I can prove that commutator is minimal subgroup such that factor group of it is abelian. I had encountered one statement as
If $H$ is a subgroup containing commutator subgroup then $H$ is normal.
I.e. we have to show that $\forall g\in G$ such that $gHg^{-1}=H$ with fact that $G'\subset H$
It is for elements in $G'$ to show condition for normality.
But how to do for elements not in $G'$ but in $H$, that is in $H\setminus G'$?