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Homework Help: |G|=pq then |G| is abelian or Z(G)=1

  1. Sep 14, 2013 #1
    1. The problem statement, all variables and given/known data
    Show that if [itex]|G|=pq[/itex] for some primes p and q, then G is abelian or Z(G)=1.

    2. Relevant equations

    |G| = pq =⇒ |Z(G)| = 1, p, q, or pq. Prove: |Z(G)| = p and |Z(G)| = q are impossible. If
    |Z(G)| = p then |G/Z(G)| =|G|/|Z(G)| =pq/p = q. But then, since G/Z(G) is cyclic of prime order,
    |G/Z(G)| = 1. Thus q = 1, a contradiction since q > 1. Similarly, |Z(G)| = q =⇒ p = 1, a
    contradiction to the definition of p

    3. The attempt at a solution

    I am trying to understand the part of the proof that says "G/Z(G) is cyclic". Why is this so?
  2. jcsd
  3. Sep 14, 2013 #2


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    Homework Helper

    I believe it's because groups of prime order are cyclic. This stems from Lagrange's theorem.
  4. Sep 14, 2013 #3
    Omg yes. Thank you, I had all those pieces just wasn't putting them together for some reason. :)
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