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Math Help - Cyclic group, relatively prime numbers proof

  1. #1
    Senior Member sfspitfire23's Avatar
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    Cyclic group, relatively prime numbers proof

    I'm a little stuck... I'm trying to prove:

    If G is a cyclic group of order n and g^m=1 where gcd(m,n)=1, then g=1.

    I know that m, n are relatively prime and thus we know that 1=xm+yn for some integers x,y. But I'm unsure where to go from here... any suggestions?
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  2. #2
    MHF Contributor

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    Re: Cyclic group, relatively prime numbers proof

    suppose the order of g is k. since g^m = 1, k is a divisor of m.

    (to see this: write m = kq + r, where 0 ≤ r < k

    then 1 = g^m = g^(kq + r) = (g^(kq))(g^r) = (g^k)^q(g^r) = (1^q)(g^r) = g^r.

    since k is the least positive integer with g^k = 1, r = 0, so m = kq).

    similarly, k divides n = |G|. since k is a common divisor of m and n, and gcd(m,n) = 1, k = 1,

    that is 1 = g^k = g^1 = g.
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