DihedralGroup

DihedralGroup[n]

represents the dihedral group of order 2n.

Details

  • The degree n of DihedralGroup[n] must be a positive integer.
  • DihedralGroup[1] is isomorphic to CyclicGroup[2] and represented by default as a permutation group on the points {1,2}.
  • DihedralGroup[2] is isomorphic to AbelianGroup[{2,2}] and represented by default as a permutation group on the points {1,2,3,4}.
  • For n3, DihedralGroup[n] is represented by default as a permutation group on the points {1,,n}.

Background & Context

  • DihedralGroup[n] represents the dihedral group of order (also denoted or ) for a given positive integer n. For , the default representation of DihedralGroup[n] is as a permutation group on the symbols . The special cases DihedralGroup[1] and DihedralGroup[2] are isomorphic to CyclicGroup[2] and AbelianGroup[{2,2}], respectively, and are represented by default as a permutation group on the symbols and , respectively.
  • Mathematically, the dihedral group consists of the symmetries of a regular -gon, namely its rotational symmetries and reflection symmetries. In particular, consists of elements (rotations) and (reflections), which combine to transform under its group operation according to the identities , , and , where addition and subtraction are performed modulo . is a permutation group, but for , the operations of reflection and rotation fail to commute in general, meaning is nonabelian for .
  • Dihedral groups are important in the analysis of regular structures, including in the determination of properties for symmetric chemical compounds and in crystallography.
  • The usual group theoretic functions may be applied to DiherdalGroup[n], including GroupOrder, GroupGenerators, GroupElements and so on. A number of precomputed properties of the dihedral group are available via FiniteGroupData[{"DihedralGroup",n},"prop"].
  • DihedralGroup is related to a number of other symbols. DihedralGroup[n] is isomorphic to the semidirect product of CyclicGroup[n] and CyclicGroup[2] (with the latter acting on the former by inversion), and for even, DihedralGroup[n] is isomorphic to the direct product of DihedralGroup[n/2] and CyclicGroup[2]. For , the dihedral group is a subgroup of the symmetric group . Other infinite families of finite groups built into the Wolfram Language that are parametrized by integers include AbelianGroup, AlternatingGroup, CyclicGroup and SymmetricGroup.

Examples

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Basic Examples  (3)

Number of elements of a dihedral group:

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Permutation generators of a dihedral group:

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Elements of a permutation representation of a dihedral group:

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Scope  (1)

Properties & Relations  (1)

Possible Issues  (1)

See Also

SymmetricGroup  AlternatingGroup  CyclicGroup  PermutationGroup

Tutorials

Introduced in 2010
(8.0)