Stabilizer of Subgroup Action is Identity

Theorem
Let $\struct {G, \circ}$ be a group whose identity is $e$.

Let $\struct {H, \circ}$ be a subgroup of $G$.

Let $*: H \times G \to G$ be the subgroup action defined for all $h \in H, g \in G$ as:
 * $\forall h \in H, g \in G: h * g := h \circ g$

The stabilizer of $x \in G$ is $\set e$:
 * $\Stab x = \set e$

Proof
From Subgroup Action is Group Action we have that $*$ is a group action.

Let $x \in G$.

Then:

Hence the result, by definition of right coset.