Center is Normal Subgroup

Theorem

The center $Z \left({G}\right)$ of any group $G$ is a normal subgroup of $G$ which is abelian.

Since $g x = x g$ for each $g \in G$ and $x \in Z \left({G}\right)$, we have $g Z \left({G}\right) = Z \left({G}\right) g$.

Thus, $Z \left({G}\right) \triangleleft G$.

$\blacksquare$

We have:

$\forall a \in G: x \in Z \left({G}\right)^a \iff a x a^{-1} = x a a^{-1} = x \in Z \left({G}\right)$

Therefore:

$\forall a \in G: Z \left({G}\right)^a = Z \left({G}\right)$

and $Z \left({G}\right)$ is a normal subgroup of $G$.

$\blacksquare$