Trivial Subgroup and Group Itself are Normal

Theorem
For every group $$G$$, the subgroups $$G$$ and $$\left\{{e}\right\}$$ are normal, where $$e$$ is the identity of $$G$$.

Proof

 * $$\forall a, g \in G: a g a^{-1} \in G$$ as $$G$$ is closed by definition.
 * $$\forall a \in G: a e a^{-1} = a a^{-1} = e$$.