Action_of_Group_on_Coset_Space_is_Group_Action_v2

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.