Some questions concerning the cofinality of Sym(κ)

Extract

Suppose that G is a group that is not finitely generated. Then the cofinality of G, written c(G), is denned to be the least cardinal λ such that G can be expressed as the union of a chain of λ proper subgroups. If κ is an infinite cardinal, then Sym(κ) denotes the group of all permutations of the set κ = {α∣α < κ}. In [1], Macpherson and Neumann proved that c(Sym(κ)) > κ for all infinite cardinals κ. In [4], we proved that it is consistent that c(Sym(ω)) and 2^ω can be any two prescribed regular cardinals, subject only to the obvious requirement that c(Sym(ω)) ≤ 2^ω. Our first result in this paper is the analogous result for regular uncountable cardinals κ.

Theorem 1.1. Let V ⊨ GCH. Let κ, θ, λ ∈ V be cardinals such that

(i) κ and θ are regular uncountable, and

(ii) κ < θ ≤ cf(λ).

Then there exists a notion of forcing ℙ, which preserves cofinalities and cardinalities, such that if G is ℙ-generic then V[G] ⊨ c(Sym(κ)) = θ ≤ λ = 2^κ.

Theorem 1.1 will be proved in §2. Our proof is based on a very powerful uniformization principle, which was shown to be consistent for regular uncountable cardinals in [2].