Droste, Manfred; Giraudet, Michèle; Macpherson, Dugald Periodic ordered permutation groups and cyclic orderings. (English) Zbl 0821.20001 J. Comb. Theory, Ser. B 63, No. 2, 310-321 (1995). It is shown that periodic ordered permutation groups satisfying certain extra conditions are very nearly simple. This is applied to several natural examples, such as the following. (i) If \(z\) denotes the map \(x \mapsto x + 1\) on \(\mathbb{R}\), and \(\text{Diff}(\mathbb{R})\) is the group of diffeomorphisms of \(\mathbb{R}\), then \(C_{\text{Diff}(\mathbb{R})}(\langle z \rangle)/\langle z \rangle\) is simple. (ii) The automorphism group of the countable homogeneous local order (one of the three countable homogeneous tournaments) is simple. Also, an extension of Rieger’s Theorem is given, relating groups of automorphisms of cyclic orderings to groups of automorphisms of total orderings. Reviewer: M.Droste (Dresden) Cited in 4 Documents MSC: 20B27 Infinite automorphism groups 20E32 Simple groups 06F15 Ordered groups 20F50 Periodic groups; locally finite groups 20F60 Ordered groups (group-theoretic aspects) 20B07 General theory for infinite permutation groups Keywords:periodic ordered permutation groups; groups of diffeomorphisms; automorphism groups; countable homogeneous local orders; countable homogeneous tournaments; cyclic orderings; groups of automorphisms; total orderings PDFBibTeX XMLCite \textit{M. Droste} et al., J. Comb. Theory, Ser. B 63, No. 2, 310--321 (1995; Zbl 0821.20001) Full Text: DOI