Congruence Modulo Normal Subgroup is Congruence Relation

Theorem
Let $\left({G, \circ}\right)$ be a group.

Let $N$ be a normal subgroup of $G$.

Then congruence modulo $N$ is a congruence relation for the group product $\circ$.

Proof
Let $x \mathcal R_N y$ denote that $x$ and $y$ are in the same coset, that is:
 * $x \mathcal R_N y \iff x N = y N$

as specified in the definition of congruence modulo $N$.

Let $x \mathcal R_N x'$ and $y \mathcal R_N y'$.

Then by

Also see

 * Compatible Relation Normal Subgroup