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