I am just in to 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 am 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!