Trivial Subgroup is Normal

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

Then the trivial subgroup $\left({\left\{{e}\right\}, \circ}\right)$ of $G$ is a normal subgroup in $G$.

Proof
First we note that the trivial group $\left({\left\{{e}\right\}, \circ}\right)$ is a subgroup of $G$.

To show $\left({\left\{{e}\right\}, \circ}\right)$ is normal in $G$:


 * $\forall a \in G: a e a^{-1} = a a^{-1} = e$

Hence the result.