Frattini's argument

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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.[1]

Frattini's Argument[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.

Statement and proof[edit]

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

so

thus

and therefore But was arbitrary,

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.)