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

TL;DR

This paper investigates the conditions under which the Fitting subgroup and soluble radical of a group are definable, establishing criteria for nilpotency and elementary class, with examples illustrating the boundaries of these properties.

Contribution

It proves that the Fitting subgroup is definable when nilpotent, classifies groups with nilpotent Fitting subgroups of bounded class as elementary, and explores similar results for the soluble radical.

Findings

Fitting subgroup is definable if nilpotent

Groups with nilpotent Fitting subgroup of class ≤ n form an elementary class

Existence of groups with definable but non-nilpotent Fitting subgroup

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.