The Cofinality of the Symmetric Group and the Cofinality of Ultrapowers

On the Cofinality of Ultrapowers

We prove some restrictions on the possible cofinalities of ultrapowers of the natural numbers with respect to ultrafilters on the natural numbers. The restrictions involve two cardinal characteristics of the continuum, the splitting number s and the groupwise density number g.

THE MINIMAL COFINALITY OF AN ULTRAPOWER OF ω AND THE COFINALITY OF THE SYMMETRIC GROUP CAN BE LARGER THAN b+

We prove the statement in the title.

The minimal cofinality of an ultrapower of ω and the cofinality of the symmetric groupcan be larger than +

We prove the statement in the title.

The Cofinality Spectrum of the Infinite Symmetric Group

The uncountable cofinality of the automorphism group of the countable universal distributive lattice

We show that the automorphism group of the countable universal distributive lattice has strong uncountable cofinality, and we adapt the method to deduce the strong uncountable cofinality of the automorphism group of the countable universal generalized boolean algebra.