Totally Nonfree Actions and the Infinite Symmetric Group

Totally Nonfree Actions and the Infinite Symmetric Group

We consider the totally nonfree (TNF) action of a groups and the corresponding adjoint invariant (AD) measures on the lattices of the subgroups of the given group. The main result is the description of all adjoint-invariant and TNF-measures on the lattice of subgroups of the infinite symmetric group SN. The problem is closely related to the theory of characters and factor representations of gro...

Twisted Symmetric Group Actions

Let K be any field, K(x1, . . . , xn) be the rational function field of n variables over K, and Sn and An be the symmetric group and the alternating group of degree n respectively. For any a ∈ K \ {0}, define an action of Sn on K(x1, . . . , xn) by σ · xi = xσ(i) for σ ∈ An and σ · xi = a/xσ(i) for σ ∈ Sn \ An. Theorem. For any field K and n = 3, 4, 5, the fixed field K(x1, . . . , xn) Sn is ra...

Closed subgroups of the infinite symmetric group

Let S = Sym(Ω) be the group of all permutations of a countably infinite set Ω, and for subgroups G1, G2 6 S let us write G1 ≈ G2 if there exists a finite set U ⊆ S such that 〈G1 ∪U 〉 = 〈G2 ∪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 ...

Nonfree Actions of Countable Groups and Their Characters

We introduce a number of definitions of nonfree actions of groups. The most important of them is that of a totally nonfree action; it is naturally related to the theory of characters of groups and their factor representations. This short note is a brief exposition of a part of a more detailed paper on this subject, which is now in preparation.

The Cofinality Spectrum of the Infinite Symmetric Group