This is a remark of Milne's Fields and Galois Theory on page 72.
This is the first time I have seen the term exponent in group theory. Though the definitions seem to be easy and I was able to easily understand the results regarding the exponent on Groupprops with their proofs, I do not see how I can convince me why this result is true.
Let $G$ be a finite and abelian group with exponent $n$.
If $G$ is finite, then $n = {\operatorname{lcm}}_{\substack g \in G}(ord(g))$. I guess that I need to put the fact that $G$ is abelian but I don't know where this is useful.
Could you please explain this result to me?
If $G$ is finite abelian, the fundamental structure theorem days that $G\simeq \mathbb{Z}/d_1\mathbb{Z}\times\cdots \times \mathbb{Z}/d_r\mathbb{Z}$ , where $r\geq 0$ and $2\leq d_1\mid d_2\cdots\mid d_r$.
It is easy to check that $\exp(G)=d_r$. Hence your assumption translates to $d_r=n$. Now $d_i\mid d_r=n$, so writing $n=d_i m_i$, we have that $\bar{m}_i\in\mathbb{Z}/n\mathbb{Z}$ has order $d_i$, so $\langle \bar{m}_i\rangle\simeq \mathbb{Z}/d_i\mathbb{Z}$. Hence $G$ is isomorphic to $\langle \bar{m}_1\rangle\times\cdots \times \langle \bar{m}_r\rangle$, which is the desired subgroup of $(\mathbb{Z}/n\mathbb{Z})^r$.