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Periodic ordered permutation groups and cyclic orderings. (English) Zbl 0821.20001

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)

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
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