Fitting’s theorem

Fitting’s Theorem states that if $G$ is a group and $M$ and $N$ are normal nilpotent subgroups (http://planetmath.org/Subgroup) of $G$, then $MN$ is also a normal nilpotent subgroup (of nilpotency class less than or equal to the sum of the nilpotency classes of $M$ and $N$).

Thus, any finite group has a unique largest normal nilpotent subgroup, called its Fitting subgroup. More generally, the Fitting subgroup of a group $G$ is defined to be the subgroup of $G$ generated by the normal nilpotent subgroups of $G$; Fitting’s Theorem shows that the Fitting subgroup is always locally nilpotent. A group that is equal to its own Fitting subgroup is sometimes called a Fitting group.

Title	Fitting’s theorem
Canonical name	FittingsTheorem
Date of creation	2013-03-22 13:51:39
Last modified on	2013-03-22 13:51:39
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	12
Author	yark (2760)
Entry type	Theorem
Classification	msc 20D25
Defines	Fitting subgroup
Defines	Fitting group