A characterization of finite symplectic polar spaces of odd prime order
Let \(p\) be a prime and let \(S = (P,L)\) be a partial linear space such that each line of \(S\) contains \(p+1\) points. A representation of \(S\) consisists of a group \(R\) and a mapping \(\psi\) from \(P\) to the set of subgroups of order \(p\) of \(R\) such that \(R\) is generated by the subgroups \(\psi(x)\), \(x \in P\), and for each line \(l \in L\) the subgroups \(\psi(x)\), \(x \in l\), are distinct and generate an elementary abelian subgroup of order \(p^ 2\). The authors show that if \(S\) is a finite polar space of rank \(r \geq 2\) and \(R\) is nonabelian, then \(p\) is odd, \(S\) is isomorphic to the symplectic polar space \(W_{2r}(p)\), and \(R\) is an extraspecial \(p\)-group of order \(p^{1+2r}\) and exponent \(p\). Moreover, the representation is unique.
scientific article; zbMATH DE number 5080377
scientific article