Frattini lifting of nilpotence
= Frattini lifting of nilpotence
{c}
Let $\Phi=\Phi(G)$ in a <finite group>. If $H\trianglelefteq G$ contains $\Phi$ and $H/\Phi$ is a <nilpotent group>, every <Sylow subgroup> $P$ of $H$ has $P\Phi\trianglelefteq G$. The <Frattini argument> gives $G=(P\Phi)N_G(P)=\Phi N_G(P)$, and the defining non-generator property of $\Phi$ forces $N_G(P)=G$. Hence $H$ is a <nilpotent group>.