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. If G is a finite group with normal subgroup H, and if P is a Sylow p-subgroup of H, then

G = NG(P)H,
where NG(P) denotes the normalizer of P in G and NG(P)H means the product of group subsets.

Proof: P is a Sylow p-subgroup of H, so every Sylow p-subgroup of H is an H-conjugate h−1Ph for some hH (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 hH

g−1Pg = h−1Ph,

so

hg−1Pgh−1 = P;

thus

gh−1NG(P),

and therefore gNG(P)H. But gG was arbitrary, so G = HNG(P) = NG(P)H. ${\displaystyle \square }$

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 NG(NG(P)), it can be shown that NG(NG(P)) = NG(P) whenever G is a finite group and P is a Sylow p-subgroup of G.
• More generally, if a subgroup MG contains NG(P) for some Sylow p-subgroup P of G, then M is self-normalizing, i.e. M = NG(M).
Proof: M is normal in H := NG(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 NH(P)M = H. Since NH(P) ≤ NG(P) ≤ M, one has the chain of inclusions MH = NH(P)MM M = M, so M = H. ${\displaystyle \square }$

References

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