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 first used it in a paper from 1885 when defining the Frattini subgroup of a group.

Statement and proof

Frattini's argument states that if a finite group G has a 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.

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

g⁻¹Pg = h⁻¹Ph,

hg⁻¹Pgh⁻¹ = P; thus

gh⁻¹ ∈ N_G(P),

and therefore g ∈ N_G(P)H. But g ∈ G was arbitrary, 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 ≤ G contains N_G(P) for some Sylow p-subgroup P of G, then M is self-normalizing, i.e. M = N_G(M).

Proof: M is normal in H := N_G(M), and P is a Sylow p-subgroup of M, so the Frattini argument applied to the group H with normal subgroup M and Sylow p-subgroup P gives N_H(P)M = H. Since N_H(P) ≤ N_G(P) ≤ M, one has the chain of inclusions M ≤ H = N_H(P)M ≤ M M = M, so M = H. $\square$

Reference

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