# Sh:823

- Bergman, G. M., & Shelah, S. (2006).
*Closed subgroups of the infinite symmetric group*. Algebra Universalis,**55**(2-3), 137–173. arXiv: math/0401305 DOI: 10.1007/s00012-006-1959-z MR: 2280223 -
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}, mrnumber = {2280223}, mrclass = {20B35 (20B07)}, 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} }