Rédei's theorem
Encyclopedia
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} where e is the identity element, then at least one of the factors is a subgroup. The theorem was proved by the Hungarian mathematician György Hajós
in 1941 using group ring
s. Rédei
later proved the statement when the factors are only required to contain the identity element and be of prime cardinality.
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.
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, Hajós's theorem states that if a finite abelian group
Abelian group
In abstract algebra, 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 their order . Abelian groups generalize the arithmetic of addition of integers...
is expressed as the Cartesian product
Cartesian product
In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
of simplexes, that is, sets of the form {e,a,a2,...,as-1} where e is the identity element, then at least one of the factors is a subgroup. The theorem was proved by the Hungarian mathematician György Hajós
György Hajós
György Hajós was a Hungarian mathematician who worked in group theory, graph theory, and geometry.-Biography:...
in 1941 using group ring
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 one-to-one with the given group. As a ring, its addition law is that of the free...
s. Rédei
László Rédei
László Rédei was a Hungarian mathematician.He graduated from the University of Budapest and initially worked as a schoolteacher...
later proved the statement when the factors are only required to contain the identity element and be of prime cardinality.
An equivalent statement on homogeneous linear forms was originally conjectured by Hermann Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...
. 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
Keller's conjecture
In geometry, Keller's conjecture is the conjecture introduced by that in any tiling of Euclidean space by identical hypercubes there are two cubes that meet face to face. For instance, as shown in the illustration, in any tiling of the plane by identical squares, some two squares must meet edge to...
is the same conjecture for non-lattice tilings, which turns out to be false in high dimensions.