User:Jim.belk/Dihedral Group Draft

Plane symmetry
An object in the plane has dihedral symmetry if its symmetry group is a dihedral group. Consider the following three examples:


 * The triskelion has rotational symmetry, but it is not symmetric under any reflections. The symmetry group of the triskelion is cyclic of order three.
 * The bullseye has complete circular symmetry, which is described by the orthogonal group O(2).
 * The pentagram has dihedral symmetry, with both rotations and translations. The symmetry group of the pentagram is the dihedral group D5.

Many geometric objects have dihedral symmetry: in addition to regular polygons, there are also star polygons, and various classical plane curves such as roses, epicycloids, hypocycloids, epitrochoids, and hypotrochoids. Many natural objects also exhibit dihedral symmetry, including snowflakes and some flowers.

Presentations

 * $$\text{1. }\langle r,s\;|\;r^n=s^2=1,\,r^s=r^{-1}\rangle$$
 * $$\text{2. }\langle s,t\;|\;s^2 = t^2 = (st)^n = 1\rangle$$

The first comes from the semidirect product, and the second is the Coxeter presentation. The second presentation is related to the following theorem:
 * Theorem: Any group generated by two elements of order two is dihedral.

Extensions

 * $$1 \rightarrow C_n \rightarrow D_n \rightarrow C_2 \rightarrow 1$$

This is the split extension associated to the semidirect product.
 * $$1 \rightarrow \{ 1,\sigma \} \rightarrow D_n \rightarrow D_{n/2} \rightarrow 1$$

This is a central extension. Only exists when n is even.

Conjugacy and centralizers

 * When n is odd, Dn has one conjugacy class of reflections.
 * When n is even, Dn has two conjugacy classes of reflections (reflections across vertex bisectors and reflections across edge bisectors). There is an outer automorphism of Dn that switches these conjugacy classes.
 * In either case, each element of the cyclic group is conjugate only to its inverse.
 * When n is even, the n/2 element of the cyclic group (representing a 180 degree rotation) is central. The center is trivial for n odd.
 * When n is odd, each reflection is centralized only by itself and the identity. For n even, the centralizer of a reflection has order four (itself, the identity, the central element, and the product of the two).
 * Each element of the cyclic group (other than the identity and the central element) is centralized by precisely the cyclic group.

Subgroups

 * Every subgroup of the dihedral group is either dihedral or cyclic.
 * There is one Ck subgroup for every k dividing n. (These are the standard subgroups of Cn.)
 * The Ck subgroup is contained in n / k different Dk subgroups (obtained by choosing any coset of Ck that consists entirely of reflections.
 * These subgroups are all conjugate when either n or n / k is odd.
 * When n is even and n / k is even, the Dk subgroups fall into two conjugacy classes (corresponding to the two conjugacy classes of reflections).

Normal subgroups

 * When n is odd, the normal subgroups of Dn are precisely Cn and its subgroups.
 * When n is even, the two copies of Dn/2 are normal as well. These are the kernels of the homomorphisms Dn &rarr; C2 defined by (r, s) &rarr; (-1, -1) and (r, s) &rarr; (-1, 1)
 * Only the cyclic subgroups are characteristic.

Commutator subgroup

 * If n is odd, the commutator subgroup of Dn is Cn, with abelianization C2.
 * If n is even, the commutator subgroup of Dn is Cn/2, with abelianization D2 (= C2&times;C2). The lifts of the C2 subgroups of the abelianization are precisely the index-two normal subgroups of Dn.

Automorphisms
The automorphism group of Dn is the group of all affine transformations of Zn (i.e. the semidirect product of Zn&times; with Zn. Specifically, if a &isin; Zn&times; and b &isin; Zn, then the transformation
 * $$f(x) = ax+b,\;\;\;\;x\in \mathbb{Z}_n$$

acts on Dn by the rule
 * $$f(r^i) = r^{ai}\;\;\;\;\text{and}\;\;\;\;f(r^i s) = r^{ai + b} s$$

Other stuff

 * Dn is solvable.
 * Dn is nilpotent if and only if n is power of 2.
 * The dihedral groups are the simplest nontrivial examples of metacyclic groups.
 * The dihedral groups are one of three infinite families of finite reflection groups. The corresponding root system consists of the vertices of a regular n-gon centered at the origin.
 * The dihedral group is a Frobenius group for odd values of n.

Snippets of text
In group theory, the dihedral group Dn is defined by the following presentation:


 * $$\langle r,s\;|\;r^n=s^2=1,\,r^s=r^{-1}\rangle$$

This group has order 2n, with elements $$\{ 1, r, r^2, \ldots, r^{n-1}, s, rs, r^2 s, \ldots, r^{n-1} s \}$$.

Product structure
The dihedral group Dn is the semidirect product of the of the cyclic subgroups $$\langle r \rangle \cong C_n$$ and $$\langle s \rangle \cong C_2$$. Extrinsically, it can be described as the semidirect product $$C_n \rtimes_\phi C_2$$, where $$\phi_s(r)=r^{-1}\,$$. This is arguably the simplest nontrivial example of a semidirect product.

Alternate presentation
The dihedral group can also be generated by a pair of reflections:


 * $$\langle s,t\;|\;s^2=t^2=(st)^n=1\rangle$$

This is the presentation for Dn as a Coxeter group. Any finite group generated by two elements of order two is dihedral.

Parity Considerations
The algebraic structure of Dn for even values of n is quite different from the structure when n is odd. This is because a regular n-gon is symmetric across the center point when n is even, but not when n is odd.

For example, when n is even, the element $$r^{n/2}$$ (representing a 180-degree rotation) is central, meaning that it commutes with every other element of Dn. The center of Dn is trivial when n is odd.

The conjugacy classes are also quite different in the even and odd cases. There are two different conjugacy classes of reflections when n is even, namely those that fix two vertices of the n-gon and those that fix the midpoints of two edges. (There is an outer automorphism of Dn that switches these two conjugacy classes.) When n is odd, any reflection fixes exactly one vertex of the n-gon, and any two reflections are conjugate by a rotation.

The subgroup generated by r is normal, and the quotient is a cyclic group of order 2. This information is summarized in the following short exact sequence:


 * $$ 1 \rightarrow C_n \rightarrow D_n \rightarrow C_2 \rightarrow 1$$

This sequence is The generator r has order n, while the generator s has order 2. Alternatively, Dn may be defined by the presentation


 * $$\langle s,t\;|\;s^2=t^2=(st)^n=1\rangle$$.

Dihedral groups Along with cyclic groups and symmetric groups, the dihedral groups are a basic class of examples in finite group theory. They have the following properties:
 * Dihedral groups are among the simplest examples of semidirect products. The dihedral group Dn fits into a short exact sequence:


 * $$0 \rightarrow C_n \rightarrow D_n \rightarrow C_2 \rightarrow 0$$

Here Cn is the (normal) cyclic subgroup generated by r, and C2 is the two-element subgroup $$\{1,s\}$$.


 * If p is prime, then C2p and Dp are the only groups of order 2p.

Generators
The dihedral group Dn is generated by the rotation $$r=R_1$$ of order n and the reflection $$s=S_0$$ of order 2. Each element of Dn can be written in terms of these generators:


 * $$R_k = r^k,\;\;\;\;S_k = r^k s$$

The

Presentation


The dihedral group Dn is generated by the rotation r = R1 and the reflection s = S0:


 * $$ r \;=\; \begin{pmatrix}

\cos 2\pi/n & -\sin 2\pi/n \\ \sin 2\pi/n & \cos 2\pi/n \end{pmatrix}$$   and    $$s\;=\;\begin{pmatrix} 1 & 0 \\   0 & -1  \end{pmatrix}$$

In particular, Rk = rk and Sk = rks. These generators satisfy the following algebraic identities:
 * $$r^n=1,\;\;\;\;s^2=1,\;\;\;\;sr=r^{-1}s$$

In the context of group theory, identities like this are known as relations. Together, these generators and relations form a presentation of the dihedral group Dn:


 * $$\mathrm{D}_n \;=\; \langle r,s \;|\; r^n=1,\,s^2=1,\,sr=r^{-1}s \rangle $$

Other Generating Sets
The dihedral group Dn can also be generated by any two adjacent reflections, e.g. $$s=S_0$$ and $$t=S_1$$. (This shows that Dn is a finite reflection group.) The product ts is the rotation r, and therefore $$R_k = (ts)^k$$ and $$S_k = (ts)^k s$$. These generators yield the following presentation:


 * $$\mathrm{D}_n \;\cong\; \langle s,t \;|\; s^2 = 1,\,t^2 = 1,\,(ts)^n = 1 \rangle $$

This is the Coxeter presentation for Dn.

Product structure
The rotations in Dn form a cyclic normal subgroup of order n. This gives rise to a short exact sequence:


 * $$1 \rightarrow C_n \rightarrow D_n \rightarrow C_2 \rightarrow 1$$

The rotations in Dn map to the identity in C2, while the reflections map to the nonidentity element. This sequence splits, making Dn a semidirect product. The group C2 acts on Cn by inversion:


 * $$\mathrm{D}_n \;\cong\; \langle r,s \;|\; r^n=1,\,s^2=1,\,srs=r^{-1} \rangle $$

This is arguably the simplest nontrivial example of a semidirect product.

Parity Considerations
In many ways, the algebraic structure of Dn for even values of n is quite different from the structure when n is odd. This is because a regular n-gon possesses point symmetry when n is even, but not when n is odd.

This manifests itself in the dihedral group in two primary ways:


 * 1) When n is even, the group Dn has a distinguished element (the [Coxeter group|Coxeter element]]) representing rotation by 180 degrees.  This element is central, meaning that it commutes with every other element of the group.  It can be described as the product of all of the reflections in Dn, taken in any order.


 * 1) When n is odd, all of the reflections in n are conjugate.  When n

Coxeter element
When n is even, Dn has a distinguished non-identity element &sigma;, known as the Coxeter element. Geometrically, the Coxeter element acts as the 180-degree rotation of the n-gon, i.e. the unique rotation of order 2. It is equal to the product of all the reflections in Dn, taken in any order.

The Coxeter element commutes with every other element of the group. When n is even, the center of Dn is the two-element subgroup $$\{1,\sigma\}$$. This gives rise to a split exact sequence:


 * $$1 \rightarrow \{ 1,\sigma \} \rightarrow D_n \rightarrow D_{n/2} \rightarrow 1$$

When n is odd, the product of all the reflections is again a reflection, with the result depending on the order of multiplication. The dihedral group has trivial center for odd values of n. It follows that Dn is nilpotent if and only if n is a power of 2.

Conjugacy and centralizers
Each rotation in Dn is centralized by the full group of rotations, and is conjugate to its inverse by any reflection. For even n, the Coxeter element is its own inverse, and is centralized by the entire group.

Dihedral subgroups
The subgroups of Dn are: The subgroup Dk can be thought of as the symmetries of an inscribed k-gon. The [[maximal subgroups of Dn
 * One copy of C2 for each reflection.
 * One copy of C k for each divisor k of ''n'.
 * One copy of Dk for each divisor k of n.

Automorphisms
The automorphisms of Dn are the maps $$\varphi_{a,b}$$ defined by:
 * $$\varphi_{a,b}(r)= r^a\;\;\;\;\mbox{and}\;\;\;\;\varphi_{a,b}(s) = r^b s$$

Here $$a$$ may be any unit in the ring $$\mathbb{Z}_n$$, and $$b$$ may be any element of $$\mathbb{Z}_n$$. The composition of two automorphisms is given by the rule:
 * $$\varphi_{a,b} \circ \varphi_{a^\prime,b^\prime} = \varphi_{aa^\prime,ab^\prime+b}$$

It follows that the automorphism group of Dn is a semidirect product $$\mathbb{Z}_n^\times \ltimes \mathbb{Z}_n$$, where $$\mathbb{Z}_n^\times$$ denotes the group of units in $$\mathbb{Z}_n$$. Alternatively, Aut(Dn) may be described as the group of affine transformations $$\varphi_{a,b}(x)=ax+b$$ of $$\mathbb{Z}_n$$.

Outer automorphisms? (Different for even and odd n.)

Conjugacy
Each rotation in Dn is conjugate only to its inverse. For n odd, the result is $$(n-1)/2$$ different conjugacy classes of rotations, each with 2 elements. For n even, we get $$n/2$$ different conjugacy classes of rotations, with the Coxeter element in a class by itself.

Subgroups
Dn possesses the following subgroups: The lattice of dihedral subgroups of Dn is isomorphic to the [[coset lattice] of
 * the cyclic subgroup Cn
 * the subgroups of Cn, namely one copy of Ck for every divisor k of n.
 * Dn has

Rotations: The centralizer of any rotation is the full rotation group. Each rotation is conjugate to its inverse, resulting in $$\frac{n-1}{2}$$ different conjugacy classes of rotations.

Reflections: The centralizer of any reflection t is just the two-element group $$\{1,t\}$$. All reflections in Dn are conjugate.

We conclude that Dn has a single conjugacy class of size 1 (the identity element), math>\frac{n-1}{2} different conjugacy classes of size 2, and a single conjugacy class of size n.

Permutation representation
Elements of a dihedral group may be thought of as permutations of the vertices of the associated polygon. For example, if we label the vertices of a regular pentagon with the numbers 0, 1, 2, 3, 4, then the ten elements of D5 can be written as follows:


 * $$\begin{matrix}

R_0 = 0, 1, 2, 3, 4 &\;\; &  S_0 = 0, 4, 3, 2, 1\\[0.2em] R_1 = 1, 2, 3, 4, 0 & & S_1 = 1, 0, 4, 3, 2\\[0.2em] R_2 = 2, 3, 4, 0, 1 & & S_2 = 2, 1, 0, 4, 3\\[0.2em] R_3 = 3, 4, 0, 1, 2 & & S_3 = 3, 2, 1, 0, 4\\[0.2em] R_4 = 4, 0, 1, 2, 3 & & S_4 = 4, 3, 2, 1, 0 \end{matrix}$$

In general, elements of Dn permute vertices labeled 0, 1, ..., n &minus; 1 according to the following rules:


 * $$R_i(j) = i + j\mod n,\;\;\;\;S_i(j) = i - j\;\mod n$$

This representation is faithful for all n > 2. In the case n = 3, any permutation of the three vertices of a triangle is possible, leading to an isomorphism between D3 and the symmetric group S3.

Other properties

 * The dihedral groups are the simplest nontrivial examples of metacyclic groups.
 * The dihedral groups are one of three infinite families of finite reflection groups. The corresponding root system consists of the vertices of a regular n-gon centered at the origin.
 * The dihedral group is a Frobenius group for odd values of n.
 * The symmetry group of a regular star polygon is always dihedral.