The cofinality spectrum of the infinite symmetric group

Let S be the group of all permutations of the set of natural numbers. The cofinality spectrum CF(S) of S is the set of all regular cardinals λ such that S can be expressed as the union of a chain of λ 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 |= GCH. Let C be a set of regular uncountable cardinals which satisfies the following conditions. (a) C contains a maximum element. (b) If μ is an inaccessible cardinal such that μ = supi(C ∩ μ), then μ ∈ C. (c) If μ is a singular cardinal such that μ = sup(C ∩ μ), then μ^- ∈ C. Then there exists a c.c.c. notion of forcing ℙ such that I^-ℙ |= CF(S) = C. We shall also investigate the connections between the cofinality spectrum and pcf theory: and show that CF (S) cannot be an arbitrarily prescribed set of regular uncountable cardinals.