### Closed subgroups of the infinite symmetric group

by Bergman and Shelah. [BmSh:823]

Algebra Universalis, 2006

Let S= Sym (omega) be the group of all permutations of
the natural numbers, and for subgroups G_1,G_2 <= 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. Given an
arbitrary subgroup G <= 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} .

Back to the list of publications