Locally soluble primitive finitary skew linear groups

Abstract

Throughout this paper D denotes a division ring and V a left vector space over D. The finitary general linear group FGL(V) or FA AutDV over V is the subgroup of AutDV of D-automorphisms g of V such that [V,g] = V(g-l) has finite (left) dimension over D. By a finitary skew linear group we mean any subgroup G of FGL(V) for any D and V. Such a G is irreducible if V is irreducible as D-G (bi)module and is primitive if whenever V = ⊕ω ∊ ΩVomega as D-module, where for all g∊G and ω∊Ω, Vωg = Vω for some ω∊Ω, we have |Ω| = 1. In [4] we showed that a primitive irreducible finitary skew linear group is finite dimensional if it is hyper locally nilpotent (that is radical in the sense of Kuros) and sometimes if it is locally soluble. Here we complete the locally soluble case and, in fact, we can be a little more general.