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


  • Abderezak Ould Houcine

    1. Mathematisches Institut und Institut für Mathematische Logik und Grundlagenforschung, Fachbereich Mathematik und Informatik, Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany
    2. École Centrale de Lyon, 6 avenue Guy de Collongue, 69134 Ecully cedex, France
    3. Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 blvd du 11 novembre 1918, 69622 Villeurbanne cedex, France
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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.