Number of Non-Isomorphic Abelian Groups

[apalmer3]

Homework Statement

Determine the number of non-isomorphic abelian groups of order
72, and list one group from each isomorphism class.

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?

 

[morphism]

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

 

[apalmer3]

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]

Yes, those are precisely all of them.

 

[apalmer3]

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]

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

Good luck on your test!

 

[apalmer3]

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]

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}).

 

[apalmer3]

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