A determinant on birational maps of Severi–Brauer surfaces

We define a determinant on the group of automorphisms of non-trivial Severi–Brauer surfaces over a perfect field. Using the generators and relations, we extend this determinant to birational maps between Severi–Brauer surfaces. By means of this determinant and a group homomorphism found in [BSY24], we can determine the abelianization of the group of birational transformations of a non-trivial Severi–Brauer surface. This is the first example of an abelianization of the group of birational transformations of a geometrically rational surface where the automorphisms are non-trivial. Using the abelianization, we find maximal subgroups of the group of birational transformations of a non-trivial Severi–Brauer surface over a perfect field.