Frattini subgroup of a finite group is nilpotent, the


The Frattini subgroupMathworldPlanetmath of a finite groupMathworldPlanetmath is nilpotent (http://planetmath.org/NilpotentGroup).

Proof.

Let Φ(G) denote the Frattini subgroup of a finite group G. Let S be a Sylow subgroup of Φ(G). Then by the Frattini argument, G=Φ(G)NG(S)=Φ(G)NG(S). But the Frattini subgroup is finite and formed of non-generators, so it follows that G=NG(S)=NG(S). Thus S is normal in G, and therefore normal in Φ(G). The result now follows, as any finite group whose Sylow subgroups are all normal is nilpotent (http://planetmath.org/ClassificationOfFiniteNilpotentGroups). ∎

In fact, the same proof shows that for any group G, if Φ(G) is finite then Φ(G) is nilpotent.

Title Frattini subgroup of a finite group is nilpotent, the
Canonical name FrattiniSubgroupOfAFiniteGroupIsNilpotentThe
Date of creation 2013-03-22 13:16:44
Last modified on 2013-03-22 13:16:44
Owner yark (2760)
Last modified by yark (2760)
Numerical id 16
Author yark (2760)
Entry type Theorem
Classification msc 20D25