group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

, Hajós's theorem states that if a finite

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...

is expressed as the

Cartesian product In mathematics, specifically set theory, the Cartesian product of two sets and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...

simplex In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

es, that is, sets of the form

\

where

e

is the

identity element In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...

, then at least one of the factors is a

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

. The theorem was proved by the Hungarian mathematician György Hajós in 1941 using

group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the gi ...

s. Rédei later proved the statement when the factors are only required to contain the identity element and be of prime

cardinality The thumb is the first digit of the hand, next to the index finger. When a person is standing in the medical anatomical position (where the palm is facing to the front), the thumb is the outermost digit. The Medical Latin English noun for thum ...

. Rédei's proof of Hajós's theorem was simplified by

Tibor Szele Tibor Szele (21 June 1918 – 5 April 1955) Hungarian mathematician, working in combinatorics and abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which ...

An equivalent statement on homogeneous linear forms was originally conjectured by

Hermann Minkowski Hermann Minkowski (22 June 1864 – 12 January 1909) was a mathematician and professor at the University of Königsberg, the University of Zürich, and the University of Göttingen, described variously as German, Polish, Lithuanian-German, o ...

. 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 In geometry, Keller's conjecture is the conjecture that in any Tessellation, tiling of -dimensional Euclidean space by identical hypercubes, there are two hypercubes that share an entire -dimensional face with each other. For instance, in any til ...

is the same conjecture for non-lattice tilings, which turns out to be false in high dimensions.

References

* * * * * * {{DEFAULTSORT:Hajos's theorem Theorems in group theory Conjectures that have been proved