What does this group theory statement ask for? This is a question that was asked in my group theory examination today:
Let $G$ be a finite cyclic group generated by an element $x$ of $G$. If $y(\ne x)\in G$ is also a generator of $G$, find the relation between the elements $x$ and $y$.
I do not think that given an arbitrary finite cyclic group one can give a nice relation between any two of its generators. For example if $\mathbb{Z}/50\mathbb{Z}$ what is the relation between 7 and 49 or 23 and 31 or say 3 and 43? I have not been able to understand clearly what kind of a relation the question asks for. 
I know that  $x=y^m$ for some $m$, and  $m$ is then coprime to the order of the group but I do not know how could this give a relation involving only $x$ and $y$.
So what is the question asking for and what in general is a relation between $x$ and $y$?

Well, there are some very basic relations between $\,x\,,\,y\,$, for example say $\,|G|=n\,$ , then:
$$\begin{align*}*&\;\;\;|x|=|y|\\**&\;\;\;y=x^k\,\,,\,\,x=y^s\;\;,\;\text{for some}\;\;k,s\in\Bbb Z\;\;\text{with}\;\;(n,k)=(n,s)=1\end{align*}$$
