  We answer two questions of Hindman, Stepr\=ans 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.
$\mathrm{cov}(\mathcal M)=\mathfrak c$), on the Boolean group, a strongly
summable ultrafilter that is not additively isomorphic to any union
ultrafilter.
