Conjugate of Subgroup is Subgroup/Proof 2

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

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

Then from Stabilizer of Coset under Group Action on Coset Space:
 * $\Stab {a H} = a H a^{-1}$

where $\Stab {a H}$ the stabilizer of $a H$ under $*$.

The result follows from Stabilizer is Subgroup.