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Wielandt-Kegel theorem (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, 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 and Sylow $q$-subgroup, both of which are nilpotent.



"Wielandt-Kegel theorem" is owned by yark.
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Other names:  Kegel-Wielandt theorem
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Cross-references: primes, order, group, solvable, subgroups, nilpotent, finite group

This is version 8 of Wielandt-Kegel theorem, born on 2006-10-02, modified 2006-10-29.
Object id is 8412, canonical name is WielandtKegelTheorem.
Accessed 874 times total.

Classification:
AMS MSC: 20D10 (Group theory and generalizations :: Abstract finite groups :: Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks)
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