In group theory, Hajós's theorem states that if a finite abelian group is expressed as the Cartesian product of simplexes, that is, sets of the form
\{e,a,a2,...,as-1\}
e
An equivalent statement on homogeneous linear forms was originally conjectured by Hermann Minkowski. A consequence is Minkowski's conjecture on lattice tilings, which says that in any lattice tiling of space by cubes, there are two cubes that meet face to face. Keller's conjecture is the same conjecture for non-lattice tilings, which turns out to be false in high dimensions.