Fitting’s theorem
Fitting’s Theorem states that if is a groupand and are normal nilpotent subgroups (http://planetmath.org/Subgroup) of ,then is also a normal nilpotent subgroup(of nilpotency class less than or equal tothe 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.