Assuming the existence of a hypermeasurable cardinal, we shall construct a model of Set Theory with a measurable cardinal \(\kappa\) such that \(2^{\kappa}=\kappa^{++}\) and the group \(Sym(\kappa)\) of all permutations of \(\kappa\) cannot be written as a union of a chain of proper subgroups of length \(<\kappa^{++}\). The proof involves the iteration of a suitably defined uncountable version of the Miller forcing poset as well as the “tuning fork” argument introduced by S.D. Friedman and K. Thompson.
Based on the joint work with S.D. Friedman