Simple automorphism groups of cycle-free partial orders

M. Droste 1 , J. K. Truss 2 ,  and R. Warren 2
  • 1 M. Droste, Institut für Algebra, Technische Universität Dresden, D-01062 Dresden, Germany. e-mail: droste@math.tu-dresden.de
  • 2 J. K. Truss and R. Warren, Department of Pure Mathematics, University of Leeds, Leeds LS2 9JT England. e-mail: pmtjkt@amsta.leeds.ac.uk and RDCWarren@compuserve.com

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.

Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


or
Log in with your institution

Journal + Issues

Forum Mathematicum is a general mathematics journal, which is devoted to the publication of research articles in all fields of pure and applied mathematics, including mathematical physics. Forum Mathematicum belongs to the top 50 journals in pure and applied mathematics, as measured by citation impact.

Search