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.

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. $$M = N_G(M)$$.