Dihedral Groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 6 December 2010

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 $$\left(x^i{y}^j\right)\left(x^k{y}^l\right)=x^{\left(i+k\right)\hspace{0.17em}mod\hspace{0.17em}n}{y}^{\left(j+l\right)\hspace{0.17em}mod\hspace{0.17em}2}.$$

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 $$o\left(1\right)=1,o\left(x^k\right)=gcd\left(k,n\right),o\left(x^{k}y\right)=2,0<k\le n-1.$$

Conjugacy classes, normal subgroups, and the center

1. The conjugacy classes of the dihedral group $D_2$ are the sets $${𝒞}_1=\left\{1\right\},{𝒞}_x=\left\{x\right\},{𝒞}_y=\left\{y\right\},{𝒞}_{xy}=\left\{xy\right\}.$$
2. If $n$ is even and $n\ne 2$, then the conjugacy classes of the dihedral group $D_n$ are the sets $$\begin{array}{ll}{𝒞}_1=\left\{1\right\},{𝒞}_{x^{n/2}}=\left\{x^{n/2}\right\},& {𝒞}_{x^k}=\left\{x^k,x^{-k}\right\},0<k<n/2,\\ {𝒞}_y=\left\{y,x^{2}y,x^{4}y,\dots ,x^{n-2}y\right\},& {𝒞}_{xy}=\left\{xy,x^{3}y,x^5,\dots ,x^{n-1}y\right\}.& \end{array}$$
3. If $n$ is even and $n\ne 2$, then the conjugacy classes of the dihedral group $D_n$ are the sets $$\begin{array}{c}{𝒞}_1=\left\{1\right\},{𝒞}_{x^k}=\left\{x^k,x^{-k}\right\},0<k<n/2,\\ {𝒞}_y=\left\{y,xy,x^{2}y,x^{3}y,\dots ,x^{n-1}y\right\}.\end{array}$$

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
2. If $n$ is even and $n\ne 2$ then the normal subgroups of the dihedral group $D_n$ are the subgroups $$⟨x^k⟩,0\le k\le n-1,⟨x^2,y⟩,⟨x^2,xy⟩.$$
3. If $n$ is odd, then the normal subgroups of the dihedral group $D_n$ are the subgroups $$⟨x^k⟩,1\le k\le n-1.$$

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 $$x^n=1,y^2=1,yx=x^{-1}y.$$

The dihedral group $D_n$ has a presentation by generators and relations by $$D_n=⟨x,y\mid x^n=1,y^2=1,yx=x^{-1}y⟩.$$

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)

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