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Cyclic group properties

  1. Oct 20, 2009 #1
    1. The problem statement, all variables and given/known data
    Zn={0,1,...,n-1}. show that an element k is a generator of Zn if and only if k and n are relatively prime.


    2. Relevant equations



    3. The attempt at a solution
    it makes sense but im having a hard time proving this.
     
  2. jcsd
  3. Oct 20, 2009 #2

    Office_Shredder

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    So you need to do two parts:
    1) If k and n are relatively prime, k generates Zn. What do you need to prove to show this?

    2) If k generates Zn, then k and n are relatively coprime. This part is probably easier done by proof by contradiction
     
  4. Oct 20, 2009 #3
    well i understand i need to show both ways but to be honest, this is all i have:
    ==>if m is in {0,1,...,n-1} is a generator, its order is n. Also, its order must be n/(m,n). Thus, n=n/(m,n) which implies (m,n)=1.
    <== if (m,n)=1 then the order of m is n/(m,n)=n/1=n.
    therefore, m is a generator of Zn.
     
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