Theorem 10.1

Statement:

Let $H$ be a subgroup of $G$.Let the relation $\sim_L$ be defined on $G$ by

(1)
\begin{align} a \sim_L b \end{align}

if and only if $a^{-1}b\in H$.
Let $\sim_R$ be defined by

(2)
\begin{align} a \sim_R b \end{align}

if and only if $ab^{-1}\in H$.
Then $\sim_L$ and $\sim_R$ are both equivalence relations on $G$.

Proof:

Reflexive:Let $a\in G$.Then $a^{-1}a=e$ and $e\in H$ since $H$ is a subgroup.Thus $a\sim_L a$.

Symmetric:Suppose $a\sim_L b$.Then $a^{-1}b\in H$.Since $H$ is a subgroup,$(a^{-1}b)^{-1}$ is in $H$ and $(a^{-1}b)^{-1}=b^{-1}a$,so $b^{-1}a$ is in $H$ and $b\sim_L a$.

Transitive:Let $a\sim_L b$ and $b\sim_L c$.Then $a^{-1}b\in H$ and $b^{-1}c\in H$.Since $H$ is a subgroup,$(a^{-1}b)(b^{-1}c)=a^{-1}c$ is in $H$,so $a\sim_L c$.