Dihedral group

Definition
The dihedral group of degree $$n$$ and order $$2n$$, denoted sometimes as sometimes as $$D_{2n}$$ (this wiki uses $$D_{2n}$$), sometimes as $$D_n$$, and sometimes as $$\operatorname{Dih}_n$$, is defined in the following equivalent ways:


 * It has the presentation (here, $$e$$ denotes the identity element):

$$\langle x,a \mid a^n = x^2 = e, xax^{-1} = a^{-1} \rangle$$


 * (For $$n \ge 3$$): It is the group of symmetries of a regular $$n$$-gon in the plane, viz., the plane isometries that preserves the set of points of the regular $$n$$-gon.

The dihedral groups arise as a special case of a family of groups called von Dyck groups. They also arise as a special case of a family of groups called Coxeter groups.

Note that for $$n = 1$$ and $$n = 2$$, the geometric description of the dihedral group does not make sense. In these cases, we use the algebraic description.

The infinite dihedral group, which is the $$n = \infty$$ case of the dihedral group and is denoted $$D_\infty$$ and is defined as:

$$\langle x,a \mid x^2 = e, xax^{-1} = a^{-1} \rangle$$.

Generalizations

 * Generalized dihedral group: This is a semidirect product of an abelian group by a cyclic group of order two acting via the inverse map.
 * q-hedral group
 * Coxeter group: Dihedral groups are Coxeter groups with two generators.

For small values
Note that all dihedral groups are metacyclic and hence supersolvable. A dihedral group is nilpotent if and only if it is of order $$2^k$$ for some $$k$$. It is abelian only if it has order $$2$$ or $$4$$.

Arithmetic functions
$$n$$ here denotes the degree, or half the order, of the dihedral group, which we denote as $$D_{2n}$$.

Subgroups
There are two kinds of subgroups:


 * Subgroups of the form $$\langle a^d \rangle$$, where $$d|n$$. The number of such subgroups equals the number of positive divisors of $$n$$, sometimes denoted $$\tau(n)$$. The subgroup generated by $$a^d$$ is a cyclic group of order $$n/d$$.
 * Subgroups of the form $$\langle a^d, a^r x \rangle$$, where $$d|n$$ and $$0 \le r < d$$. The number of such subgroups equals the sum of all positive divisors of $$n$$, sometimes denoted $$\sigma(n)$$. The subgroup of the above form is a dihedral group of order $$2n/d$$.

In particular, all subgroups of the dihedral group are either cyclic or dihedral.

Also note that the dihedral group has subgroups of all orders dividing its order. This is true more generally for all finite supersolvable groups.

Groups having the dihedral group as quotient
The dicyclic group, also called the binary dihedral group, of order $$4n$$, has the dihedral group of order $$2n$$ as a quotient -- in fact the quotient by its center, which is a cyclic subgroup of order two. the presentation for the dicyclic group is given by:

$$\langle a,x \mid a^n = x^2 = (ax)^2 \rangle$$.

Dicyclic groups whose order is a power of $$2$$ are termed generalized quaternion groups.

Internal links

 * Element structure of dihedral groups
 * Subgroup structure of dihedral groups
 * Linear representation theory of dihedral groups
 * Group cohomology of dihedral groups

Textbook references

 * , Page 23-27, Section 1.2 Dihedral Groups (the entire section discusses dihedral groups from a number of perspectives)
 * , Page 24 (definition introduced in paragraph)
 * , Page 78, Exercise 34 (a) (definition introduced in exercise)
 * , Page 54, Problem 17
 * , Page 6 (definition introduced informally, in paragraph, using the geometric perspective)
 * , Page 42, under The symmetry group of the regular n-gon
 * , Page 50 (definition introduced as a subgroup of the symmetric group)