Definition 1. Let $G$ be a group and $N$ be a normal subgroup. The binary operation $*$ on $G / N$
\[aN * bN = (ab)N\]is well-defined and $(G / N, *)$ is a group, called the quotient group of $G$ by $N$.
Proposition 1. If $G$ is a group and $N$ is a normal subgroup,
\[G / N = N \backslash G.\]Proposition 2. Let $G$ be a group and $H$ be a subgroup. If we define two equivalence relations on $G$:
\[\begin{align*} a \sim_H b \enspace&\lrimp\enspace ab^{-1} \in H; \\ a \prerel{H}{\sim} b \enspace&\lrimp\enspace a^{-1}b \in H, \end{align*}\]then for every $a \in G$:
- \[[a]_{\sim_H} = aH.\]
- \[[a]_\prerel{H}{\sim} = Ha.\]
If $H$ is normal:
- \[{\sim_H} = {\prerel{H}{\sim}}.\]
- \[G / H = G / {\sim_H}.\]