# Number of Non-Isomorphic Abelian Groups

1. Homework Statement
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?

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morphism
Homework Helper
Do you know the fundamental theorem of finite abelian groups? If you really understand it, this should be straightforward.

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?

morphism
Homework Helper
Yes, those are precisely all of them.

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*)

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

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!

morphism
Homework Helper
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}).

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