A remark on the definability of the Fitting subgroup and the soluble radical

Abstract

Let $G$ be an arbitrary group. We show that if the Fitting subgroup of $G$ is nilpotent then it is definable. We show 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.