# Birational automorphisms of Severi-Brauer surfaces

## TL;DR

This paper investigates the structure of finite groups acting birationally on non-trivial Severi-Brauer surfaces, establishing bounds on their composition and providing explicit order limits when roots of unity are present.

## Contribution

It proves that such groups have a normal abelian subgroup of index at most 3 and derives explicit bounds for their orders over fields containing all roots of unity.

## Key findings

- Finite groups have a normal abelian subgroup of index ≤ 3.
- Explicit bounds for group orders are provided when the field contains all roots of unity.
- The results apply to non-trivial Severi-Brauer surfaces over characteristic zero fields.

## Abstract

We prove that a finite group acting by birational automorphisms of a non-trivial Severi-Brauer surface over a field of characteristic zero contains a normal abelian subgroup of index at most 3. Also, we find an explicit bound for orders of such finite groups in the case when the base field contains all roots of 1.