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 used it in a paper from 1885 when defining the Frattini subgroup of a group. The argument was taken by Frattini, as he himself admits, from a paper of Alfredo Capelli dated 1884.

Frattini's argument

Statement

If G is a finite group with normal subgroup H, and if P is a Sylow p-subgroup of H, then where N_G(P) denotes the normalizer of P in G, and N_G(P)H means the product of group subsets.

Proof

The group P is a Sylow p-subgroup of H, so every Sylow p-subgroup of H is an H-conjugate of P, that is, it is of the form h^{-1}Ph for some h \in H (see Sylow theorems). Let g be any element of G. Since H is normal in G, the subgroup g^{-1}Pg is contained in H. This means that g^{-1}Pg is a Sylow p-subgroup of H. Then, by the above, it must be H-conjugate to P: that is, for some h \in H and so Thus and therefore. But g \in G was arbitrary, and so

Applications