Birational automorphism groups of Severi–Brauer surfaces over the field of rational numbers

In this talk we will discuss finite subgroups in the group of birational automorphisms of Severi–Brauer surfaces. We will prove that the only non-trivial finite subgroups of birational automorphisms group of a non-trivial Severi–Brauer surfaces over the field of rational numbers are $\mathbb{Z}/3\mathbb{Z}$ and $(\mathbb{Z}/3\mathbb{Z})^2.$ Also we will discuss $3$-subgroups in the birational automorphisms group for a Severi–Brauer surface over any field of characteristic zero.