# 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. Truth to be told, the argument was take by Frattini, as he himself admits, from a paper of Alfredo Capelli dated 1884.[1]

## Frattini's Argument

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

${\displaystyle G=N_{G}(P)H}$

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

### Statement and proof

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

${\displaystyle g^{-1}Pg=h^{-1}Ph}$

so

${\displaystyle hg^{-1}Pgh^{-1}=P}$

thus

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

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

## 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 ${\displaystyle N_{G}(N_{G}(P))}$, it can be shown that ${\displaystyle N_{G}(N_{G}(P))=N_{G}(P)}$ whenever ${\displaystyle G}$ is a finite group and ${\displaystyle P}$ is a Sylow ${\displaystyle p}$-subgroup of ${\displaystyle G}$.
• More generally, if a subgroup ${\displaystyle M\leq G}$ contains ${\displaystyle N_{G}(P)}$ for some Sylow ${\displaystyle p}$-subgroup ${\displaystyle P}$ of ${\displaystyle G}$, then ${\displaystyle M}$ is self-normalizing, i.e. ${\displaystyle M=N_{G}(M)}$.