  We study finite $p$-subgroups of birational automorphism groups. By virtue of
boundedness theorem of Fano 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 $p$-subgroups of maximal rank for $p > R(n)$. Some related
applications on Jordan property are discussed.

Algebraic Geometry