PB∞-subgroups of Butler groups

It is well known [BF] that in the constructible universe (V=L) the class ofB ₁-groups is closed under prebalanced subgroups. A similar attempt at the increasing countable unions of prebalanced subgroups has been done in [B]. Here we give an affirmative answer to a similar question concerning the so-calledP B ^∞-subgroups. This research was initiated while the first author was on a short visit at the University of Colorado, Colorado Springs, in February 1994 and supported by Charles University grant GAUK 45/94.