# If a finite group G is nilpotent then G/F is abelian, where F is the Frattini subgroup of G

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??

• In a nilpotent group all maximal subgroups are normal and have prime index. So any commutator lies in all maximal subgroups. – Derek Holt Aug 26 '14 at 8:15
• @DerekHolt Thanks for the help..... – Ripan Saha Aug 26 '14 at 8:32