$G$ is a finite group of permutations of the set $X$. Suppose $G$ acts transitively on $X$. Then $X$ is a finite set, and $|X|$ divides $|G|$.
So since $G$ acts transitively on $X$, there exists $x\in X$ such that $Gx = X$ for all $x \in X$. I don't really understand what is going on here with the concept of $X$ being a "single $G$-orbit" as my book says.
I think what is happening is that we take a single element from $X$ and we compose this with each element from the group of permutations $G$, then we get that the set of elements in $Gx$ is just $X$. We do this with each element of $X$. But if that is the case then there is some correspondence between $Gx$ and $X$ where $Gx$ is finite because $G$ is finite and therefore $X$ is finite.
I'm stuck moving forward from the transitive definition. Any help is appreciated.