Representations of Finite Groups by Posets of Small Height

We characterize the finite groups as the automorphism groups of the finite height one posets with at most four orbits. We also prove that for each n ≥ 8 , the cyclic group Z n is isomorphic to the automorphism group of a finite height one poset with at most two orbits. As a consequence, for each n , we determine the minimum size of the posets whose automorphism groups are isomorphic to Z n .