A note on unbounded strongly measure zero subgroups of the Baer–Specker group

We show that it is consistent with ZFC that there exist:(1)An unbounded (with respect to ⩽∗) and strongly measure zero subgroup of ZN, but without the Menger property.(2)An unbounded (with respect to ⩽∗) and strongly measure zero subgroup of ZN with the Menger property which does not have the Rothberger property. This answers the last two problems which remained from a classification project of Machura and Tsaban.