Examine Structurally Distinct Abelian Groups with Primary Decomposition Thm

[dumbQuestion]

I am just into reviewing abstract algebra and came across a theorem I'd forgotten:

http://en.wikipedia.org/wiki/Finitely-generated_abelian_group#Primary_decomposition

(I linked to the theorem instead of writing it here just because I'm not sure how to write all those symbols here) Anyway, this seems super useful to me because there are a lot of theorems about the group of integers modulo n and their direct sums, (for example, I know if a is a generator of Zn, then a^m is a generator of Zn <=> m and n are relatively prime), and these groups are easier to conceptualize. So the primary decomposition theorem seems like a nice way to be able to take any general finite abelian group and put it in terms of a direct sum of these "Easy to deal with groups" of integers modulo m.But I have a question. All the questions in my book along the lines of "how many structurally distinct finite abelian groups of order n are there?" make use of the primary decomposition theorem and I don't understand why. For example, say I have a finite abeliian group G of order 12. The prime decompositioin of 12 is just (2)²(3) so using the primary decomposition theorem I know G is isomorphic to Z₂ + Z₂ + Z₃ and it's also isomorphic to Z₄ + Z₃. But since it's isomorphic to both of these groups, wouldn't that mean these groups are isomorphic to each other as well? I mean, isomorphism preserves all the group structures, subgroups, etc., so all these groups would be pretty much structurally identical right? So how are they mutually non-isomorphic? I'm so confused!

 

[dumbQuestion]

Nevermind I understand my mistake. The prime decomposition theorem doesn't say a group G will be isomorphic to ALL Of those different groups, just that it's isosmorphic to one of them. I feel really stupid now. Ok, another question then, how do we know which one it is isomorphic to? Is there another theorem that tells us? Or do we just kind of use process of elimination by examining the different possibilities and finding which one it's structurally similar to?And one other question, why isn't it the case that in the example I gave, Z4 + Z3 and Z2 + Z2 + Z3 are not isomorphic? I mean I know I can go to the groups, maybe see for example that Z4+Z3 is cyclic while Z2+Z2+Z3 is not and so I'd know they aren't isomorphic because of that. But what's the more general reasoning? I imagine it has something to do with the fact that for example, all the elements in {4,3} are not relatively prime to all the elements in {2,3}. is there a theorem that says that Zm + Zn is isomorphic to Zk + Zh if say m is relatively prime with k and h and n is also relatively prime with k and h?

 

[Robert1986]

There is a theorem that says $m,m$ are co-prime iff $Z_m \times Z_n \simeq Z_{mn}$. (Here I am abusing notation and writing the direct product for a direct sum.) So, that is why $Z_2 \times Z_2$ is not isomorphic to $Z_4$. But, as you said, every finite abelian group can be written as the direct sum of factors like $Z_{p^a}$ where $p$ is prime. For each group, these $p^a$ are called the elementary divisors of G and any two groups are isomorphic iff they have the same elementary divisors.

For example, if $G_1 = Z_2 \times Z_2 \times Z_3$ then the elementary divisors are $(2,2,3)$ and if $G_2 = Z_4 \times Z_3$ then the elementary divisors are $(2^2,3)$.

 

[dumbQuestion]

Thank you so much this is exactly what I was looking for!

 

[blue_raver22]

I can say that the primary decomposition theorem is a powerful tool in understanding the structure of finite abelian groups. It allows us to break down a complicated group into simpler, more familiar groups, making it easier to study and analyze.

Regarding your question about the isomorphism between different decompositions of the same group, it is important to note that while these groups may have the same underlying structure, they may have different elements and operations. For example, Z2 + Z2 + Z3 and Z4 + Z3 may both have the same number of elements and subgroups, but the elements themselves may be different. This can lead to different group properties and behaviors, making them non-isomorphic.

In fact, the primary decomposition theorem tells us that every finite abelian group can be decomposed into a direct sum of cyclic groups, but the number and order of these cyclic groups can vary, resulting in different structurally distinct groups.

I hope this helps clarify your confusion and shows the significance of the primary decomposition theorem in understanding finite abelian groups. Keep exploring and learning about abstract algebra – it is a fascinating field with many applications in mathematics and beyond.