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Some questions concerning the cofinality of Sym(κ)

  • James D. Sharp (a1) and Simon Thomas (a2)
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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 = {αα < 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 2ω can be any two prescribed regular cardinals, subject only to the obvious requirement that c(Sym(ω)) ≤ 2ω. 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 = 2κ.

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].

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Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
References
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[1]Macpherson, H. D. and Neumann, P. M., Subgroups of infinite symmetric groups, Journal of the London Mathematical Society, ser. 2, vol. 42 (1990), pp. 6484.
[2]Mekler, A. H. and Shelah, S., Uniformizationprinciples, this Journal, vol. 54 (1989), pp. 441459.
[3]Sharp, J. D. and Thomas, S., Unbounded families and the cofinality of the infinite symmetric group, Archive for Mathematical Logic (to appear).
[4]Sharp, J. D. and Thomas, S., Uniformization problems and the cofinality of the infinite symmetric group, preprint, 1993.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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