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**Statement:**

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

(1)if and only if $a^{-1}b\in H$.

Let $\sim_R$ be defined by

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 $a\sim_L c$.