    Quote Originally Posted by kalagota View Post
    you don't actually answered the question..
    Yes I have answered the question. If you know that (ab)^n=e\;\Leftrightarrow \;(ba)^n=e, it plainly follows that ab and ba have the same order.
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  8. #8
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    Quote Originally Posted by kalagota View Post
    you don't actually answered the question.. of course i know what that means.. the question was how are we sure that n is the smallest integer that satisfies (ba)^n = e
    Look in my first reply, I deliberately leave some point for the questioner to answer.
    I showed that ord(ba) \leqslant n, correct?
    Now suppose that ord(ba) = m.
    As Opalg pointed out the same argument shows that n \leqslant m, so n = m.
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  9. #9
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    ............

    what does the 'e' mean?
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  10. #10
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    Quote Originally Posted by anncar View Post
    what does the 'e' mean?
    Identity element.
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  11. #11
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    Quote Originally Posted by anncar View Post
    what does the 'e' mean?
    OH! My goodness!
    We never know with what level we are dealing. Do we?
    It is dangerous to assume anything.
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  12. #12
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    Quote Originally Posted by ThePerfectHacker View Post
    Identity element.
    what is an identity element? Please explain
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  13. #13
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    actually it seems self explanotory(sp?) now...kinda..but please explain..
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  14. #14
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    alright guys. I checked on google and understand it now. nevermind..
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