Frattini's argument

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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.

[edit] Statement and proof

Frattini's argument states that if a finite group G has a normal subgroup H and if H has a Sylow p-subgroup P 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 its H-conjugates h⁻¹Ph are also Sylow p-subgroups belonging to H. H is normal in G, so the action of a g ∈ G by conjugacy sends P in H to one of its H-conjugates (see Sylow theorems), i.e.

g⁻¹Pg = h⁻¹Ph,

so

hg⁻¹Pgh⁻¹ = P, thus

gh⁻¹ ∈ N_G(P),

therefore g ∈ N_G(P)H. But g ∈ G was arbitrary, so G = HN_G(P) = N_G(P)H.

[edit] 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.