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
where NG(P) denotes the normalizer of P in G.
- G = NG(P)H,
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;
- gh−1 ∈ NG(P),
and therefore g ∈ NG(P)H. But g ∈ G was arbitrary, so G = HNG(P) = NG(P)H.
- 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 M ≤ G 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 M ≤ H = NH(P)M ≤ M M = M, so M = H.
- Hall, Marshall (1959). The theory of groups. New York, N.Y.: Macmillan. (See Chapter 10, especially Section 10.4.)