LOCALLY FINITE FINITARY SKEW LINEAR GROUPS

Abstract

Let $V$ be a vector space over the division ring $D$ of infinite dimension. We study locally finite, primitive groups $G$ of finitary linear automorphisms of $V$. We show that the derived group $G'$ of $G$ is infinite, simple, and lies in every non-trivial normal subgroup of $G$, and that $G' \le G \le {\rm Aut\,}G'$. Moreover if ${\rm char\,}D = 0$, then $G$ is either the finitary symmetric group or the alternating group on some infinite set. If $D$ is commutative, that is, if $D$ is a field, then all these results are known and are the consequence of the collective work of a number of people: in particular (in alphabetical order) V.~V.~Belyaev, J.~I.~Hall, F.~Leinen, U.~Meierfrankenfeld, R.~E.~Phillips, O.~Puglisi, A.~Radford and quite probably others. 2000 Mathematics Subject Classification: 20H25, 20H20, 20F50.