1. Introduction

A class [Xscr ] of groups is said to be countably recognizable, if every group all of whose countable subgroups are contained in countable [Xscr ]-subgroups is itself an [Xscr ]-group. Many examples of such classes are discussed in section 8·3 of [20]. In the present work we are concerned with the question of how far countable recognizability can be obtained for classes of finitary linear groups. Recall that a group is said to be finitary [ ]-linear if it is isomorphic to a subgroup of FGL[ ](V), the group of all invertible [ ]-linear transformations α of the [ ]-vector space V with the property that the image of the endomorphism α−idV has finite [ ]-dimension. This generalizes the notion of linearity. A survey about features of finitary linear groups is given in [18].