3-designs from PGL(2, q)

The group PGL(2, q), q = pⁿ, p an odd prime, is 3-transitive on the projective line and therefore it can be used to construct 3-designs. In this paper, we determine the sizes of orbits from the action of PGL(2, q) on the k-subsets of the projective line when k is not congruent to 0 and 1 modulo p. Consequently, we find all values of λ for which there exist 3-(q + 1, k, λ) designs admitting PGL(2, q) as automorphism group. In the case p = 3 (mod 4), the results and some previously known facts are used to classify 3-designs from PSL(2, p) up to isomorphism.