User:Jim.belk/Dihedral Group Draft

Definition

Dihedral symmetry

Plane symmetry

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

Triskelion	Bullseye	Pentagram

• 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.

Symmetry in three dimensions

Dihedral groups in group theory

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, D_n has one conjugacy class of reflections.
• When n is even, D_n has two conjugacy classes of reflections (reflections across vertex bisectors and reflections across edge bisectors). There is an outer automorphism of D_n 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 C_k subgroup for every k dividing n. (These are the standard subgroups of C_n.)
• The C_k subgroup is contained in n / k different D_k subgroups (obtained by choosing any coset of C_k 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 D_k subgroups fall into two conjugacy classes (corresponding to the two conjugacy classes of reflections).

Normal subgroups

• When n is odd, the normal subgroups of D_n are precisely C_n and its subgroups.
• When n is even, the two copies of D_n/2 are normal as well. These are the kernels of the homomorphisms D_n → C₂ defined by (r, s) → (-1, -1) and (r, s) → (-1, 1)
• Only the cyclic subgroups are characteristic.

Commutator subgroup

• If n is odd, the commutator subgroup of D_n is C_n, with abelianization C₂.
• If n is even, the commutator subgroup of D_n is C_n/2, with abelianization D₂ (= C₂×C₂). The lifts of the C₂ subgroups of the abelianization are precisely the index-two normal subgroups of D_n.

Automorphisms

The automorphism group of D_n is the group of all affine transformations of Z_n (i.e. the semidirect product of Z_n^× with Z_n. Specifically, if a ∈ Z_n^× and b ∈ Z_n, then the transformation

$f(x)=ax+b,\;\;\;\;x\in \mathbb {Z} _{n}$

acts on D_n by the rule

$f(r^{i})=r^{ai}\;\;\;\;{\text{and}}\;\;\;\;f(r^{i}s)=r^{ai+b}s$

Other stuff

• D_n is solvable.
• D_n 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.

Generalizations

See also

• examples of groups
• symmetry group
• dicyclic group
• quasidihedral group
• Verhoeff algorithm
• Point groups in three dimensions
• Dihedral symmetry in three dimensions
• List of planar symmetry groups

Snippets of text

In group theory, the dihedral group D_n 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 D_n 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 D_n as a Coxeter group. Any finite group generated by two elements of order two is dihedral.

Parity Considerations

The algebraic structure of D_n 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 D_n. The center of D_n 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 D_n 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, D_n 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 D_n fits into a short exact sequence:

$0\rightarrow C_{n}\rightarrow D_{n}\rightarrow C_{2}\rightarrow 0$

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

• If p is prime, then C_2p and D_p are the only groups of order 2p.

Generators

The dihedral group D_n is generated by the rotation $r=R_{1}$ of order n and the reflection $s=S_{0}$ of order 2. Each element of D_n can be written in terms of these generators:

$R_{k}=r^{k},\;\;\;\;S_{k}=r^{k}s$

The

Presentation

An illustration of the relation $sr=r^{-1}s$ in D₈

The dihedral group D_n 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, R_k = r^k and S_k = r^ks. 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 D_n:

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

Other Generating Sets

The dihedral group D_n can also be generated by any two adjacent reflections, e.g. $s=S_{0}$ and $t=S_{1}$. (This shows that D_n 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 D_n.

Product structure

The rotations in D_n 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 D_n map to the identity in C₂, while the reflections map to the nonidentity element. This sequence splits, making D_n a semidirect product. The group C₂ acts on C_n 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 D_n 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 D_n 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 D_n, 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, D_n has a distinguished non-identity element σ, 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 D_n, taken in any order.

The Coxeter element commutes with every other element of the group. When n is even, the center of D_n 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 D_n is nilpotent if and only if n is a power of 2.

Conjugacy and centralizers

Each rotation in D_n 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.

Subgroup structure

Dihedral subgroups

The subgroups of D_n are:

• One copy of C₂ for each reflection.
• One copy of C_{k for each divisor k of n'.}
• One copy of D_k for each divisor k of n.

The subgroup D_k can be thought of as the symmetries of an inscribed k-gon. [[The maximal subgroups of D_n

Automorphisms

The automorphisms of D_n 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 D_n 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(D_n) 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 D_n 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

D_n possesses the following subgroups:

• the cyclic subgroup C_n
• the subgroups of C_n, namely one copy of C_k for every divisor k of n.
• D_n has

The lattice of dihedral subgroups of D_n is isomorphic to the [[coset lattice] of

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 D_n are conjugate.

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

Permutation representation

The group D₅ permutes the vertices of the pentagon.

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 D_n permute vertices labeled 0, 1, ..., n − 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.