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Group theory problem

  1. Oct 18, 2009 #1
    Hi

    I have a problem I just can't seem to solve, even though the solution shouldn't be too hard

    Let G be a finite abelian group and let p be a prime.
    Suppose that any non-trivial element g in G has order p. Show that the order of G must be p^n for some positive integer n.

    Anyone got any ideas about how to approach this??

    thanks,
     
  2. jcsd
  3. Oct 18, 2009 #2
    Suppose there is another prime q that divides the order of the group and show there must be an element of order q.
     
  4. Oct 18, 2009 #3
    but is it the case that for all factors of the order of a group there is an element of that order?? i am soo confused..
     
  5. Oct 18, 2009 #4
    You know Lagranges theorem..? Consider the subgroup generated by g,- what's his order?. Well, if you like carefully at what " generates" means, youll see that the order of the subgroup generated by g is also p.
     
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