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Math Help - abstract algebra

  1. #1
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    abstract algebra

    |x| = order of x

    Assume |x|=n and |y|=m. Suppose that x and y commute: xy = yx. Prove that |xy| divides the least common multiple of m and n.

    Can anyone help?
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  2. #2
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    Quote Originally Posted by dori1123 View Post
    |x| = order of x
    Assume |x|=n and |y|=m. Suppose that x and y commute: xy = yx. Prove that |xy| divides the least common multiple of m and n.
    if \text{lcm}(m,n)=\ell, then obviously x^{\ell}=y^{\ell}=1. thus since xy=yx, we have: (xy)^{\ell}=x^{\ell}y^{\ell}=1. hence |xy| \mid \ell. \ \ \Box


    Remark: here are three related questions: suppose G is an abelian group and |x|=n, \ |y|=m.

    Q1: show that it's possible to have |xy| \neq \text{lcm}(m,n). (by giving an example!)

    Q2: what can we say about |xy| if \gcd(m,n)=1?

    and a much more interesting question:

    Q3: show that G has an element of order \text{lcm}(m,n).
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  3. #3
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    Quote Originally Posted by NonCommAlg View Post
    Remark: here are three related questions: suppose G is an abelian group and |x|=n, \ |y|=m.

    Q1: show that it's possible to have |xy| \neq \text{lcm}(m,n). (by giving an example!)
    Please give a hint. Thank you.
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