On a Question of Ross and Stromberg

A topological group G is said an [AFG] -group if G contains an increasing sequence of finite subgroups having dense union.In this paper it is proved that the identity component G 0 of a locally compact [AFG] -group G is a pro-torus.This partially answers an old open question posed by Ross and Stromberg in Pacific J. Math., 20 (1967), 135-147.Other results included in this paper give a necessary and sufficient condition for an almost connected Lie group to be an [AFG]-group.