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].