Frattini argument ($G=N_G(P)H$)

If $H$ is a normal subgroup of a finite group $G$ and $P$ a Sylow subgroup of $H$, thenNormality puts every $g^{-1}Pg$ inside $H$, and Sylow conjugacy there gives $g^{-1}Pg=h^{-1}Ph$. Hence $g{h}^{-1}$ normalizes $P$, giving the factorization. This reduces questions about $G$ to a normaliser and its normal subgroup.