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Math Help - Order of Group Elements

  1. #1
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    Order of Group Elements

    Hi, I need help solving these problems:

    Let a, b, and c be elements of a group G. Prove the following:

    1. If ak =e where k is odd, then the order of a is odd.

    3.The order of ab is the same as the order of ba. (Hint: if (ba)n = (baba…..b)a = e. Let (baba….b) =x then a is the inverse of x. Thus, ax =e.)

    2. Let x =a1a2… an, and let y be a product of the same factors, permuted cyclically. (That is, y = akak+1…ana1…ak-1.) Then ord(x) = ord(y).

    Thanks!
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  2. #2
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    Quote Originally Posted by steph615 View Post
    1. If ak =e where k is odd, then the order of a is odd.
    You probably mean a^k = e. If d is the order of a then d|k. So d cannot be even, i.e. it must be odd.

    3.The order of ab is the same as the order of ba. (Hint: if (ba)n = (baba…..b)a = e. Let (baba….b) =x then a is the inverse of x. Thus, ax =e.)
    (ab)^n = 1 \implies (ab)...(ab) = 1 \implies a (ba)^{n-1} b = 1 \implies (ba)^{n-1} = a^{-1}b^{-1}
    But a^{-1}b^{-1} = (ba)^{-1} thus (ba)^{n-1} \implies (ab)^{-1} \implies (ba)^n = 1.
    (Why is this not a complete proof?)

    2. Let x =a1a2… an, and let y be a product of the same factors, permuted cyclically. (That is, y = akak+1…ana1…ak-1.) Then ord(x) = ord(y).
    [/FONT]
    Use induction, with the basic case covered in #2.
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  3. #3
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    Re

    Ok, the proof for ord(ab)=ord(ba) is not complete because don't you have to show that n is the smallest positive integer? How then would I show that?
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