Irreducible locally nilpotent finitary skew linear groups

Abstract

Let V be a left vector space over the arbitrary division ring D and G a locally nilpotent group of finitary automorphisms of V (automorphisms g of V such that dim D V ( g -1)<∞) such that V is irreducible as D-G bimodule. If V is infinite dimensional we show that such groups are very rare, much rarer than in the finite-dimensional case. For example we show that if dim D V is infinite then dim D V = |G| = ℵ 0 and G is a locally finite q -group for some prime q ≠ char D . Moreover G is isomorphic to a finitary linear group over a field. Examples show that infinite-dimensional such groups G do exist. Note also that there exist examples of finite-dimensional such groups G that are not isomorphic to any finitary linear group over a field. Generally the finite-dimensional examples are more varied.