Strongly Summable Ultrafilters, Union Ultrafilters, and the Trivial Sums Property

We answer two questions of Hindman, Steprans, and Strauss; namely, we prove that every strongly summable ultrafilter on an abelian group is sparse and has the trivial sums property. Moreover, we show that in most cases the sparseness of the given ultrafilter is a consequence of its being isomorphic to a union ultrafilter. However, this does not happen in all cases; we also construct (assuming Martin's Axiom for countable partial orders, i.e., cov(M) = c), a strongly summable ultrafilter on the Boolean group that is not additively isomorphic to any union ultrafilter.