Automorphisms of unitary block designs

In this paper, we discuss the geometry and determine the automorphism group of the unital or unitary block design associated with the threedimensional unitary group. If V is a three-dimensional vector space over the field with q2 elements and b is a non degenerate Hermitian bilinear form on V, we let X denote the family of isotropic one-dimensional subs paces of V with respect to b. Then X has 1 + q3 points, and the three-dimensional projective unitary groups, PSU(3, q) and PGU(3, q), act on X as doubly-transitive permutation groups. There is a naturally arising family of subsets of X, d, forming a unitary block design on X. Each member of d is the set of isotropic one-dimensional subs paces contained in a fixed nonisotropic two-dimensional subspace of V. Our principal result is: