Conjugacy Action on Identity

Theorem
Let $$G$$ be a group whose identity is $$e$$.

For the Conjugacy Action, $$\left|{\operatorname{Orb} \left({e}\right)}\right| = 1$$ and thus $$\operatorname{Stab} \left({e}\right) = G$$.

Proof
$$ $$ $$

So the only conjugate of $$e$$ is $$e$$ itself.

Thus $$\operatorname{Orb} \left({e}\right) = \left\{{e}\right\}$$ and $$\left|{\operatorname{Orb} \left({e}\right)}\right| = 1$$.

From the Orbit-Stabilizer Theorem, it follows immediately that $$\operatorname{Stab} \left({e}\right) = G$$.