Reflection group

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.

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. (r_i r_j)^{c_{ij}} = 1, expressing the fact that the product of the reflections r i and r j in two hyperplanes H i and H j meeting at an angle \pi/c_{ij} is a rotation by the angle 2\pi/c_{ij} fixing the subspace H i  ∩ H j of codimension 2. Thus, viewed as an abstract group, every reflection group is a Coxeter group. ===Notes=== ===Bibliography=== {{Citation |authorlink=H.S.M. Coxeter |first=H.S.M. |last=Coxeter |title=The complete enumeration of finite groups of the form r_i^2=(r_ir_j)^{k_{ij}}=1 |journal=J. London Math. Soc. |volume=10 |year=1935 |pages=21–25 |doi=10.1112/jlms/s1-10.37.21 }} ===Textbooks=== ==External links==

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