The Cofinality Spectrum of the Infinite Symmetric Group

Abstract

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 A such that S can be expressed as the union of a chain of i 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 t GCH. Let C be a set of regular uncountable cardinals which satisfies the jollowing coalitions. (a) C contains a maximum element. (b) Iju is an inaccessible cardinal such that ui = sup(C n iu), then ,u E C. (c) I'li is a singular cardinal such that pi = sup(C n iu), then i + E C. Then there exists a ce..c. notion offorcing P such that VP t CF(S) = C. We shall also investigate the connections between the cofinality spectrum and pef theory; and show that CF(S) cannot be an arbitrarily prescribed set of regular uncountable cardinals.