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.

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

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

so

thus

and therefore g ∈ N_G(P)H. But g ∈ G was arbitrary, so G = HN_G(P) = N_G(P)H. ${\displaystyle \square }$

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