Abelian group
first reformulated Keller's conjecture in terms of factorizations of abelian groups. He shows that if there is a counterexample to the conjecture, then it can be assumed to be a periodic tiling of cubes with an integer side length and integer vertex positions; thus, in studying the conjecture, it is sufficient to consider tilings of this special form. In this case, the group of integer translations, modulo the translations that preserve the tiling, forms an abelian group, and certain elements of this group correspond to the positions of the tiles. Hajós defines a family of subsets of an abelian group to be a "factorization" if each element of the group has a unique expression as a sum , where each belongs to . With this definition, Hajós' reformulated conjecture is that whenever an Abelian group has a factorization in which the first set may be arbitrary but each subsequent set takes the special form } for some element of , then at least one element must belong to (the difference set of with itself).