Condition for Group to Act Effectively on Left Coset Space

Theorem
Let $G$ be a group whose identity is $e$.

Let $H$ be a subgroup of $G$.

Then $G$ acts effectively on the left coset space $G / H$ :
 * $\displaystyle \bigcap_{a \mathop \in G} H^a = \set e$

where $H^a$ denotes the conjugate of $H$ by $a$.

Proof
$G$ acts effectively on the left coset space $G / H$ $a H \mapsto g a H$ is faithful, :