To show a finite group G is nilpotent Let G be a finite group and G' denotes it's commutator.If order of G' is 2.Then show that G is Nilpotent.
What I have tried:G/G' is abelian,so it is nilpotent again G' is nilpotent as it's order is 2.But from this I can't conclude that G is nilpotent.So I am trying to show that all it's p-sylow subgroups are normal.From Which I can say that G is nilpotent.But I am not able to show this.
Any help will be appreciated.

You need to show that the derived subgroup $G^\prime$ is central.  To this end, take any commutator $[a,b]$ and any $g\in G$. Since $G^\prime$ is normal, and has order $2$, it must be that $g^{-1}[a,b]g = [a,b]$, so $[a,b]$ commutes with $g$.  And, since $g$ was arbitrary, it follows that $G^\prime\leq Z(G)$.  Thus, $G$ is nilpotent.
