The subject of my paper belongs to the field of mathematics, specifically group theory. I study a fundamental group of some three-dimensional manifold, which is a space similar to our 3D space with three perpendicular axis, however more complicated. This group is the group of all the braids on three strands. It can be studied purely algebraically. The braid group is rather complicated, so proving properties of this group directly can also be quite complex. We can therefore study a reasonably large subgroups of this group in order to get the properties of the whole group in the sequel. The most important subgroup of this type is given by considering only braids, that does not change the order of strands.