Let G be a non-abelian finite group, Z(G) be the center of G and $\displaystyle C_{G}(x)$ be the centralizer of the element x in G. Let $\displaystyle x \in G \setminus Z(G)$, so we have $\displaystyle |G| \geq 2|C_{G}(x)|$. Now how can I prove that $\displaystyle |G \setminus C_{G}(x)|>(|G|-|Z(G)|)/2$ ?