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In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include Weyl groups and crystallographic Coxeter groups. While the orthogonal group is generated by reflections (by the Cartan–Dieudonné theorem), it is a continuous group (indeed, Lie group), not a discrete group, and is generally considered separately.

Attributes	Values
rdf:type	yago:Abstraction100002137 yago:Group100031264 Band yago:WikicatReflectionGroups
rdfs:label	Gruppo di riflessione (it) Reflection group (en) Grupa odbić (pl)
rdfs:comment	In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include Weyl groups and crystallographic Coxeter groups. While the orthogonal group is generated by reflections (by the Cartan–Dieudonné theorem), it is a continuous group (indeed, Lie group), not a discrete group, and is generally considered separately. (en) Un gruppo di riflessione, in geometria e nella teoria dei gruppi, è un gruppo discreto generato dall'insieme delle riflessioni di uno spazio euclideo di dimensione finita. Tra i gruppi di riflessione si annoverano anche i gruppi di Weyl e i gruppi cristallografici di Coxeter. (it) Grupa odbić – dyskretna grupa odwzorowań generowana przez symetrie względem hiperpowierzchni przestrzeni euklidesowej lub przestrzeni hiperbolicznej lub innej jednospójnej przestrzeni Riemanna o stałej . (pl)
dcterms:subject	Coxeter groups Reflection groups
Wikipage page ID	2422023 (xsd:integer)
Wikipage revision ID	1059751052 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press Cartan–Dieudonné theorem Lie group Complex reflection group Chevalley–Shephard–Todd theorem General linear group Geometry N-sphere Frieze group Fundamental domain Springer Publishing Symmetry group Wallpaper group Affine group American Mathematical Monthly Euclidean space Field (mathematics) Finite field Discrete group Isometry group Kaleidoscope Tessellation Reflection (mathematics) Regular polytope Riemannian manifold Riemannian symmetric space Coxeter group Hyperbolic space Triangle group ADE classification Coxeter groups Reflection groups Dihedral group Platonic solid Group theory H. S. M. Coxeter Orthogonal group Rotation Shear mapping Point groups in three dimensions Point groups in two dimensions Weyl group Hyperplane arrangement Coxeter diagram Group presentation
Link from a Wikipage to an external page	https://books.google.com/books%3Fid=ODfjmOeNLMUC http://www.math.rutgers.edu/~goodman/pub/monthly.pdf
sameAs	Reflection group Reflection group Reflection group Reflection group Reflection group Reflection group Reflection group
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Citation_needed dbt:Commonscatinline dbt:Harvtxt dbt:Refbegin dbt:Refend dbt:Reflist dbt:Sfnp dbt:Eom
id	Reflection_group (en)
title	Reflection group (en)
has abstract	In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent copies of a regular polytope is necessarily a reflection group. Reflection groups also include Weyl groups and crystallographic Coxeter groups. While the orthogonal group is generated by reflections (by the Cartan–Dieudonné theorem), it is a continuous group (indeed, Lie group), not a discrete group, and is generally considered separately. (en) Un gruppo di riflessione, in geometria e nella teoria dei gruppi, è un gruppo discreto generato dall'insieme delle riflessioni di uno spazio euclideo di dimensione finita. Tra i gruppi di riflessione si annoverano anche i gruppi di Weyl e i gruppi cristallografici di Coxeter. (it) Grupa odbić – dyskretna grupa odwzorowań generowana przez symetrie względem hiperpowierzchni przestrzeni euklidesowej lub przestrzeni hiperbolicznej lub innej jednospójnej przestrzeni Riemanna o stałej . (pl)
gold:hypernym	Group
prov:wasDerivedFrom	wikipedia-en:Reflection_group?oldid=1059751052&ns=0
page length (characters) of wiki page	7226 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Reflection_group
is Link from a Wikipage to another Wikipage of	Dynkin diagram Kyoji Saito (2,3,7) triangle group Ruth Kellerhals Peter Orlik Glossary of Lie groups and Lie algebras Leech lattice Ideal triangle Building (mathematics) Mirrors and Reflections Discrete geometry History of group theory Kazhdan's property (T) Reflection (mathematics)

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