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...copies of a regular polytope is necessarily a reflection group. Reflection groups also include [[Weyl group]]s and crystallographic [[Coxeter group]]s. While ...a set of ''affine reflections'' of ''E'' (without the requirement that the reflection hyperplanes pass through the origin). ...

{{Short description|Unique element of maximal length in a finite Coxeter group}} ...roup''' is the unique element of maximal [[length function|length]] in a [[finite Coxeter group]] with respect to the chosen generating set consisting of sim ...

...ion in terms of root data, which mirrors the classification of compact Lie groups, but with the integers replaced by the [[p-adic integers|''p''-adic integer ...thematics)|homology]], and such the pointed [[loop space]] ''G = ΩBG'' has finite mod ''p'' homology. One sometimes also refer to the ''p''-compact group by ...

...ription|One of several theorems linking the sizes of different ideal class groups}} {{For|reflection principles in set theory|Reflection principle}} ...

..._∞ group. They all have 2 generators. The first has two parallel reflection lines, the second two 2-fold gyrations, and the last has one mirror and one ...the symmetry of an [[apeirogon]] alternating two edge lengths, containing reflection points at the center of each edge.]] ...

...a different, although related, meaning: a '''generic polynomial''' for a [[finite group]] ''G'' and a [[field (mathematics)|field]] ''F'' is a [[monic polyno ...to the [[inverse Galois problem]] for that group. However, not all Galois groups have generic polynomials, a counterexample being the [[cyclic group]] of or ...

In three dimensional [[geometry]], there are four infinite series of [[point groups in three dimensions]] (''n''≥1) with ''n''-fold rotational or reflectional They are the finite [[symmetry group]]s on a [[Cone (geometry)|cone]]. For ''n'' = ∞ they corre ...

...s for the axial [[point groups in three dimensions|three-dimensional point groups]], with the addition of reflections in the axial coordinate. They are also == Discrete groups == ...

...[[scalar transformation]]s of ''V'' – these consist of the identity and [[reflection through the origin]]. These scalars are quotiented out because they act [[t ...his odd/even distinction occurs throughout the structure of the orthogonal groups. By analogy with GL/SL and GO/SO, the projective orthogonal group is also s ...

{{About|reflection in geometry|reflexivity of [[binary relation]]s|reflexive relation}} [[File:SimmetriainvOK.svg|right|thumb|A reflection through an axis.]] ...

{{Lie groups|Homogeneous spaces}} In the theory of [[algebraic groups]], a '''Borel subgroup''' of an [[algebraic group]] ''G'' is a maximal [[Za ...

Frieze groups are two-dimensional [[line group]]s, having repetition in only one directio |+The seven frieze groups ...

...|Topological}}{{Short description|Isometry group of Euclidean space}}{{Lie groups |Other}} ...on (mathematics)|reflections]] of <math>\mathbb{E}^n</math>; and arbitrary finite combinations of them. The Euclidean group can be seen as the [[symmetry gr ...

...verses. (Note that inverses are only needed if the group is infinite; in a finite group, the inverse of an element can be expressed as a power of that elemen ...math>\langle x\rangle</math> equals the entire group <math>G</math>. For [[finite group]]s, it is also equivalent to saying that <math>x</math> has [[order ( ...

...metry in three dimensions''' is one of three infinite sequences of [[point groups in three dimensions]] which have a [[symmetry group]] that as an abstract g ...vertical plane, or as the restriction to the plane of a rotation about the reflection line, by 180°. In 3D, the two operations are distinguished: the group ''D<s ...

...wed with additional [[operation (mathematics)|operation]]s and [[axiom]]s. Groups recur throughout mathematics, and the methods of group theory have influenc * [[Direct product of groups]] ...

...(mathematics)|finite group]] acting on a [[finite-dimensional vector space|finite-dimensional]] [[complex numbers|complex]] [[vector space]] that is generate ...groups (the [[Coxeter group]]s or [[Weyl group]]s, including the symmetry groups of [[regular polyhedron|regular polyhedra]]). ...

{{About|symmetry groups of geometric objects}} ...y group]] of the ambient space. This article mainly considers [[symmetry]] groups in [[Euclidean geometry]], but the concept may also be studied for more gen ...

...berg groups]] in [[Lie theory]] that yield [[finite simple group]]s over [[finite field]]s. * {{cite journal|title=Finite reflection groups|journal=[[Transactions of the American Mathematical Society]]|year=1959|vol ...

{{Group theory sidebar |Finite}} ...generated abelian groups|Fundamental Theorem of Finitely Generated Abelian Groups]]. It was named '''''Vierergruppe''''' ({{IPA|de|ˈfiːʁɐˌɡʁʊpə|lang|De-Viere ...