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Concept
Dihedral group
Summary
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, D_n or Dih_n refers to the symmetries of the n-gon, a group of order 2n. In abstract algebra, D_2n refers to this same dihedral group. This article uses the geometric convention, D_n.
A regular polygon with sides has different symmetries: rotational symmetries and reflection symmetries. Usually, we take here. The associated rotations and reflections make up the dihedral group . If is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex. If is even, there are axes of symmetry connecting the midpoints of opposite sides and axes of symmetry connecting opposite vertices. In either case, there are axes of symmetry and 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 on a stop sign:
The first row shows the effect of the eight rotations, and the second row shows the effect of the eight reflections, in each case acting on the stop sign with the orientation as shown at the top left.
As with any geometric object, the composition of two symmetries of a regular polygon is again a symmetry of this object. With composition of symmetries to produce another as the binary operation, this gives the symmetries of a polygon the algebraic structure of a finite group.
The following Cayley table shows the effect of composition in the group D3 (the symmetries of an equilateral triangle). r0 denotes the identity; r1 and r2 denote counterclockwise rotations by 120° and 240° respectively, and s0, s1 and s2 denote reflections across the three lines shown in the adjacent picture.
Official source
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