Stabilizer of Element of Group Acting on Itself is Trivial

Theorem
Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Let $*$ be the group action of $\left({G, \circ}\right)$ on itself by the rule:


 * $\forall g, h \in G: g * h = g \circ h$

Then the stabilizer of an element $x \in G$ is given by:
 * $\operatorname{Stab} \left({x}\right) = \left\{{e}\right\}$

Proof
Let $g \in \operatorname{Stab} \left({x}\right)$.

Then:

Hence the result, by definition of trivial subgroup.

Also see

 * Group Acts on Itself
 * Orbit of Element of Group Acting on Itself is Group