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