Finite $$p$$-Groups of Birational Automorphisms and Characterizations of Rational Varieties

We prove that there exists a constant $$R(n)$$ such that a rationally connected variety of dimension $$n$$ over an algebraically closed field is rational if its birational automorphism group contains a $$p$$ -subgroup of maximal rank for some prime number $$p>R(n)$$ . This answers a question of Prokhorov and Shramov. Some applications related to the Jordan property are discussed.