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 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.

is a finite group with normal subgroup

is a Sylow p-subgroup of

denotes the normalizer of

means the product of group subsets.

The group

is a Sylow

, so every Sylow

, that is, it is of the form

{\displaystyle h^{-1}Ph}

(see Sylow theorems).

be any element of

is normal in

, the subgroup

is contained in

This means that

is a Sylow

was arbitrary, and so