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

  • 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, it can be shown that whenever is a finite group and is a Sylow -subgroup of.
  • More generally, if a subgroup contains for some Sylow -subgroup of, then is self-normalizing, i.e..