A problem about the order of element in a group Assume $a \in G$, where $G$ is a group and element $a$'s order is $mn$ with $(m,n) = 1$, the problem is to prove  there exist elements $b$ and $c$ in the group where $a = bc = cb$ , $b$'s order is $m$ and $c$'s order is $n$ and $b,c$ are unique.
I have proven that the existence of b and c using $ms+nt = 1$, but still don't know how to prove the uniqueness.

Hint: $\mathbb Z_{mn}\cong \mathbb Z_m\times \mathbb Z_n$ where $(m,n)=1$. The uniqnes comes from the propery of "direct product".
