Frattini argument

Statement

Let $G$ be a finite group, let $H$ be a normal subgroup of $G$, and let $P$ be a Sylow p-subgroup of $H$. Then $G = N_G(P) H$, where $N_G(P)$ is the normaliser of $P$ in $G$.

Remarks

For $g \in G$, the subgroup $gPg^{-1}$ lies in $H$, because $H$ is normal, and it is a Sylow $p$-subgroup of $H$. By the second Sylow theorem in $H$, there is $h \in H$ with $gPg^{-1} = hPh^{-1}$, so $h^{-1}g \in N_G(P)$. The argument is used in the proof that a finite nilpotent group is the direct product of its Sylow subgroups.

History

Giovanni Frattini used the argument in a paper from 1885. He took it, as he said himself, from a paper of Alfredo Capelli from 1884.

Explanations

L1 human

Conjugation by an element of $G$ moves $P$ to another Sylow subgroup of $H$. An element of $H$ can move it back. So each element of $G$ is an element of $H$ times an element that normalises $P$.