Let \(G\) be a group that is not finitely generated. (a) The cofinality of \(G\), denoted by \(\text{cf}(G)\), is the smallest cardinal \(\lambda\) for which there is a chain of proper subgroups \((U_i)_{i\in\lambda}\) of \(G\) such that \(G=(\bigcup U_i)_{i\in\lambda}\).
(b) The strong cofinality of \(G\), denoted \(\text{scf}(G)\), is the smallest cardinal \(\lambda\) for which there is a chain of proper subsets \((U_i)_{i\in\lambda}\) of \(G\) such that for each \(i\in\lambda\), \(U_i=U_i^{-1}\) and \(U_i\cdot U_i\subseteq U_j\) for some \(j\in\lambda\), and \(G=(\bigcup U_i)_{i\in\lambda}\).
In this paper a sufficient criterion is found for certain permutation groups \(G\) to have uncountable strong cofinality. The main result of this paper 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 \(\text{BSym}(\mathbb{Q})\) of all bounded permutations of the rationals \(\mathbb{Q}\) which has uncountable cofinality but countable strong cofinality.