Suppose $H$ is a Sylow subgroup of $G$ and let $J$ be a subgroup of $G$ which contains $H$. If $H$ is normal in $J$, and if $J$ is normal in $G$, prove that $H$ is normal in $G$.
I do not quite see how to use the fact that $H$ is a Sylow subgroup of $G$ in order to prove this. Can anyone help?
Recall that Frattini's Argument states:
Given a finite group $G$ with normal subgroup $J$ and Sylow subgroup $H$, we have $$ G = N_G(H)J $$ where $N_G(H)$ is the normalizer of $H$ in $J$.
Since your question satisfies the hypotheses of Frattini's Argument, we indeed have $$ G = N_G(H)J.$$ However, since $H$ is normal in $J$, we know that $J \leq N_G(H)$. This implies that $N_G(H)J = N_G(H)$, i.e. $$G = N_G(H).$$ Since the normalizer is the whole group, $H$ is normal in $G$.
