Uniqueness in the Fundamental Theorem of Finite Abelian Groups

Question

The statement of the fundamental theorem of finite abelian groups I have is as follows:

Let $G$ be a finite abelian group. Then G can be written as $$\mathbb{Z}_{n_1} \oplus \cdots \oplus \mathbb{Z}_{n_s}$$ where each $n_{i+1} | n_i$ for $1 \le i \le s-1 $ and each $n_i \ge 2$. Moreover, this decomposition is unique.

I am unsure exactly what the uniqueness part means. For example, take $G$ to be a group of order 72. I can write down two compositions $$\mathbb{Z}_{24} \oplus \mathbb{Z}_{3}$$ $$\mathbb{Z}_{12} \oplus \mathbb{Z}_{6}$$ which satisfy the theorem, but do not appear to be unique, i.e. I have two distinct integer sequences $\{24, 3\}$ and $\{12, 6\}$. What am I missing?

Answer 1

Those two groups are not isomorphic. It fails to be a counterexample because the theorem claims the uniqueness of a decomposition for every fixed finitely generated abelian group G. (You can clearly see that they aren’t isomorphic since the first of the two has an element of order 24 and the second one doesn’t).