Here is a paraphrased version of problem 4B.4 in Isaacs's Finite Group Theory:
Let $G$ be a group and $X,Y$ subgroups of $G$, such that $Y$ centralizes $[X,Y]$. If $X$ is normal in $G$, show $[X,Y]$ is abelian.
Now of course we can assume $G=\langle X,Y\rangle$, and thus $K=[X,Y]$ is normal in $G$. But then $C_G(K)$ is normal in $G$ too, and in the quotient $G/C_G(K)$, $\bar{K}=[\bar{X},\bar{Y}]=[\bar{X},\bar{1}]=\bar{1}$, so $K\subset C_G(K)$ and $K$ is abelian.
So my question is:
Why do we need to assume $X$ is normal in $G$?