Generating the infinite symmetric group using a closed subgroup and the least number of other elements
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- by J. D. Mitchell, M. Morayne and Y. Péresse PDF
- Proc. Amer. Math. Soc. 139 (2011), 401-405 Request permission
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 $|H|\in \{0,1,\mathfrak {d},\mathfrak {c}\}$.References
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Additional Information
- J. D. Mitchell
- Affiliation: Mathematics Institute, University of St. Andrews, North Haugh, St. Andrews, Fife, KY16 9SS, Scotland
- MR Author ID: 691066
- 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
- Received by editor(s): February 4, 2010
- Published electronically: September 21, 2010
- Communicated by: Julia Knight
- © Copyright 2010 American Mathematical Society
- 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
- MathSciNet review: 2736324