Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Cyclic abelian group of order pq

  1. Jul 3, 2009 #1
    I'm looking at the exercises of Hungerfod's Algebra. Some looks easy but it seems the proofs are not so obvious. Here's one I'm particularly having a hard time solving:

    Let G be an abelian group of order pq with (p,q)=1. Assume that there exists elements a and b in G such that |a|= p and |b| = q. Show that G is cyclic.

    Help anyone?
     
  2. jcsd
  3. Jul 4, 2009 #2

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    What's the order of ab?
     
  4. Jul 5, 2009 #3
    Oh yeah. |ab|=pq because p and q are relatively prime. Whice means ab will generate the whole of G. And hence G is cyclic. Thanks.
     
  5. Nov 11, 2009 #4
    For this problem, in order for the group G to be cyclic, is the abelian condition necessary? In other words, if the problem is restated as: "if a finite group of order pq, where p and q are distinct primes, the the group is cyclic", is it still true?

    The reason I asked this question is that in my proof, I didn't see why we need the group to be abelian. Thanks!
     
  6. Nov 11, 2009 #5
    It's absolutely necessary. Consider the permutation group on three letters (ie. S3), then this is a group of order 6 = 2 *3 and is clearly not cyclic (and definitely not abelian either). However we do have this result:

    If G is a group order pq, pq distinct primes say P < q and p does not divide q-1, then G is abelian, hence cyclic. The hard part is proving it's abelian and the cyclic part follows from your initial problem.

    There's also a bit more interesting of a problem:

    If G is a group of order pq as above and p does q-1, then G is the unique nonabelian group of order pq.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Cyclic abelian group of order pq
  1. Abelian group order (Replies: 4)

Loading...