# 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 FittingsTheorem 2013-03-22 13:51:39 2013-03-22 13:51:39 yark (2760) yark (2760) 12 yark (2760) Theorem msc 20D25 Fitting subgroup Fitting group