Skip to main content
×
×
Home

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.

Copyright
References
Hide All
[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.
[4]Holland, W.C., ‘The lattice-ordered group of automorphisms of an ordered set’, Michigan Math. J. 10 (1963), 399408.
[5]Holland, W.C., ‘Group equations which hold in lattice-ordered groups’, Sympos. Math. 21 (1977), 365378.
[6]McCleary, S.H., ‘O–primitive ordered permutation groups II’, Pacific J. Math. 49 (1973), 431443.
[7]McCleary, S.H., ‘Free lattice-ordered groups represented as o–2-transitive l–permutation groups’, Trans. Amer. Math. Soc. 290 (1985), 6979.
[8]McCleary, S.H., ‘An even better representation for free lattice-ordered groups’, Trans. Amer. Math. Soc. 290 (1985), 81100.
[9]McCleary, S.H., ‘Free lattice-ordered groups’, in Lattice-ordered groups: advances and techniques, Editors Glass, A.M.W. and Holland, W.C., pp. 206227 (Kluwer Academic Publishers, Dordrecht, 1989).
[10]McDonough, T.P., ‘A permutation representation of a free group’, Quart. J. Math. Oxford Ser 2 28 (1977), 353356.
[11]Mura, R.B. and Rhemtulla, A., Orderable groups (Marcel Dekker, New York, 1977).
[12]White, S., ‘The group generated by xx + 1 and xxp is free’, J. Algebra 118 (1988), 408422.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed