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Math Help - the order of two elements in group

  1. #1
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    the order of two elements in group

    Let G be an abelian group and let g,h in G, assume that |g|,|h| (the order of g and h) are both finite,with hcf(|g|,|h|)=1.then Prove that |g+h|=|g|*|h|
    i dont know how to relate the order of g,h to the order of |g+h| could somone give me any hints ??
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    Re: the order of two elements in group

    Start summing g+h with g+h. you'll get m(g+h), where m is an integer. since hcf(|g|,|h|)=1, the intersection of <g> and <h> is the {0}. thus, m(g+h)=mg+mh is always different from 0 unless both mg=0 and mh=0. That happens when m=lcm(|g|,|h|)=|g|*|h|.
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    Re: the order of two elements in group

    why hcf(|g|,|h|)=1 implies the intersection of <g> and <h> is the {0}????
    Last edited by cummings123321; November 27th 2012 at 08:42 AM.
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    Re: the order of two elements in group

    Suppose it isn't true, let t be an element of the intersection. t=ag=bh. we have t*|t|=(ag)*|t|=(bh)*|t|=0, the orders of g and h are a|t| and b|t|, which is a contradiction cuz they are supposed to have hcf=1
    Last edited by Link; November 27th 2012 at 08:56 AM.
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    Re: the order of two elements in group

    You need the theorem that if ng = 0, then |g| divides n. Using it, show that |g + h| divides |g| * |h|. Next, you need to prove that both |g| and |h| divide |g + h|. Since hcf(|g|, |h|) = 1, this would imply that |g| * |h| divides |g + h| and so |g + h| = |g| * |h|.

    To prove that |g| divides |g + h|, consider |g + h|(g + h) = 0 and multiply both sides by the number (not a group element) |h|. Then use the fact above and (a generalization) of the Euclid's lemma.
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    Re: the order of two elements in group

    Quote Originally Posted by Link View Post
    Suppose it isn't true, let t be an element of the intersection. t=ag=bh. we have t*|t|=(ag)*|t|=(bh)*|t|=0, the orders of g and h are a|t| and b|t|
    The last fact does not follow.
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    Re: the order of two elements in group

    let |g| = k, |h| = m.

    consider (km)(g+h) = (km)g + (km)h = (mk)g + (km)h (since km = mk in the the ring of integers)

    = m(kg) + k(mh) = m(0) + k(0) = 0 + 0 = 0 (we are using the fact that an abelian group has a natural interpretation as a Z-module (the "laws of exponents" with "powers" meaning "repeated sums")).

    since (km)(g + h) = 0, |g + h| divides km.

    now suppose we write |g + h| = t.

    this means that t(g +h) = tg + th = 0.

    thus -th = tg, which shows that -th is in both <h> and <g>. so we'll be done if we can show that <g> ∩ <h> = {0}

    (because this will show that -th = tg = 0, and thus th = -0 = 0, so k divides t, and m divides t).

    by lagrange, |<g> ∩ <h>| divides <g> = k, and <h> = m. but gcd(k,m) = 1, so |<g> ∩ <h>| = 1, and thus must be the trivial subgroup.
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    Re: the order of two elements in group

    thank you i see , many thanks 2 u!!!
    Last edited by cummings123321; November 27th 2012 at 02:51 PM.
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    Re: the order of two elements in group

    Quote Originally Posted by cummings123321 View Post
    why hcf(|g|, |h|) = 1, this would imply that |g| * |h| divides |g + h|????
    You first prove that |g| and |h| divide |g + h|, as describe in post #5.
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  10. #10
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    Re: the order of two elements in group

    yes,i get it now ,thanks a lot
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