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.[1]
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 (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
- 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
References
- M. Brescia, F. de Giovanni, M. Trombetti, "The True Story Behind Frattini’s Argument", Advances in Group Theory and Applications 3, doi:10.4399/97888255036928
- Hall, Marshall (1959). The theory of groups. New York, N.Y.: Macmillan. (See Chapter 10, especially Section 10.4.)