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Number of Non-Isomorphic Abelian Groups

  1. Apr 7, 2008 #1
    1. The problem statement, all variables and given/known data
    Determine the number of non-isomorphic abelian groups of order
    72, and list one group from each isomorphism class.


    3. The attempt at a solution

    72 = 2^3*3^2
    3= 1+1+1= 2+1= 3 (3)

    2= 1+1= 2 (2)

    3*2 = 6

    And then I get lost on the listing of a group from each isomorphism class.... Help?
     
  2. jcsd
  3. Apr 7, 2008 #2

    morphism

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    Do you know the fundamental theorem of finite abelian groups? If you really understand it, this should be straightforward.
     
  4. Apr 7, 2008 #3
    The fundamental theorem of finite abelian groups states that every finite abelian group can be expressed as the direct sum of cyclic subgroups of prime-power order.

    Z_2 x Z_2 x Z_2 x Z_3 x Z_3
    Z_2 x Z_2 x Z_2 x Z_9
    Z_2 x Z_4 x Z_3 x Z_3
    Z_2 x Z_4 x Z_9
    Z_8 x Z_3 x Z_3
    Z_8 x Z_9

    Yes?
     
  5. Apr 7, 2008 #4

    morphism

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    Yes, those are precisely all of them.
     
  6. Apr 7, 2008 #5
    Okay. Thanks so much! I was trying to make it more complicated than it needed to be, I think. :-D

    This is a side question that kind of pertains to this. Why is Z_3 x Z_3 not isomorphic to Z_9?

    Thanks again for your help, Morphism! You're a lifesaver! (I have a test in less than 10 hours... *sigh*)
     
  7. Apr 7, 2008 #6

    morphism

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    Well, for instance, notice that Z_3 x Z_3 doesn't have an element of order 9.

    Good luck on your test!
     
  8. Apr 7, 2008 #7
    Thanks for the luck! I'm probably going to need it.

    Is there a simple way to test whether or not something has an element of a certain order (i.e. Z_3 x Z_3 not having an element of order 9)? I want to make sure I understand this fully before morning.

    Thanks again!
     
  9. Apr 7, 2008 #8

    morphism

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    Try to prove that the order of an arbitrary element (x,y) in GxH is lcm{o(x), o(y)}. Can you generalize this to the direct product of n groups?

    So in our case, the largest possible order an element of Z_3 x Z_3 can have is 6 (= lcm{2,3}).
     
  10. Apr 7, 2008 #9
    Good Morning, Morphism!

    I'm looking at this last post, and I'm still a bit confused (forgive me). How did you decide that for Z_3 x Z_3 we were using lcm{2,3}?

    Thank you so much for your time and patience. :-D
     
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