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.M. Brescia, F. de Giovanni, M. Trombetti
"The True Story Behind Frattini’s Argument"
''Advances in Group Theory and Applications'' 3
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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

:$G = N_(P)H$

where $N_(P)$ denotes the normalizer of $P$ in $G$ and $N_(P)H$ 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 $h^Ph$, for some $h \in H$ (see Sylow theorems). Let $g$ be any element of $G$. Since $H$ is normal in $G$, the subgroup $g^Pg$ is contained in $H$. This means that $g^Pg$ is a Sylow $p$-subgroup of $H$. Then by the above, it must be $H$-conjugate to $P$: that is, for some $h \in H$

:$g^Pg = h^Ph$,

and so

:$hg^Pgh^ = P$.

Thus,

:$gh^ \in N_(P)$,

and therefore $g \in N_(P)H$. 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. $M = N_G(M)$.

External links

Frattini's Argument on ProofWiki

References

* (See Chapter 10, especially Section 10.4.)

{{DEFAULTSORT:Frattini's Argument
Lemmas in group theory
Articles containing proofs