On the non-existence of certain transitive actions of solvable groups on symplectic spreads

Abstract

We prove the following result. Let q be a power of an odd prime and let Sp(2m,q) denote the symplectic group of degree 2m over F_q. Then if q=1 mod 4, no solvable subgroup of Sp(2m,q) acts transitively on a complete symplectic spread defined on the underlying vector space of dimension 2m, unless m=1 and q=5. The solvable group Sp(2,3) of order 24 acts transitively on a complete symplectic spread defined on a two-dimensional vector space over F_5. By contrast, when q=3 mod 4, there is a metacyclic group of order 2m(q^m+1) that acts transitively on a complete symplectic spread.