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 first used it in a paper from 1885 when defining the Frattini subgroup of a group.

tatement and proof

Frattini's argument states that if a finite group "G" has a normal subgroup "H" and if "H" has a Sylow "p"-subgroup "P" then

:"G" = "N"_"G"("P")"H",

where "N"_"G"("P") denotes the normalizer of "P" in "G".

Proof: "P" is a Sylow "p"-subgroup of "H", so its "H"-conjugates "h"⁻¹"Ph" are also Sylow "p"-subgroups belonging to "H". "H" is normal in G, so the action of a "g" ∈ "G" by conjugacy sends "P" in "H" to one of its "H"-conjugates (see Sylow theorems), i.e.

:"g"⁻¹"Pg" = "h"⁻¹"Ph",

so :"hg"⁻¹"Pgh"⁻¹ = "P", thus

:"gh"⁻¹ ∈ "N"_"G"("P"),

therefore "g" ∈ "N"_"G"("P")"H". But "g" ∈ "G" was arbitrary, so "G" = "HN"_"G"("P") = "N"_"G"("P")"H". $square$

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 "N"_"G"("N"_"G"("P")), it can be shown that "N"_"G"("N"_"G"("P")) = "N"_"G"("P") whenever "G" is a finite group and "P" is a Sylow "p"-subgroup of "G".
* More generally, if a subgroup "M" ≤ "G" contains "N"_"G"("P") for some Sylow "p"-subgroup "P" of "G", then "M" is self-normalizing, "i.e." "M" = "N"_"G"("M").::Proof: "M" is normal in "H" := "N"_"G"("M"), and "P" is a Sylow "p"-subgroup of "M", so the Frattini argument applied to the group "H" with normal subgroup "M" and Sylow "p"-subgroup "P" gives "N"_"H"("P")"M" = "H". Since "N"_"H"("P") ≤ "N"_"G"("P") ≤ "M", one has the chain of inclusions "M" ≤ "H" = "N"_"H"("P")"M" ≤ "M" "M" = "M", so "M" = "H". $square$