Group is Normal in Itself

Theorem
Let $\struct {G, \circ}$ be a group.

Then $\struct {G, \circ}$ is a normal subgroup of itself.

Proof
First we note that $\struct {G, \circ}$ is a subgroup of itself.

To show $\struct {G, \circ}$ is normal in $G$:
 * $\forall a, g \in G: a \circ g \circ a^{-1} \in G$

as $G$ is closed by definition.

Hence the result.