Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-07-08T01:31:04.055Z Has data issue: false hasContentIssue false
  • English
  • Français

Commutators in groups of order-preserving permutations

Published online by Cambridge University Press:  18 May 2009

Manfred Droste  and
R. M. Shortt
Manfred Droste
Affiliation:
Fachbereich 6-Mathematik, Universität GHS Essen, 4300 Essen 1, Germany
R. M. Shortt
Affiliation:
Department of Mathematics, Wesleyan University, Middletown, Connecticut 06457, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let (S, ≤) be a poset (partially ordered set), A(S) = Aut(S, ≤) its automorphism group and G ⊆ A(S) a subgroup. In the literature, various authors have studied sufficient conditions on G and the structure of (S, ≤) which imply that G is simple or perfect. Let us call (S, ≤) doubly homogeneous if each isomorphism between two 2-subsets of 5 extends to an isomorphism of (S, ≤). Higman [8] proved that if (S, ≤) is a doubly homogeneous chain then B(S), the group of all automorphisms of (S, ≤) with bounded support, is simple, and each element of B(S) is a commutator in B(S). Droste, Holland and Macpherson [5] showed that if (S, ≤) is a doubly homogeneous tree then its automorphism group again contains a unique simple normal subgroup in which each element is a commutator. Dlab [3] established similar results for various groups of locally linear automorphisms of the reals. Further results in this direction are contained in Glass [7]. It is the aim of this note to establish a common generalization and sharpening of the previously mentioned results.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 33 , Issue 1 , January 1991 , pp. 55 - 59
Copyright
Copyright © Glasgow Mathematical Journal Trust 1991

References

REFERENCES

1.Adeleke, S. and Neumann, P. M., On infinite Jordan groups, manuscript, 1987.Google Scholar
2.Anderson, R. D., The algebraic simplicity of certain groups of homeomorphisms, Amer. J. Math. 80 (1958), 955–963.Google Scholar
3.Dlab, V., On a family of simple ordered groups, J. Austral. Math. Soc. 8 (1968), 591–608.Google Scholar
4.Droste, M., Structure of partially ordered sets with transitive automorphism groups, Memoirs Amer. Math. Soc. 334 (1985).Google Scholar
5.Droste, M., Holland, W. C. and Macpherson, H. D., Automorphism groups of infinite semilinear orders (I), Proc. London Math. Soc. (3) 58 (1989), 454–478.Google Scholar
6.Droste, M., Holland, W. C. and Macpherson, H. D., Automorphism groups of infinite semilinear orders (II), Proc. London Math. Soc. (3) 58 (1989), 479–494.CrossRefGoogle Scholar
7.Glass, A. M. W., Ordered permutation groups, London Math. Soc. Lecture Note Series 55 (Cambridge University Press, 1981).Google Scholar
8.Higman, G., On infinite simple permutation groups, Publ. Math. Debrecen 3 (1954), 221–226.Google Scholar
9.Maroli, J., Tree permutation groups (Ph.D. thesis, Bowling Green State University, Bowling Green, Ohio, 1989).Google Scholar
10.McCleary, S. H., Some simple homeomorphism groups having nonsolvable outer automorphism groups, Comm. Algebra 6 (1978), 483–496.Google Scholar