Ultrafilters and partial products of infinite cyclic groups

We consider, for infinite cardinals κ and α le; κ⁺, the group Π(κ, <α) of sequences of integers, of length κ, with nonzero entries in fewer than α positions. Our main result tells when Π(κ, <α) can be embedded in Π(λ, <β). The proof involves some set-theoretic results, one about families of finite sets and one about families of ultrafilters.