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Highly transitive representations of free groups and free products

  • A.M.W. Glass (a1) and Stephen H. McCleary (a1)
Abstract

A permutation group is highly transitive if it is n–transitive for every positive integer n. A group G of order-preserving permutations of the rational line Q is highly order-transitive if for every α1 < … < αn and β1 < … < βn in Q there exists gG such that αig = βi, i = 1, …, n. The free group Fn(2 ≤ η ≤ אo) can be faithfully represented as a highly order-transitive group of order-preserving permutations of Q, and also (reproving a theorem of McDonough) as a highly transitive group on the natural numbers N. If G and H are nontrivial countable groups having faithful representations as groups of order-preserving permutations of Q, then their free product G * H has such a representation which in addition is highly order-transitive. If G and H are nontrivial finite or countable groups and if H has an element of infinite order, then G * H can be faithfully represented as a highly transitive group on N. Some of the representations of Fη on Q can be extended to faithful representations of the free lattice-ordered group Lη.

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Copyright
References
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[1]Dixon, J.D., ‘Most finitely generated permutation groups are free’, Bull. London Math. Soc. 22 (1990), 222226.
[2]Fuchs, L., Partially ordered algebraic systems (Pergamon Press, New York, 1963).
[3]Glass, A.M.W., ‘Free products of lattice-ordered groups’, Proc. Amer. Math. Soc. 101 (1987), 1116.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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