Universal sequences for the order-automorphisms of the rationals

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, Khelif states that every countable subset of Aut(Q, <=) is contained in an N-generated subgroup of Aut(Q, <=) for some fixed N is an element of 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), ... is an element of Aut(Q, <=) there is a homomorphism phi : {a, b}(+) -> Aut(Q, <=) where (w(i)) phi = 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.