Frattini argument
= Frattini argument
{c}
{title2=$G=N_G(P)H$}
{wiki}
If $H$ is a <normal subgroup> of a finite group $G$ and $P$ a <Sylow subgroup> of $H$, then
$$
G=N_G(P)H.
$$
Normality puts every $g^{-1}Pg$ inside $H$, and Sylow conjugacy there gives $g^{-1}Pg=h^{-1}Ph$. Hence $gh^{-1}$ normalizes $P$, giving the factorization. This reduces questions about $G$ to a <normaliser> and its normal subgroup.