Dihedral Group

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 of order n differs in geometry and abstract algebra. In geometry, D_n or Dih_n refers to the symmetries of the n-gon, a group of order 2n. In abstract algebra, D_n refers to the dihedral group of order n. The geometric convention is used in this article.

A regular polygon with ${\displaystyle n}$ sides has ${\displaystyle 2n}$ different symmetries: ${\displaystyle n}$ rotational symmetries and ${\displaystyle n}$ reflection symmetries. Usually, we take ${\displaystyle n}$≥3 here.The associated rotations and reflections make up the dihedral group ${\displaystyle D_{n}}$ . If ${\displaystyle n}$ is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If ${\displaystyle n}$ is even, there are n/2 axes of symmetry connecting the midpoints of opposite sides and ${\displaystyle n/2}$ axes of symmetry connecting opposite vertices. In either case, there are ${\displaystyle n}$ axes of symmetry and ${\displaystyle 2n}$ elements in the symmetry group. Reflecting in one axis of symmetry followed by reflecting in another axis of symmetry produces a rotation through twice the angle between the axes.

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