Let $G$ be a finite group and $x\in G$. Then $|C_G(x)|\geq |G/G'|$

Question

Let $G$ be a finite group and $x\in G$.
Then $$|C_G(x)|\geq |G/G'|$$ or equivalently, $|G'|\geq |G:C_G(x)|$

By Orbit-Stabilizer Theorem, $|G:C_G(x)|$ equals the size of conjugacy class of $x$, $(x)$. I am trying to get an injection from $G'$ to $(x)$ but I don't think there is any obvious or natural map for that.

Answer 1

Let $gxg^{-1}\in (x)$.
Note that $gxg^{-1}x^{-1}G'=G'$.
So we have $gxg^{-1}\in xG'$.
Thus $(x)\subseteq xG'$ and hence $$|G:C_G(x)|=|(x)|\leq |xG'|=|G'|$$

Again thanks for the hint by Steve D.