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 $y$?

Your answer sounds fairly reasonable. If $y$ is also a generator then the order of $y$ is the same as the order of $x$. Since $G$ is cyclic $y=x^k$ for some $k<|G|$ but $y^m=e$ since it generates $G$. This will indicate $k$ is relatively prime to $m$. Towards a contradiction suppose otherwise, suppose $m=kl$ then $y^l=(x^{k})^l=x^{kl}=x^m=e$ which says $|y|=l \neq m$. Hence $m$ and $k$ are relatively prime.
As an example, if $G = <c>$ for $|G|=10$ then $c,c^3,c^7,c^9$ are the possible generators. I would wager your professor was fishing for something like what you say in your post.
