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Groups of order-automorphisms of the rationals with prescribed scale type. (English) Zbl 0681.20003

A group G of order-preserving permutations has scale type (m,n) if it is m-transitive and uniquely n-transitive. The largest m and smallest n, if they exist, are the degrees of homogeneity and uniqueness, respectively. One says \(n=\infty\) if every nonidentity permutation fixes only finitely many points. It is proved that, given \(m<n\leq \infty\), there is a group G of order-preserving permutations of the rationals with scale type (m,n). It is striking that the free group of countable rank acts on a countable set in such a way that it is m-transitive for every m and its degree of uniqueness is \(\infty.\)
Let G be a permutation group on a countably infinite set X satisfying n- point uniqueness. The author defines a \((2n+2)\)-ary relation R as follows. Let \(O_ 1,O_ 2,..\). be the orbits of G on \((n+1)\)-tuples of distinct points. Then a \((2n+2)\)-tuple \((x_ 0,...,x_ n,y_ 0,...,y_ n)\) satisfies R if and only if there exists i such that \((x_ 0,...,x_ n)\) is in \(O_ i\) and \((y_ 0,...,y_ n)\) is in \(O_{i+1}\). Then \(G=Aut(X,R)\). The author then proves: Let \({\mathfrak M}\) be a countable homogeneous relational structure, G its automorphism group and \({\mathfrak C}\) the class of all structures isomorphic to finite substructures of \({\mathfrak M}\). Then the following are equivalent: 1. \({\mathfrak C}\) has the strong amalgamation property. 2. The stabilizer in G of a finite tuple has no additional fixed points. 3. The stabilizer in G of a finite tuple has no additional finite orbits. Further, in these cases, for any positive integer n, there is a subgroup H of Aut(\({\mathfrak M})\) such that H is a free group of countable rank and the scale type of H is \((n-1,n).\) The results and techniques will be of wide interest to those interested in ordered algebraic structures.
Reviewer: S.P.Hurd

MSC:

20B27 Infinite automorphism groups
20B22 Multiply transitive infinite groups
06A06 Partial orders, general
06F15 Ordered groups
08A30 Subalgebras, congruence relations
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
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