I am currently learning about modular forms from J.P. Serre's A Course in Arithmetic.
$H$ is the open upper half plane, and $G$ is the modular group, acting on H by linear fractional transformation. We have defined $D$ to be a subset of $H$ formed by all points $z$ such that $|z|\geq 1$ and $|\Re(z)|\leq \frac{1}{2}$. We are trying to prove that $D$ is a fundamental domain for the action of $G$ on $H$. To that ends, we have the statement of the following theorem:
"Suppose that two distinct points $z,z'$ of $D$ are congruent modulo $G$. Then $\Re(z)=\pm\frac{1}{2}$ and $z=z'\pm 1$, or $|z|=1$ and $z'=-\frac{1}{z}$."
What is the precise definition of "congruent modulo $G$" here? What would be an example of such points?
If a group $G$ acts on a set $S$, two elements, $s$, $s'$ in $S$ are said to be congruent modulo $G$ if there is a group element $g\in G$ such that $g.s = s'$. In other words, $s$ and $s'$ are in the same $G$-orbit.