Fitting's Theorem states that if $G$ is a group and $M$ and $N$ are normal nilpotent subgroups 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.