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??

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

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