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 $N$.
If $G$ is a group and $N$ is a normal subgroup,
\[G/N = N\backslash G.\]
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:
- \[{\sim_H} = {\prerel{H}{\sim}}.\]
- \[G/H = G/{\sim_H}.\]