Action of Group on Coset Space is Group Action

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

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

Then $G$ acts on the left coset space $G/H$ by the rule:
 * $\forall g \in G: g * \paren {g' H} = \paren {g g'} H$

Proof
As $H$ is a subset of $G$, the result follows from Group Action on Subset of Group.