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′ ⩽ G ⩽ Aut G′. Moreover if 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.