## 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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### References:

 [1] Journal of the London Mathematical Society 48 pp 204– (1993) [2] DOI: 10.1007/978-1-4613-9754-0_16 [3] DOI: 10.1016/0168-0072(90)90057-9 · Zbl 0713.03024 [4] Cardinal arithmetic 29 (1994) · Zbl 0848.03025 [5] DOI: 10.1007/978-94-011-2080-7_25 [6] Set theory, an introduction to independence proofs (1980) · Zbl 0443.03021 [7] Sets, Graphs and Numbers 60 pp 637– (1991) [8] DOI: 10.1007/BF01269875 · Zbl 0818.03027 [9] DOI: 10.1305/ndjfl/1040511341 · Zbl 0824.03027 [10] Journal of the London Mathematical Society 42 pp 64– (1990) [11] American Mathematical Society Bulletin, New Series 26 pp 197– (1992) · Zbl 0771.03017
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