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 $G$.

Let $*: G \times G / H \to G / H$ be the action on the (left) coset space:
 * $\forall g \in G, \forall g' H \in G / H: g * \paren {g' H} := \paren {g g'} H$

Then $G$ is a group action.

Proof
demonstrating that $*$ satisfies.

Then:

demonstrating that $*$ satisfies.