Dihedral group

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In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections.[1] Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.

See also: Dihedral symmetry in three dimensions.

Contents

Notation

There are two competing notations for the dihedral group associated to a polygon with n sides. In geometry the group is denoted Dn, while in algebra the same group is denoted by D2n to indicate the number of elements.

In this article, Dn (and sometimes Dihn) refers to the symmetries of a regular polygon with n sides.

Definition

Elements

A regular polygon with n sides has 2n different symmetries: n rotational symmetries and n reflection symmetries. The associated rotations and reflections make up the dihedral group Dn. If n is odd each axis of symmetry connects the mid-point of one side to the opposite vertex. If n is even there are n/2 axes of symmetry connecting the mid-points of opposite sides and n/2 axes of symmetry connecting opposite vertices. In either case, there are n axes of symmetry altogether and 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. The following picture shows the effect of the sixteen elements of D8 on a stop sign:

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