Metrizable groups and strict o-boundedness

We show that for metrizable topological groups being a strictly o-bounded group is equivalent to being a Hurewicz group. In [5] Hernandez, Robbie and Tkachenko ask if there are strictly o-bounded groups G and H for which G x H is not strictly o-bounded. We show that for metrizable strictly o-bounded groups the answer is no. In the same paper the authors also ask if the product of an o-bounded group with a strictly o-bounded group is again an o-bounded group. We show that if the strictly o-bounded group is metrizable, then the answer is yes.