Let G be a finite abelian group of order g: We determine, for all 1pr; spg; the minimal size mGðr; sÞ ¼ minjA þ Bj of sumsets A þ B; where A and B range over all subsets of G of cardinality r and s; respectively. We do so by explicit construction. Our formula for mGðr; sÞ shows that this function only depends on the cardinality of G;