Dihedral group

[3] 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.

The latter comes from the Greek word hédra, which means "face of a geometrical solid".

is odd, each axis of symmetry connects the midpoint of one side to the opposite vertex.

axes of symmetry connecting the midpoints of opposite sides and

[7] The following Cayley table shows the effect of composition in the group D3 (the symmetries of an equilateral triangle).

[7] In general, the group Dn has elements r0, ..., rn−1 and s0, ..., sn−1, with composition given by the following formulae: In all cases, addition and subtraction of subscripts are to be performed using modular arithmetic with modulus n. If we center the regular polygon at the origin, then elements of the dihedral group act as linear transformations of the plane.

This lets us represent elements of Dn as matrices, with composition being matrix multiplication.

For example, the elements of the group D4 can be represented by the following eight matrices: In general, the matrices for elements of Dn have the following form: rk is a rotation matrix, expressing a counterclockwise rotation through an angle of 2πk/n.

sk is a reflection across a line that makes an angle of πk/n with the x-axis.

has the presentation In particular, Dn belongs to the class of Coxeter groups.

The dark vertex in the cycle graphs below of various dihedral groups represents the identity element, and the other vertices are the other elements of the group.

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.

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).

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.

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).

The properties of the dihedral groups Dn with n ≥ 3 depend on whether n is even or odd.

For example, the center of Dn consists only of the identity if n is odd, but if n is even the center has two elements, namely the identity and the element rn/2 (with Dn as a subgroup of O(2), this is inversion; since it is scalar multiplication by −1, it is clear that it commutes with any linear transformation).

In the case of 2D isometries, this corresponds to adding inversion, giving rotations and mirrors in between the existing ones.

m. Therefore, the total number of subgroups of Dn (n ≥ 1), is equal to d(n) + σ(n), where d(n) is the number of positive divisors of n and σ(n) is the sum of the positive divisors of n. See list of small groups for the cases n ≤ 8.

All the reflections are conjugate to each other whenever n is odd, but they fall into two conjugacy classes if n is even.

Algebraically, this is an instance of the conjugate Sylow theorem (for n odd): for n odd, each reflection, together with the identity, form a subgroup of order 2, which is a Sylow 2-subgroup (2 = 21 is the maximum power of 2 dividing 2n = 2[2k + 1]), while for n even, these order 2 subgroups are not Sylow subgroups because 4 (a higher power of 2) divides the order of the group.

) = {ax + b | (a, n) = 1} and has order nϕ(n), where ϕ is Euler's totient function, the number of k in 1, ..., n − 1 coprime to n. It can be understood in terms of the generators of a reflection and an elementary rotation (rotation by k(2π/n), for k coprime to n); which automorphisms are inner and outer depends on the parity of n. D9 has 18 inner automorphisms.

The 18 inner automorphisms provide rotation of the mirrors by multiples of 20°, and reflections.

As abstract group there are in addition to these, 36 outer automorphisms; e.g., multiplying angles of rotation by 2.

The 10 inner automorphisms provide rotation of the mirrors by multiples of 36°, and reflections.

Compare the values 6 and 4 for Euler's totient function, the multiplicative group of integers modulo n for n = 9 and 10, respectively.