In a non-abelian group, if $C(a)=\langle a\rangle$ then $a\not\in Z(G)$. Suppose $G$ is a non-abelian group and $a∈G$. Prove that if $C(a)= \langle a \rangle$ then $a\not\in Z(G)$. I just don't understand this proof at all. Would someone mind walking me through the entire proof?

Since $C(a) = \langle a\rangle$ is abelian and $G$ is not, $C(a) \neq G$. Now $a\notin Z(G)$, because otherwise by definition $C(a) = G$.
