page_title: Dihedral group
section_title: The dihedral group as symmetry group in 2D and rotation group in 3D
hierarchical_section_title: Dihedral group / The dihedral group as symmetry group in 2D and rotation group in 3D
context_page_description: In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.
  The notation for the dihedral group differs in geometry and abstract algebra. In geometry, Dₙ or Dihₙ refers to the symmetries of the n-gon, a group of order 2n. In abstract algebra, D₂ₙ refers to this same dihedral group. The geometric convention is used in this article.
context_section_description: An example of abstract group Dₙ, and a common way to visualize it, is the group of Euclidean plane isometries which keep the origin fixed. These groups form one of the two series of discrete point groups in two dimensions. Dₙ consists of n rotations of multiples of 360°/n about the origin, and reflections across n lines through the origin, making angles of multiples of 180°/n with each other. This is the symmetry group of a regular polygon with n sides (for n ≥ 3; this extends to the cases n = 1 and n = 2 where we have a plane with respectively a point offset from the "center" of the "1-gon" and a "2-gon" or line segment).
  Dₙ is generated by a rotation r of order n and a reflection s of order 2 such that
  In geometric terms: in the mirror a rotation looks like an inverse rotation.
  In terms of complex numbers: multiplication by  and complex conjugation.
  In matrix form, by setting
  and defining  and  for  we can write the product rules for  Dₙ as
  (Compare coordinate rotations and reflections.)
  The dihedral group D₂ is generated by the rotation r of 180 degrees, and the reflection s across the x-axis. The elements of D₂ can then be represented as {e, r, s, rs}, where e is the identity or null transformation and rs is the reflection across the y-axis.
  D₂ is isomorphic to the Klein four-group.
  For n > 2 the operations of rotation and reflection in general do not commute and Dₙ is not abelian; for example, in D₄, a rotation of 90 degrees followed by a reflection yields a different result from a reflection followed by a rotation of 90 degrees.
  Thus, beyond their obvious application to problems of symmetry in the plane, these groups are among the simplest examples of non-abelian groups, and as such arise frequently as easy counterexamples to theorems which are restricted to abelian groups.
  The 2n elements of Dₙ can be written as e, r, r², ... , rⁿ⁻¹, s, r s, r²s, ... ,  rⁿ⁻¹s. The first n listed elements are rotations and the remaining n elements are axis-reflections (all of which have order 2). The product of two rotations or two reflections is a rotation; the product of a rotation and a reflection is a reflection.
  So far, we have considered Dₙ to be a subgroup of O(2), i.e. the group of rotations (about the origin) and reflections (across axes through the origin) of the plane. However, notation Dₙ is also used for a subgroup of SO(3) which is also of abstract group type Dₙ: the proper symmetry group of a regular polygon embedded in three-dimensional space (if n ≥ 3). Such a figure may be considered as a degenerate regular solid with its face counted twice. Therefore, it is also called a dihedron (Greek: solid with two faces), which explains the name dihedral group (in analogy to tetrahedral, octahedral and icosahedral group, referring to the proper symmetry groups of a regular tetrahedron, octahedron, and icosahedron respectively).
caption: D4 is nonabelian (x-axis is vertical here).
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