Wielandt-Kegel theorem

Theorem.

If a finite group is the product of two nilpotent subgroups, then it is solvable.

That is, if $H$ and $K$ are nilpotent subgroups of a finite group $G$, and $G=HK$, then $G$ is solvable.

This result can be considered as a generalization of Burnside’s $p$-$q$ Theorem (http://planetmath.org/BurnsidePQTheorem), because if a group $G$ is of order $p^m{q}^n$, where $p$ and $q$ are distinct primes, then $G$ is the product of a Sylow $p$-subgroup (http://planetmath.org/SylowPSubgroup) and Sylow $q$-subgroup, both of which are nilpotent.

Title	Wielandt-Kegel theorem
Canonical name	WielandtKegelTheorem
Date of creation	2013-03-22 16:17:37
Last modified on	2013-03-22 16:17:37
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	11
Author	yark (2760)
Entry type	Theorem
Classification	msc 20D10
Synonym	Kegel-Wielandt theorem