A Geometric Characterization of Rational Groups

We give a geometric characterization of finite rational groups. In particular, we prove that a finite group is rational if and only if there exists a finite geometry $\mathrm{\Gamma }$ of type $I$ and action of $G$ on $\mathrm{\Gamma }$ as a group of automorphisms such that if $g$ and $h$ are elements of $G$ fixing the same number of flags of type $J$ for all subsets $J$ of $I$, then $g$ and $h$ are conjugate in $G$.