Homomorphic images of R-factorizable groups

It is well known that every R-factorizable group is omega-narrow, but not vice versa. One of the main problems regarding R-factorizable groups is whether this class of groups is closed under taking continuous homomorphic images or, alternatively, whether every omega-narrow group is a continuous homomorphic image of an R-factorizable group. Here we show that the second hypothesis is definitely false. This result follows from the theorem stating that if a continuous homomorphic image of an R-factorizable group is a P-group, then the image is also R-factorizable.