Dihedral Groups

## Dihedral Groups

Definition. The Dihedral group, ${D}_{n}$ is the set ${D}_{n}=\left\{1,x,{x}^{2},\dots ,{x}^{n-1},y,xy,{x}^{2}y,\dots ,{x}^{n-1}y\right\}$ with the operation given by $xi yj xk yl = x i+k mod n y j+l mod 2 .$

HW: Show that the order of the dihedral group ${D}_{n}$ is $2n$.

The orders of the elements in the dihedral group ${D}_{n}$ are $ο1=1, οxk= gcdk,n, οxky=2, 0

## Conjugacy classes, normal subgroups, and the center

1. The conjugacy classes of the dihedral group ${D}_{2}$ are the sets $𝒞1= 1 , 𝒞x= x , 𝒞y= y , 𝒞xy= xy .$
2. If $n$ is even and $n\ne 2$, then the conjugacy classes of the dihedral group ${D}_{n}$ are the sets $𝒞1= 1 , 𝒞xn/2 = xn/2 , 𝒞xk= xk,x-k , 0
3. If $n$ is even and $n\ne 2$, then the conjugacy classes of the dihedral group ${D}_{n}$ are the sets $𝒞1= 1 , 𝒞xk= xk,x-k , 0

Let $⟨a,b,\dots ⟩$ denote the subgroup generated by the elements $a,b,\dots .$

1. The normal subgroups of the dihedral group ${D}_{2}$ are the subgroups $⟨x⟩, ⟨y⟩, ⟨xy⟩.$
2. If $n$ is even and $n\ne 2$ then the normal subgroups of the dihedral group ${D}_{n}$ are the subgroups $⟨xk⟩, 0≤k≤n-1, ⟨x2,y⟩, ⟨x2,xy⟩.$
3. If $n$ is odd, then the normal subgroups of the dihedral group ${D}_{n}$ are the subgroups $⟨xk⟩, 1≤k≤n-1.$

1. The center of the dihedral group ${D}_{2}$ is the subgroup $Z\left({D}_{2}\right)={D}_{2}.$
2. If $n$ is even and $n\ne 2$,then the center of the dihedral group ${D}_{n}$ is the subgroup $Z\left({D}_{n}\right)=\left\{1,{x}^{n/2}\right\}.$
3. If $n$ is odd, then the center of the dihedral group ${D}_{n}$ is the subgroup $Z\left({D}_{n}\right)=⟨1⟩.$

## The action of ${D}_{n}$ on an $n$-gon

Let $F$ be an $n$-gon with vertices ${v}_{0},{v}_{1},\dots ,{v}_{n-1}$ numbered counterclockwise around $F$. Then there is an action of the group ${D}_{n}$ on the $n$-gon $F$ such that

1. $x$ acts by rotating the $n$-gon by an angle of $2\pi /n$;
2. $y$ acts by reflecting about the line which contains the vertex ${v}_{0}$ and the center of $F$.

## Generators and relations

1. The dihedral group ${D}_{n}=\left\{1,x,{x}^{2},\dots ,{x}^{n-1},y,yx,y{x}^{2},\dots ,y{x}^{n-1}\right\}$ is generated by the elements $x$ and $y$.
2. The elements $x$ and $y$ in Dn satisfy the relations $xn=1, y2=1, yx=x-1y.$

The dihedral group ${D}_{n}$ has a presentation by generators and relations by $Dn= ⟨ x,y∣ xn=1, y2=1, yx=x-1y⟩.$

## References

[Bou] N. Bourbaki, Groupes et Algèbres de Lie, Masson, Paris, 1990.

[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)