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.[1]

Frattini's argument edit

Statement edit

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 edit

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   (see Sylow theorems). 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 edit

  • 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.  .

External links edit

References edit

  • Hall, Marshall (1959). The theory of groups. New York, N.Y.: Macmillan. (See Chapter 10, especially Section 10.4.)