Fitting's Theorem states that if is a group and and are normal nilpotent subgroups of , then is also a normal nilpotent subgroup (of nilpotency class less than or equal to the sum of the nilpotency classes of and ).

Thus, any finite group has a unique largest normal nilpotent subgroup, called its Fitting subgroup. More generally, the Fitting subgroup of a group is defined to be the subgroup of generated by the normal nilpotent subgroups of ; 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.