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

## TL;DR

This paper investigates the conditions under which solvable groups can act transitively on symplectic spreads, proving non-existence results for most cases and identifying specific exceptions based on the prime power q.

## Contribution

It establishes new non-existence results for transitive actions of solvable groups on symplectic spreads, with explicit exceptions for certain parameters q and m.

## Key findings

- No solvable subgroup acts transitively when q ≡ 1 mod 4, except for (m,q) = (1,5).
- The solvable group Sp(2,3) acts transitively on a symplectic spread over F_5.
- When q ≡ 3 mod 4, a specific metacyclic group acts transitively on the spread.

## 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.