Symmetric designs and projective special linear groups of dimension at most four

In this article, we study symmetric ( v , k , lambda ) designs admitting a flag-transitive and point-primitive automorphism group G whose socle is isomorphic to a projective special linear group of dimension at most four. We, in particular, determine all such possible parameters ( v , k , lambda ) and show that such a design belongs to one of two infinite families of point-hyperplane designs or it is isomorphic to a design with parameters (7, 3, 1), (7, 4, 2), (11, 5, 2), (11, 6, 3), (15, 8, 4), (35, 18, 9), or (56, 45, 18).