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Math Help - True or False? If a and b are elements of an abelian group, then order of (ab)= lcm

  1. #1
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    True or False? If a and b are elements of an abelian group, then order of (ab)= lcm

    Dear friends,

    May i check if the answer should be false?

    True or false? If a and b are elements of an abelian group, then order of (ab)= lcm (order of (a), order of (b))

    Thanks,
    weijing
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  2. #2
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    Re: True or False? If a and b are elements of an abelian group, then order of (ab)= l

    ord(ab) and lcm(ord(a), ord(b)) have a divisibility relationship:

    (ab)^{lcm(ord(a), ord(b))} \ \overset{(abelian)}{=} \ (a)^{lcm(ord(a), ord(b))}(b)^{lcm(ord(a), ord(b))}

    = (a)^{ord(a)k_1}(b)^{ord(b) k_2} = ((a)^{ord(a)})^{k_1} \ ((b)^{ord(b)})^{k_2} = (1)^{k_1}(1)^{k_2} = 1.

    so that ord(ab) always divides lcm(ord(a), ord(b).

    But are they always equal? (Hint: think about inverses.)
    Last edited by johnsomeone; October 16th 2012 at 08:17 AM.
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