Partitions and sums with inverses in Abelian groups

Let G   be an Abelian group and let A={x∈G:2x≠0} be infinite. We construct a partition {Am:m<ω} of A   such that whenever (xn)n<ω(xn)n<ω is a one-to-one sequence in A  , g∈Gg∈G and m<ωm<ω, one has(g+FSI((xn)n<ω))∩Am≠∅,(g+FSI((xn)n<ω))∩Am≠∅, whereFSI((xn)n<ω)={∑n∈FεnFxn:F∈Pf(ω) and εnF∈{1,−1} for all n∈F} and Pf(ω)Pf(ω) is the set of finite nonempty subsets of ω.