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Math Help - Meaning of a theorem

  1. #1
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    Meaning of a theorem

    This is a simple theorem I have read online:

    Let G be a group and let g, h \in G be commuting elements of finite orders m, n, respectively,
    where gcd(m,n)=1. Then (g) \cap (h) is the trivial subgroup.
    With the following proof:
    Let x \in (g) \cap (h). Then |x| is a divisor of |g| and of |h|, and so |x| is a divisor of
    gcd(|g|,|h|)=1. Thus x is the identity of G.
    The question is: What does "commuting element" mean? If it's the ordinary meaning (Like in ab=ba).
    What is the role in the proof and why cannot the above theorem be proven for non-commuting elements?
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  2. #2
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    Re: Meaning of a theorem

    Your definition (ab=ba) of a commuting element does hold here.

    It is important as elements don't have to commute inside a group. This proof only holds when elements do commute.
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  3. #3
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    Re: Meaning of a theorem

    the hypothesis that g and h commute is not needed, here. <g>∩<h> is a subgroup of <g> and <h>. by lagrange, its order is a common divisor of <g> and <h>, and thus of |g| and |h|.

    but this means that |<g>∩<h>| divides gcd(|g|,|h|) = gcd(m,n) = 1, so |<g>∩<h>| = 1, that is: <g>∩<h> is trivial.

    the condition that gh = hg may be important, however, in something else that the uses the proof, or that the proof builds upon. without knowing the context, i cannot say for sure.
    Thanks from emakarov
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