Let $G$ be a finite nilpotent group. Consider $F$ the Frattini subgroup of $G$, that is, intersection of all maximal subgroup of $G$. Prove that $G/F$ is abelian.
What I am trying that G/F is abelian if and only if [G,G] is contained in F.so I have to show that all maximal subgroup contains [G,G] in case of G is nilpotent. Am I going to right direction??