Residually finite subgroups of some countable McLain groups

The McLain groups are groups of finitary linear transformations that, besides being of considerable interest in their own right, constitute a most useful source of counterexamples to conjectures arising in the investigation of locally nilpotent groups. In this article it is shown that certain torsion-free McLain groups have subgroups “of periodic index” that are residually finite. This has a bearing on the question as to which McLain groups might embed in simple locally soluble-by-finite groups.