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On the Countable Recognition of Finitary Groups

A stage in the classification of the periodic simple finitary linear groups by J. I. Hall and others, is a partial reduction to the countable case; specifically a simple group is isomorphic to a finitary linear group if and only if its countable subgroups are isomorphic to finitary linear groups. It is natural to ask, how common is it for a group to have a faithful finitary linear (or possibly only skew linear) representation, whenever each of its countable subgroups has such a representation? In very recent work F. Leinen and O. Puglisi have shown that periodic primitive finitary linear groups are countably recognizable, see [4]. It is easy to see that every abelian group is isomorphic to a finitary linear group, for example over the subfield of the complexes C generated by all roots of unity. Thus the first non‐obvious case, at the opposite extreme to the simple groups (and also in some sense to the primitive groups), is that of the nilpotent groups. We show the following, which, it should be noted, covers the case of periodic nilpotent groups.