It is consistent with ZFC that B₁-groups are not B₂

Both, B₁-groups and B₂-groups are natural generalizations of finite rank Butler groups to the infinite rank case and it is known that every B₂-group is a B₁-group. Moreover, assuming V = L it was proven that the two classes coincide. Here we demonstrate that it is undecidable in ZFC whether or not all B₁-groups are B₂-groups. Using Cohen forcing we prove that there is a model of ZFC in which there exists a B₁-group that is not a B₂-group.