Frattini's argument

In group theory, a branch of mathematics, Frattini's argument is an important lemma in the structure theory of finite groups. It is named after Giovanni Frattini, who used it in a paper from 1885 when defining the Frattini subgroup of a group. The argument was taken by Frattini, as he himself admits, from a paper of Alfredo Capelli dated 1884.^[1]

Frattini's argument

Statement

If $G$ is a finite group with normal subgroup $H$ , and if $P$ is a Sylow p-subgroup of $H$ , then

G=N_{G}(P)H,

where $N_{G}(P)$ denotes the normalizer of $P$ in $G$ , and $N_{G}(P)H$ means the product of group subsets.

Proof

The group $P$ is a Sylow $p$ -subgroup of $H$ , so every Sylow $p$ -subgroup of $H$ is an $H$ -conjugate of $P$ , that is, it is of the form $h^{-1}Ph$ for some $h\in H$ (see Sylow theorems). Let $g$ be any element of $G$ . Since $H$ is normal in $G$ , the subgroup $g^{-1}Pg$ is contained in $H$ . This means that $g^{-1}Pg$ is a Sylow $p$ -subgroup of $H$ . Then, by the above, it must be $H$ -conjugate to $P$ : that is, for some $h\in H$

g^{-1}Pg=h^{-1}Ph,

and so

hg^{-1}Pgh^{-1}=P.

Thus

gh^{-1}\in N_{G}(P),

and therefore $g\in N_{G}(P)H$ . But $g\in G$ was arbitrary, and so $G=HN_{G}(P)=N_{G}(P)H.\ \square$

Applications

Frattini's argument can be used as part of a proof that any finite nilpotent group is a direct product of its Sylow subgroups.
By applying Frattini's argument to $N_{G}(N_{G}(P))$ , it can be shown that $N_{G}(N_{G}(P))=N_{G}(P)$ whenever $G$ is a finite group and $P$ is a Sylow $p$ -subgroup of $G$ .
More generally, if a subgroup $M\leq G$ contains $N_{G}(P)$ for some Sylow $p$ -subgroup $P$ of $G$ , then $M$ is self-normalizing, i.e. $M=N_{G}(M)$ .