ANZAHL THEOREMS IN FINITE SINGULAR SYMPLECTIC, UNITARY AND ORTHOGONALGEOMETRIES

Transitive sets of subspaces of a vector space over a finite field und er extended symplectic, unitary and orthogonal groups are determined a nd their cardinals are computed. The number of subspaces in a transiti ve set contained in a given subspace of the ordinary symplectic, unita ry or orthogonal geometries is also computed. These numbers have obvio us applications in block designs.