Let $G$ be a finite abelian group with identity $e$. Prove that $\exists$ an element $x \in G$ such that ord y | ord x for all $y \in G$

Question

My attempt:

I think somehow we have to show that if $|x|=\operatorname{lcm}\{|x_i|: x_i \in G\}$ (as the group is finite abelian) then such an element actually lies in $G-\{e\}$. But I don't know how to show that such an element indeed lies in $G$.

Thanks for any help!

Answer 1

If $G = \bigoplus_{1 \leq i \leq k} \mathbb{Z}_{n_i}$ with $n_i | n_{i+1}$, consider the element $(0,0, \cdots ,1)$ with order $n_k$. It is clear that $\exp G = n_k$.