A remark on the definability of the Fitting subgroup and the soluble radical
Let G be an arbitrary group. We show that if the Fitting subgroup of G is nilpotent then it is definable. We prove also that the class of groups whose Fitting subgroup is nilpotent of class at most n is elementary. We give an example of a group (arbitrary saturated) whose Fitting subgroup is definable but not nilpotent. Similar results for the soluble radical are given; that is for the subgroup generated by all normal soluble subgroups of G.
definable fitting subgroup group soluble radical. msc (2010) 20f14 20f16 20f17 20f18 20f19 03cxx