Fitting’s theorem

Fitting’s Theorem states that if $G$ is a groupand $M$ 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 tothe 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.