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The cofinality spectrum of the infinite symmetric group. (English) Zbl 0889.03037

Summary: Let \(S\) be the group of all permutations of the set of natural numbers. The cofinality spectrum \(\text{CF}(S)\) of \(S\) is the set of all regular cardinals \(\lambda\) such that \(S\) can be expressed as the union of a chain of \(\lambda\) proper subgroups. This paper investigates which sets \(C\) of regular uncountable cardinals can be the cofinality spectrum of \(S\). The following theorem is the main result of this paper.
Theorem. Suppose that \(V\models \text{GCH}\). Let \(C\) be a set of regular uncountable cardinals which satisfies the following conditions.
(a) \(C\) contains a maximum element.
(b) If \(\mu\) is an inaccessible cardinal such that \(\mu= \sup(C\cap \mu)\), then \(\mu\in C\).
(c) If \(\mu\) is a singular cardinal such that \(\mu= \sup(C\cap \mu)\), then \(\mu^+\in C\).
Then there exists a c.c.c. notion of forcing \(\mathbb{P}\) such that \(V^{\mathbb{P}}\models \text{CF}(S)= C\).
We shall also investigate the connections between the cofinality spectrum and pcf theory, and show that \(\text{CF}(S)\) cannot be an arbitrarily prescribed set of regular uncountable cardinals.

MSC:

03E05 Other combinatorial set theory
20B35 Subgroups of symmetric groups
03E55 Large cardinals
03E35 Consistency and independence results
03E50 Continuum hypothesis and Martin’s axiom
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