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Generating the infinite symmetric group using a closed subgroup and the least number of other elements


Authors: J. D. Mitchell, M. Morayne and Y. Péresse
Journal: Proc. Amer. Math. Soc. 139 (2011), 401-405
MSC (2010): Primary 20B07; Secondary 54H11
DOI: https://doi.org/10.1090/S0002-9939-2010-10694-9
Published electronically: September 21, 2010
MathSciNet review: 2736324
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Abstract: Let $ S_{\infty}$ denote the symmetric group on the natural numbers $ \mathbb{N}$. Then $ S_{\infty}$ is a Polish group with the topology inherited from $ \mathbb{N}^{\mathbb{N}}$ with the product topology and the discrete topology on $ \mathbb{N}$. Let $ \mathfrak{d}$ denote the least cardinality of a dominating family for $ \mathbb{N}^{\mathbb{N}}$ and let $ \mathfrak{c}$ denote the continuum. Using theorems of Galvin, and Bergman and Shelah we prove that if $ G$ is any subgroup of $ S_{\infty}$ that is closed in the above topology and $ H$ is a subset of $ S_{\infty}$ with least cardinality such that $ G\cup H$ generates $ S_{\infty}$, then $ \vert H\vert\in \{0,1,\mathfrak{d},\mathfrak{c}\}$.


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Additional Information

J. D. Mitchell
Affiliation: Mathematics Institute, University of St. Andrews, North Haugh, St. Andrews, Fife, KY16 9SS, Scotland
Email: jdm3@st-and.ac.uk

M. Morayne
Affiliation: Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
Email: michal.morayne@pwr.wroc.pl

Y. Péresse
Affiliation: Mathematics Institute, University of St. Andrews, North Haugh, St. Andrews, Fife, KY16 9SS, Scotland
Email: yhp1@st-and.ac.uk

DOI: https://doi.org/10.1090/S0002-9939-2010-10694-9
Received by editor(s): February 4, 2010
Published electronically: September 21, 2010
Communicated by: Julia Knight
Article copyright: © Copyright 2010 American Mathematical Society