Quotient Group

Given a Group $G$ and a Normal Subgroup $N⊴G$, we define the quotient group as the set of all left-Coset of $N$.

$$G/N\equiv \{gN\mid g\in G\}$$

Its group operation is defined as

$$(gN)(hN)\equiv (gh)N$$

Equivalence relation defined by $N$

A quotient group induces an equivalence relation upon its base group, since, by thinking about $N$ as the set of “identity elements”, then all elements “connected” by an element of $N$ will be equivalent.