Stabilizer of Coset under Group Action on Coset Space

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 the stabilizer of $a H$ under $*$ is given by:
 * $\Stab {a H} = a H a^{-1}$

Proof
It is established in Action of Group on Coset Space is Group Action that $*$ is a group action.

Then: