Simple automorphism groups of cycle-free partial orders

Abstract

The purpose of this paper is to show that the automorphism groups of many of the ‘cycle-free’ partial orders studied in [Warren, R.: The structure of k-CS-transitive cycle-free partial orders. Memoirs of the American Mathematical Society (1997), to appear] and [Creed, P., Truss, J. K. and Warren, R.: The structure of k-CS-transitive cycle-free partial orders having infinite chains, to appear] are simple. This contrasts strongly with the situation for trees, of which they form a natural generalization. It was shown in [Droste, M., Holland, W.C. and Macpherson, H.D.: Automorphism groups of infinite semilinear orders (I) and (II). Proc. London Math. Soc. 58 (1989), 454–478 and 479–494] that the automorphism group of any sufficiently transitive tree has at least normal subgroups. All the infinite chain cycle-free partial orders studied in [Creed, P., Truss, J. K. and Warren, R.: The structure of k-CS-transitive cycle-free partial orders having infinite chains, to appear] have simple automorphism groups. The finite chain case is more involved; where the ordering on chains of the Dedekind-MacNeille completion can be expressed as a lexicographic product by a non-trivial discrete (transitive) ordering (respected by the group), the automorphism group is not simple. For both finite and infinite chain cases the simple automorphism groups split into two classes: those where there is a bound (≤ 2) on the number of conjugates required to express one non-identity element in terms of another, and those in which there is no such bound.