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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.
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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

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