Show $a\in G$ is contained in $Z(G)$ iff $Z(a)=G$ for center and centralizer?

Question

The center of a group $G$ is defined as the set $Z(G):= \{a\in G\mid \forall b\in G : ab=ba\}$ and the centralizer of an element $a\in G$ is defined as the set $Z(a) := \{b\in G\mid ab=ba\}$.

How can one show that an element $a\in G$ is contained in $Z(G)$ iff $Z(a)=G$ ?

Answer 1

$$Z(a)=G\iff \forall\,x\in G\;,\;\;ax=xa\iff a\in Z(G)$$