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.

Frattini's Argument

Statement

If is a finite group with normal subgroup, and if is a Sylow p-subgroup of, then

where denotes the normalizer of in and means the product of group subsets.

Proof

The group is a Sylow -subgroup of, so every Sylow -subgroup of is an -conjugate of, that is, it is of the form , for some . Let be any element of. Since is normal in, the subgroup is contained in. This means that is a Sylow -subgroup of. Then by the above, it must be -conjugate to : that is, for some
and so

Thus,
and therefore. But was arbitrary, and so

Applications