Closed subgroups of the infinite symmetric group
| dc.creator | Bergman, George M. | |
| dc.creator | Shelah, Saharon | |
| dc.date | 2004-01-22 | |
| dc.date | 2005-05-27 | |
| dc.date.accessioned | 2026-07-07T08:06:13Z | |
| dc.date.available | 2026-07-07T08:06:13Z | |
| dc.description | Let S=Sym(Ω) be the group of all permutations of a countably infinite set Ω, 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 < G_1\cup U > = < G_2\cup U >. It is shown that the subgroups closed in the function topology on S lie in precisely four equivalence classes under this relation. Which of these classes a closed subgroup G belongs to depends on which of the following statements about pointwise stabilizer subgroups G_{(Γ)} of finite subsets Γ\subseteqΩholds: (i) For every finite set Γ, the subgroup G_{(Γ)} has at least one infinite orbit in Ω. (ii) There exist finite sets Γsuch that all orbits of G_{(Γ)} are finite, but none such that the cardinalities of these orbits have a common finite bound. (iii) There exist finite sets Γsuch that the cardinalities of the orbits of G_{(Γ)} have a common finite bound, but none such that G_{(Γ)}=\{1\}. (iv) There exist finite sets Γsuch that G_{(Γ)}=\{1\}. Some questions for further investigation are discussed. | |
| dc.description | 33 pages. See also http://math.berkeley.edu/~gbergman/papers and http://shelah.logic.at (pub. 823). To appear, Alg.Univ., issue honoring W.Taylor. Main results as before (greater length due to AU formatting), but some new results in §\S11-12. Errors in subscripts between displays (12) and (13) fixed. Error in title of orig. posting fixed. 1 ref. added | |
| dc.identifier | https://arxiv.org/abs/math/0401305 | |
| dc.identifier | http://arxiv.org/abs/math/0401305 | |
| dc.identifier | Algebra Universalis 55 (2006) 137-173 | |
| dc.identifier | doi:10.1007/s00012-006-1959-z | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/130530 | |
| dc.subject | Group Theory | |
| dc.subject | 20B07 (primary), 22F50 (secondary) | |
| dc.title | Closed subgroups of the infinite symmetric group | |
| dc.type | text |