Frattini subgroup of a finite group is nilpotent, the
Let denote the Frattini subgroup of a finite group . Let be a Sylow subgroup of . Then by the Frattini argument, . But the Frattini subgroup is finite and formed of non-generators, so it follows that . Thus is normal in , and therefore normal in . 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 , if is finite then is nilpotent.
|Title||Frattini subgroup of a finite group is nilpotent, the|
|Date of creation||2013-03-22 13:16:44|
|Last modified on||2013-03-22 13:16:44|
|Last modified by||yark (2760)|