### On a conjecture of G. Hajós

A. D. Sands
1974 Glasgow Mathematical Journal
1. Introduction. The purpose of this note is to provide by means of an example a negative answer to a conjecture of Haj6s [3] concerning the factorization of finite abelian groups. This question is also raised as Problem 81 in Fuchs [2]. If S, T are subsets of an additive abelian group G their sum S+ T is said to be direct if s t + t x = s 2 +1 2 implies s t = s 2 , t x = t 2 , where s t e S, t t e T. If the sum is direct and S+ T = G, then we have a factorization of G. All sums considered in
more » ... ums considered in this note are direct. A subset S of G is said to be periodic if there exists heG, h^Q, with S+h = S. If H = {heG\S+h = S}, then H is a subgroup of G and we have S = H+S^ for some subset S i . When Haj6s discovered that neither factor in a factorization of certain finite abelian groups G need be periodic he asked the following weaker question. Is every factorization G = S+ T of a finite abelian group G quasi-periodic in the sense that one factor, say T, is a disjoint union of subsets T t (1 ^ i• £ m, m > 1), such that there is a subgroup H of G of order m with S+T t = S+Ti+hj, where H = {h t \ 1 ^ i £ m} ? Clearly, if T is periodic, the factorization is quasi-periodic with the set of periods of T, including 0, as the subgroup H. Example. We give the following example of a non-quasi-periodic factorization. The construction is provided by a special case of a technique of de Bruijn [1], despite the closing remark of that paper.