Ultraproducts of Finite Groups

There are two major types of ultrafilters, principal ultrafilters and non-principal ultrafilters. A principal ultrafilter is one that is generated by any single element a ∈ S, and has the form 〈a〉 = {A ⊆ S | a ∈ A}. However, these principal ultrafilters are not incredibly interesting, and will be used only as an example in this document. On the other hand, we have non-principal ultrafilters. We can say that an ultrafilter ψ is non-prinicpal if it contains all cofinite subsets of S, that is all subsets whose complement is finite. From this, we can infer that ∩(A ∈ ψ) = ∅. If instead, we had x ∈ ∩(A ∈ ψ), then ψ = 〈x〉. In this document, we will be considering ultrafilters over the set of natural numbers, N. Except where noted otherwise, we will use ψ to denote a nonprincipal ultrafilter over N.