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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. Truth to be told, the argument was take by Frattini, as he himself admits, from a paper of Alfredo Capelli dated 1884.
If is a finite group with normal subgroup , and if is a Sylow p-subgroup of , then
Statement and proof
is a Sylow -subgroup of , so every Sylow -subgroup of is an -conjugate 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 therefore But was arbitrary,
- 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. .
- Hall, Marshall (1959). The theory of groups. New York, N.Y.: Macmillan. (See Chapter 10, especially Section 10.4.)