LetSbe the group of all permutations of the set of natural numbers. The cofinality spectrumCF(S)ofSis the set of all regular cardinalsλsuch thatScan be expressed as the union of a chain ofλproper subgroups. This paper investigates which setsCof regular uncountable cardinals can be the cofinality spectrum ofS. 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)Ifμis an inaccessible cardinal such (...) thatμ= sup(C∩μ),thenμ∈C.(c)ifμis a singular cardinal such thatμ= sup(C∩μ),thenμ+∈C.Then there exists a c.c.c. notion of forcing ℙ such that Vℙ⊨ CF(S) = C.We shall also investigate the connections between the cofinality spectrum andpcftheory; and show thatCF(S)cannot be an arbitrarily prescribed set of regular uncountable cardinals. (shrink)

We show that under Martin's Axiom, the cofinalitycf(Aut(Γ)) of the automorphism group of the random graph Γ is 2ω.

Assuming CH, let be the saturated random graph of cardinality ω1. In this paper we prove that it is consistent that and can be any two prescribed regular cardinals subject only to the requirement.

We show that under Martin's Axiom, the cofinality cf(Aut(Γ)) of the automorphism group of the random graph Γ is 2 ω.

We study the relationship between the cofinality $c(Sym(\omega))$ of the infinite symmetric group and the cardinal invariants $\frak{u}$ and $\frak{g}$ . In particular, we prove the following two results. Theorem 0.1 It is consistent with ZFC that there exists a simple $P_{\omega_{1}}$ -point and that $c(Sym(\omega)) = \omega_{2} = 2^{\omega}$ . Theorem 0.2 If there exist both a simple $P_{\omega_{1}}$ -point and a $P_{\omega_{2}}$ -point, then $c(Sym(\omega)) = \omega_{1}$.

We show that g ≤ c(Sym(ω)) where g is the groupwise density number and c(Sym(ω)) is the cofinality of the infinite symmetric group. This solves (the second half of) a problem addressed by Thomas.