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Universal sequences for the order-automorphisms of the rationals
- Publication Year :
- 2014
-
Abstract
- In this paper, we consider the group 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, Kh\'elif states that every countable subset of Aut$(\mathbb{Q}, \leq)$ is contained in an $N$-generated subgroup of 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 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 $f_1, f_2, \ldots\in\,$Aut$(\mathbb{Q}, \leq)$ there is a homomorphism $\phi:\{a,b\}^{+}\to$ 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 Aut$(\mathbb{Q}, \leq)$ is uncountable, or equivalently that Aut$(\mathbb{Q}, \leq)$ has uncountable cofinality and Bergman's property.<br />Comment: Updated to clarify some parts of the proof
- Subjects :
- Discrete mathematics
Rational number
General Mathematics
20B27, 20B07 (primary) 20E15 (secondary)
010102 general mathematics
T-NDAS
Order (ring theory)
Group Theory (math.GR)
Automorphism
01 natural sciences
Mathematics::Group Theory
0103 physical sciences
FOS: Mathematics
010307 mathematical physics
QA Mathematics
0101 mathematics
BDC
QA
Mathematics - Group Theory
Group theory
R2C
Mathematics
Subjects
Details
- Language :
- English
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....ed38bd0b520c8befd7b9eb5bca8139f6