what am i misunderstanding here? Dummit and Foote p.161

Let $G$ be an abelian group of order $n&gt;1$.
Let $n={p_1}^{a_1}\cdots {p_k}^{a_k}$ be the prime factorzation.
Then, $G\cong \mathbb{Z}_{{p_1}^{a^1}}\times\cdots\times\mathbb{Z}_{{p_k}^{a^k}}$

Doesn't this mean that there is a unique abelian group of order $n$ up to isomorphism?
What am i misunderstanding here?
The statement in Dummit and Foote says

$$G \cong A_1 \times A_2 \times \dots \times A_k,\text{ where } |A_i| = p_i^{\alpha_i}.$$

It does not say $A_i \cong \mathbb{Z}/p_i^{\alpha_i}\mathbb{Z}$. Indeed, there can be many abelian groups of the same prime-powered order, for example $\mathbb{Z}/p^2\mathbb{Z}$ and $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$.
The prime factors does not have to be unique.