Assuming the existence of a P₂κ-hypermeasurable cardinal, we construct a model of Set Theory with a measurable cardinal κ such that and the group Sym(κ) of all permutations of κ cannot be written as the union of a chain of proper subgroups of length < κ⁺⁺. The proof involves iteration of a suitably defined uncountable version of the Miller forcing poset as well as the “tuning fork” argument introduced by the first author and K. Thompson [J. Symbolic Logic 73 (2008)].
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm207-2-1,
author = {Sy-David Friedman and Lyubomyr Zdomskyy},
title = {Measurable cardinals and the cofinality of the symmetric group},
journal = {Fundamenta Mathematicae},
volume = {209},
year = {2010},
pages = {101-122},
zbl = {1196.03063},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm207-2-1}
}
Sy-David Friedman; Lyubomyr Zdomskyy. Measurable cardinals and the cofinality of the symmetric group. Fundamenta Mathematicae, Tome 209 (2010) pp. 101-122. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm207-2-1/