My book says this:
Suppose that $G$ is a group of permutations of a set $X$. We shall show that the group structure of $G$ leads naturally to a partition of $X$.
Define a relation $\sim$ on $X$ by the rule
$ x \sim y \iff g(x) = y \text{ for some } g \in G $
Lets say that $X$ is the set $\{ 1, 2, 3 \}$, then a group of permutations of $X$ could be $\{ (1)(2)(3), (123), (132) \} = H$.
As I've understood it, group elements act on group elements. For example, $(123) \cdot (132) = (1)(2)(3) \in H$.
But what does $g(x) \mid g ∈ G$ mean? I assume $x, y \in X$ which would be either $1$, $2$ or $3$ in the example, but $g$ doesn't have the binary operation defined over elements of $X$?
I'm sorry if this is very basic :(