Publication Date:
2016-07-29
Description:
In this paper, we consider the group $ \hbox {Aut}( \mathbb {Q}, \leq )$ 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 $ \hbox {Aut}( \mathbb {Q}, \leq )$ is contained in an $N$ -generated subgroup of $ \hbox {Aut}( \mathbb {Q}, \leq )$ for some fixed $N\in \mathbb {N}$ . We show that the least such $N$ is 2. Moreover, for every countable subset of $ \hbox {Aut}( \mathbb {Q}, \leq )$ , 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,\ldots $ over $\{a,b\}$ such that for every sequence $f_1, f_2, \ldots \in \hbox {Aut}( \mathbb {Q}, \leq )$ there is a homomorphism $\phi :\{a,b\}^{+}\to \hbox {Aut}( \mathbb {Q}, \leq )$ 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 $ \hbox {Aut}( \mathbb {Q}, \leq )$ is uncountable, or equivalently that $ \hbox {Aut}( \mathbb {Q}, \leq )$ has uncountable cofinality and Bergman's property.
Print ISSN:
0024-6107
Electronic ISSN:
1469-7750
Topics:
Mathematics
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