On the generalized Fitting group of locally finite, finitary groups

Abstract

Let G be a locally finite, finitary group and F (G) the group generated by the Hirsch-Plotkin radical of G and the components of G. Our main theorem asserts that CG (F (G)) F (G). 1 Introduction The main purpose of this paper is to extend the concept of the generalized Fitting group from finite groups to locally finite, finitary groups. Recall that a group G is called locally finite if every finite subset of G lies in a finite subgroup. G is called finitary if there exist a field K and a faithful KG-module V so that [V; g] is finite dimensional for all g 2 G. G is quasi-simple if it is perfect, and G=Z(G) is simple. A component of G is a non-trivial quasi-simple, subnormal subgroup of G. The layer E(G) is defined as the group generated by the components of G. It is easy to see that distinct components of a group commute. Thus E(G)=Z(E(G)) is semisimple, that is the ( restricted) direct product of simple groups. Assume for the moment that G is finite. Then the Fitting group..