Hajós's theorem

Hajós's theorem   (Redirected from Rédei's theorem) Jump to navigation Jump to search 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 {displaystyle {e,a,a^{2},dots ,a^{s-1}}} where {displaystyle 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 rings. Rédei later proved the statement when the factors are only required to contain the identity element and be of prime cardinality. Rédei's proof of Hajós's theorem was simplified by Tibor Szele.

In this lattice tiling of the plane by congruent squares, the green and violet squares meet edge-to-edge as do the blue and orange squares.

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.

References Hajós, Georg (1941), "Über einfache und mehrfache Bedeckung des {displaystyle n} -dimensionalen Raumes mit einem Würfelgitter", Mathematische Zeitschrift (in German), 47: 427–467, doi:10.1007/BF01180974, Zbl 0025.25401 Minkowski, H. (1907), Diophantische Approximationen (in German), Leipzig, p. 28 Rédei, L. (1965), "Die neue Theorie der endlichen abelschen Gruppen und Verallgemeinerung des Hauptsatzes von Hajós", Acta Mathematica Academiae Scientiarum Hungaricae (in German), 16: 329–373, doi:10.1007/BF01904843, MR 0186729 Stein, S. K. (1974), "Algebraic tiling", American Mathematical Monthly, 81: 445–462, doi:10.2307/2318582, JSTOR 2318582, MR 0340063 Stein, Sherman K.; Szabó, Sándor (1994), "The group theoretic version of Minkowski's conjecture; more about the algebraic version of Minkowski s conjecture", Algebra and Tiling: Homomorphisms in the Service of Geometry, Carus Mathematical Monographs, vol. 25, Mathematical Association of America, pp. 23–28, ISBN 978-0-88385-028-2, MR 1311249 Szele, T. (1949), "Neuer vereinfachter Beweis des gruppentheoretischen Satzes von Hajós", Publicationes Mathematicae Debrecen (in German), 1: 56–62, MR 0032620 Categories: Theorems in group theoryConjectures that have been proved