Existence of Steiner systems that admit automorphisms with large cycles

For every integer k ≥ 2, we construct infinite families of Steiner systems S(2, k, ν) that have (1) an automorphism that permutes the v points in a single-cycle of length ν, (2) an automorphism that fixes one point and permutes the remaining ν – 1 points in a single cycle, or (3) an automorphism that fixes one point and permutes the remaining points in two cycles, of lengths r = (ν - 1)/(k – 1) and ν – r – 1. The designs we construct with property (2) are also resolvable.