UNCOUNTABLE COFINALITIES OF PERMUTATION GROUPS

Abstract

A sufficient criterion is found for certain permutation groups $G$ to have uncountable strong cofinality, that is, they cannot be expressed as the union of a countable, ascending chain $(H_i)_{i\in\o}$ of proper subsets $H_i$ such that $H_iH_i \subseteq H_{i+1}$ and $H_i\,{=}\,H_i^{-1}$. This is a strong form of uncountable cofinality for $G$, where each $H_i$ is a subgroup of $G$. This basic tool comes from a recent paper by Bergman on generating systems of the infinite symmetric groups, which is discussed in the introduction. The main result is a theorem which can be applied to various classical groups including the symmetric groups and homeomorphism groups of Cantor's discontinuum, the rationals, and the irrationals, respectively. They all have uncountable strong cofinality. Thus the result also unifies various known results about cofinalities. A notable example is the group BSym ($\Q$) of all bounded permutations of the rationals $\Q$ which has uncountable cofinality but countable strong cofinality.