# Sh:823

• Bergman, G. M., & Shelah, S. (2006). Closed subgroups of the infinite symmetric group. Algebra Universalis, 55(2-3), 137–173.
• Abstract:
Let S={\rm Sym}(\omega) be the group of all permutations of the natural numbers, and for subgroups G_1,G_2\leq S let us write G_1\approx G_2 if there exists a finite set U\subseteq S such that \langle G_1\cup U\rangle=\langle G_2\cup U\rangle. It is shown that the subgroups closed in the function topology on S lie in precisely four equivalence classes under this relation. Given an arbitrary subgroup G\leq S, which of these classes the closure of G belongs to depends on which of the following statements about pointwise stabilizer subgroups G_{(\Gamma)} of finite subsets \Gamma\subseteq\omega holds:

(i) For every finite set \Gamma, the subgroup G_{(\Gamma)} has at least one infinite orbit in \omega.

(ii) There exist finite sets \Gamma such that all orbits of G_{(\Gamma)} are finite, but none such that the cardinalities of these orbits have a common finite bound.

(iii) There exist finite sets \Gamma such that the cardinalities of the orbits of G_{(\Gamma)} have a common finite bound, but none such that G_{(\Gamma)}=\{1\}.

(iv) There exist finite sets \Gamma such that G_{(\Gamma)}=\{1\}.

• published version (37p)
Bib entry
@article{Sh:823,
author = {Bergman, George Mark and Shelah, Saharon},
title = {{Closed subgroups of the infinite symmetric group}},
journal = {Algebra Universalis},
fjournal = {Algebra Universalis},
volume = {55},
number = {2-3},
year = {2006},
pages = {137--173},
issn = {0002-5240},
doi = {10.1007/s00012-006-1959-z},
mrclass = {20B35 (20B07)},
mrnumber = {2280223},
mrreviewer = {Wolfgang Knapp},
doi = {10.1007/s00012-006-1959-z},
note = {\href{https://arxiv.org/abs/math/0401305}{arXiv: math/0401305}},
arxiv_number = {math/0401305},
specialissue = {Special issue dedicated to Walter Taylor}
}