Normalizer of Center is Group

Theorem
$$Z \left({G}\right) = \left\{{x \in G: N_G \left({x}\right) = G}\right\}$$

That is, the center of a group $$G$$ is the set of elements $$x$$ of $$G$$ such that the normalizer of $$x$$ is the whole of $$G$$.

Proof
$$N_G \left({x}\right)$$ is the normalizer of the set $$\left\{{x}\right\}$$. Thus: