Birational Automorphism Groups of Severi-Brauer Surfaces Over the Field of Rational Numbers

We prove that the only non-trivial finite subgroups of birational automorphism group of non-trivial Severi-Brauer surfaces over the field of rational numbers are Z/3Z and (Z/3Z)(2). Moreover, we show that (Z/3Z)(2) is contained in Bir(V) for any Severi-Brauer surface V over a field of characteristic different from 2 and 3, and (Z/3Z)(3) is contained in Bir(V) for any Severi-Brauer surface V over a field of characteristic different from 2 and 3, which contains a non-trivial cube root of unity.