## Abstract

In this paper, we consider the group Aut(Q,≤) of order-automorphisms of the rational numbers, proving a result analogous to a theorem of Galvin's for the symmetric group. In an announcement, Khélif states that every countable subset of Aut(Q,≤) is contained in an

As a corollary to the main theorem in this paper, we obtain a result of Droste and Holland showing that the strong cofinality of Aut(Q,≤) is uncountable, or equivalently that Aut(Q,≤) has uncountable cofinality and Bergman's property.

*N*-generated subgroup of Aut(Q,≤) for some fixed*N*∈ N. We show that the least such*N*is 2. Moreover, for every countable subset of Aut(Q,≤), we show that every element can be given as a prescribed product of two generators without using their inverses. More precisely, suppose that a and b freely generate the free semigroup {a,b}^{+}consisting of the non-empty words over a and b. Then we show that there exists a sequence of words w_{1}, w_{2},... over {a,b} such that for every sequence f_{1}, f_{2}, ... ∈ Aut(Q,≤) there is a homomorphism φ : {a,b}^{+}→ Aut(Q,≤) where (w_{i})φ=f_{i}for every i.As a corollary to the main theorem in this paper, we obtain a result of Droste and Holland showing that the strong cofinality of Aut(Q,≤) is uncountable, or equivalently that Aut(Q,≤) has uncountable cofinality and Bergman's property.

Original language | English |
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Pages (from-to) | 21-37 |

Number of pages | 17 |

Journal | Journal of the London Mathematical Society |

Volume | 94 |

Issue number | 1 |

Early online date | 13 May 2016 |

DOIs | |

Publication status | Published - Aug 2016 |