A characterization of ¯nite symplectic polar spaces of odd prime order

Abstract

A su±cient condition for the representation group for a nonabelian representation (De¯nition 1.1) of a ¯nite partial linear space to be a ¯nite p-group is given (Theorem 2.9). We characterize ¯nite symplectic polar spaces of rank r at least two and of odd prime order p as the only ¯nite polar spaces of rank at least two and of prime order admitting nonabelian representations. The representation group of such a polar space is an extraspecial p-group of order p1+2r and of exponent p (Theorems 1.5 and 1.6).