Anzahl theorems in finite singular symplectic, unitary and orthogonal geometries

Transitive sets of subspaces of a vector space over a finite field under extended symplectic, unitary and orthogonal groups are determined and their cardinals are computed. The number of subspaces in a transitive set contained in a given subspace of the ordinary symplectic, unitary or orthogonal geometries is also computed. These numbers have obvious applications in block designs. 1. The symplectic case Let F, be the finite field with q elements, where q is a power of a prime. Put K,= where The set of (2v+l) x (2v+ 1) nonsingular matrices T over Fq satisfying TK,T’= K,, (1)