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Math Help - Relatively Prime Proof

  1. #1
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    Relatively Prime Proof

    Assume that a is an element of order n in a group G. Prove that m and n are relatively
    prime if and only if a^m has order n.

    Thanks for your help...
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  2. #2
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    You don't need to prove anything, this follows from the definitions: If a is an element of order n in a group and m is a positive integer, then <a^k>=<a^{gcd(n,m)}> and |a^m|=\frac{n}{gcd(n,m)}. So if n and m are relatively prime then |a^m|= \frac{n}{gcd(m,n)}=\frac{n}{1}=n.

    To see why that definition holds, let d=gcd(n,m), clearly (a^d)^{n/d}=a^n=e, so that | a^d| \leq n/d. Also if i is a positive integer less than n/d. then (a^d)^i \ne e by definition of |a|. And since d=gcd(n,m) you get |a^m|=|<a^m>|=|<a^{gcd(n,m)}>|=|a^{gcd(n,m)}| = \frac{n}{gcd(n,m)}.
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