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 asa 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.

单词	WielandtKegelTheorem
释义	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 asa 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.
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