A quotient group induces an equivalence relation upon its base group

up:: Quotient Group

Given a Group $G$ and a Normal Subgroup $N⊴G$, we can create an Equivalence Relation in $G$ as

$$g\sim h⟺\exists n\in N\mid g=hn⟺g{h}^{-1}\in N$$

That is, elements of $N$ act as identity elements in the Quotient Space.

We can identify each Equivalence Class in this quotient space as a left coset of $N$

$$[g]\in G/\sim \leftrightarrow gN\in G/N$$