Solution to a problem of Sands on the factorization of groups

Using elementary methods, a positive answer is given to a question of A. D. Sands concerning factorizations of abelian groups. We then indicate how our approach to Sands’s question has its roots in a result on the ergodic theory of infinite measure preserving transformations due to Eigen, Hajian and Ito. 1. SANDS’S PROBLEM AND AN ELEMENTARY SOLUTION In [2], A.D. Sands notes that a positive answer to the following problem would simplify the proofs in [2]. Our Theorem 1.1 which is proved using elementary arguments, provides (as a special case) such a positive answer. Here is Sands’s problem: Problem(Sands). Suppose that G = A + B is a factorization of the group G, and that the subset A isjinite. Suppose that a subset C of G exists such that IAl = ICI and the sum of C and B is direct. Does it follow that G = C + B? The terminology from [2] is as follows: the “group” G is an additive abelian group; the cardinality of a set C is denoted 1 Cl; for subsets A, B of G their sum is A + B = {a + b : a E A, b E B}. When each element of A + B is expressed uniquely in this way, the sum is called direct and we write A + B = A $ B. A factorization of G, means that for some sets A and B, G = A 8 B. In this note, we write -A = {-a : a E A} and A A = A + (-A), the difiv-