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 ${\displaystyle G}$ 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.

Proof

The group ${\displaystyle P}$ is a Sylow ${\displaystyle p}$-subgroup of ${\displaystyle H}$, so every Sylow ${\displaystyle p}$-subgroup of ${\displaystyle H}$ is an ${\displaystyle H}$-conjugate of ${\displaystyle P}$, that is, it is of the form ${\displaystyle h^{-1}Ph}$ 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 ${\displaystyle h\in H}$

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

and 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, and so ${\displaystyle G=HN_{G}(P)=N_{G}(P)H.\ \square }$

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. ${\displaystyle M=N_{G}(M)}$.

External links

• Frattini's Argument on ProofWiki

References

1. M. Brescia, F. de Giovanni, M. Trombetti, "The True Story Behind Frattini’s Argument", Advances in Group Theory and Applications 3, doi:10.4399/97888255036928

• Hall, Marshall (1959). The theory of groups. New York, N.Y.: Macmillan. (See Chapter 10, especially Section 10.4.)