Keller's conjecture for certain finite groups

Abstract

In 1930, Keller [3] conjectured that in any tiling of Euclidean n-space by translates of the unit cube, some pair of translates share a face. In 1949, Hajos [2] reduced this conjecture to a certain statement about finite abelian groups. If G is a finite abelian group written additively, g is an element of G, and n 3 2 is an integer, we denote the set (0, g, . . . . (n 1) g} by L-s, nl. If A,, . . . . A, are subsets of G, we say that A, + ... + A, is a factorization of G if each element of G has exactly one representation as a sum a,+ ... +a,, where a,~A~for i= 1, . . . . m. Also, if B and C are subsets of G, we let B C denote the set of all differences b c between elements b E B and c E C. Hajos’ statement then says that for all finite abelian groups G, if H + [xi, r,] + ... + [x,, r,n] is a factorization of G, then for some iE { 1, . . . . m}, we have rixi~ HH. In this paper we adapt a method of Corradi and Szabo [l] to prove this form of Keller’s conjecture in two known cases and to prove it for two new classes of abelian groups.