The Dihedral Group $D_4$ of Order Eight

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

Last updates: 26 January 2011

The dihedral group $D_4$ of order eight

The groups $D_4$ is as in the following table.

Set	Operation
$D_4=\left\{1,x,x^2,x^3,y,xy,x^{2}y,x^{3}y\right\}$	$x^i{y}^j{x}^k{y}^l=x^{\left(i-k\right)mod\hspace{0.17em}4}{y}^{\left(j+l\right)mod\hspace{0.17em}2}$

The complete multiplication tables for $D_4$ is as follows.

Multiplication table

$D_4$ $1$ $x$ $x^2$ $x^3$ $y$ $xy$ $x^{2}y$ $x^{3}y$ $1$ $1$ $x$ $x^2$ $x^3$ $y$ $xy$ $x^{2}y$ $x^{3}y$ $x$ $x$ $x^2$ $x^3$ $1$ $xy$ $x^{2}y$ $x^{3}y$ $y$ $x^2$ $x^2$ $x^3$ $1$ $x$ $x^{2}y$ $x^{3}y$ $y$ $xy$ $x^3$ $x^3$ $1$ $x$ $x^2$ $x^{3}y$ $y$ $xy$ $x^{2}y$ $y$ $y$ $x^{3}y$ $x^{2}y$ $xy$ $1$ $x^3$ $x^2$ $x$ $xy$ $xy$ $y$ $x^{3}y$ $x^{2}y$ $x$ $1$ $x^3$ $x^2$ $x^{2}y$ $x^{2}y$ $xy$ $y$ $x^{3}y$ $x^2$ $x$ $1$ $x^3$ $x^{3}y$ $x^{3}y$ $x^{2}y$ $xy$ $y$ $x^3$ $x^2$ $x$ $1$

Center	Abelian	Conjugacy classes	Subgroups
$Z\left(D_4\right)=\left\{1,x^2\right\}$	No	${𝒞}_1=\left\{1\right\}$	$H_0=D_4$
		${𝒞}_{x^2}=\left\{x^2\right\}$	$H_1=\left\{1,x,x^2,x^3\right\}$
		${𝒞}_y=\left\{y,x^{2}y\right\}$	$H_2=\left\{1,x^2,y,x^{2}y\right\}$
		${𝒞}_{xy}=\left\{xy,x^{3}y\right\}$	$H_3=\left\{1,x^2,xy,x^{3}y\right\}$
		${𝒞}_x=\left\{x,x^3\right\}$	$H_4=\left\{1,x^2\right\}$
			$H_5=\left\{1,y\right\}$
			$H_6=\left\{1,xy\right\}$
			$H_7=\left\{1,x^{2}y\right\}$
			$H_8=\left\{1,x^{3}y\right\}$
			$H_9=\left\{1\right\}$

Element $g$	Order $o\left(g\right)$	Centralizer $Z_g$	Conjugacy Class ${𝒞}_g$
$1$	$1$	$D_4$	${𝒞}_1$
$x$	$4$	$H_1$	${𝒞}_x$
$x^2$	$2$	$D_4$	${𝒞}_{x^2}$
$x^3$	$4$	$H_1$	${𝒞}_x$
$y$	$2$	$H_2$	${𝒞}_y$
$xy$	$2$	$H_3$	${𝒞}_{xy}$
$x^{2}y$	$2$	$H_2$	${𝒞}_y$
$x^{3}y$	$2$	$H_3$	${𝒞}_{xy}$

	Generators	Relations
$D_4$	$x,y$	$x^4=y^2=1$
		$yx=x^{-1}y$

Subgroups $H_i$	Structure	Index	Normal	Quotient group
$H_0=D_4$	$H_0=D_4$	$\left[D_4:D_4\right]=1$	Yes	$D_4/H_0\cong ⟨1⟩$
$H_1=\left\{1,x,x^2,x^3\right\}$	$H_1\cong C_4$	$\left[D_4:H_1\right]=2$	Yes	$D_4/H_1\cong C_2$
$H_2=\left\{1,x^2,y,x^{2}y\right\}$	$H_2\cong C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2$	$\left[D_4:H_2\right]=2$	Yes	$D_4/H_2\cong C_2$
$H_3=\left\{1,x^2,xy,x^{3}y\right\}$	$H_3\cong C_2\hspace{0.17em}\times \hspace{0.17em}{C}_2$	$\left\{D_4:H_3\right\}=2$	Yes	$D_4/H_3\cong C_2$
$H_4=\left\{1,x^2\right\}$	$H_4\cong C_2$	$\left[D_4:H_4\right]=4$	No
$H_5=\left\{1,y\right\}$	$H_5\cong C_2$	$\left[D_4:H_5\right]=4$	No
$H_6=\left\{1,xy\right\}$	$H_6\cong C_2$	$\left[D_4:H_6\right]=4$	No
$H_7=\left\{1,x^{2}y\right\}$	$H_7\cong C_2$	$\left[D_4:H_7\right]=4$	No
$H_8=\left\{1,x^{3}y\right\}$	$H_8\cong C_2$	$\left[D_4:H_8\right]=4$	No
$H_9=\left\{1\right\}$	$H_9=⟨1⟩$	$\left[D_4:⟨1⟩\right]=1$	Yes	$D_4/H_9\cong D_4$

Subgroups $H_i$	Left Cosets	Right Cosets
$H_0=D_4$	$D_4=x{D}_4=x^3{D}_4=y{D}_4$	$D_4=D_{4}x=D_4{x}^2=D_4{x}^3=D_{4}y$
	$=xy{D}_4=x^{2}y{D}_4=x^{3}y{D}_4$	$=D_{4}xy=D_4{x}^{2}y=D_4{x}^{3}y$
$H_1=\left\{1,x,x^2,x^3\right\}$	$H_1=x{H}_1=x^2{H}_1=x^3{H}_1$	$H_1=H_{1}x=H_1{x}^2=H_1{x}^3$
	$=\left\{1,x,x^2,x^3\right\}$	$=\left\{1,x,x^2,x^3\right\}$
	$y{H}_1=xy{H}_1=x^{2}y{H}_1=x^{3}y{H}_1$	$H_{1}y=H_{1}xy=H_1{x}^{2}y=H_1{x}^{3}y$
	$=\left\{y,xy,x^{2}y,x^{3}y\right\}$	$=\left\{y,xy,x^{2}y,x^{3}y\right\}$
$H_2=\left\{1,x^2,y,x^{2}y\right\}$	$H_2=x^2{H}_2=y{H}_2=x^{2}y{H}_2$	$H_2=H_2{x}^2=H_{2}y=H_2{x}^{2}y$
	$=\left\{1,x^2,y,x^{2}y\right\}$	$=\left\{1,x^2,y,x^{2}y\right\}$
	$x{H}_2=x^3{H}_2=xy{H}_2=x^{3}y{H}_2$	$H_{2}x=H_2{x}^3=H_{2}xy=H_2{x}^{3}y$
	$=\left\{x,x^3,xy,x^{3}y\right\}$	$=\left\{x,x^3,xy,x^{3}y\right\}$
$H_3=\left\{1,x^2,xy,x^{3}y\right\}$	$H_3=x^2{H}_3=xy{H}_3=x^{3}y{H}_3$	$H_3=H_3{x}^2=H_{3}xy=H_3{x}^{3}y$
	$=\left\{1,x^2,xy,x^{3}y\right\}$	$=\left\{1,x^2,xy,x^{3}y\right\}$
	$x{H}_3=x^3{H}_3=y{H}_3=x^{2}y{H}_3$	$H_{3}x=H_3{x}^3=H_{3}y=H_3{x}^{2}y$
	$=\left\{x,x^3,y,x^{2}y\right\}$	$=\left\{x,x^3,y,x^{2}y\right\}$
$H_4=\left\{1,x^2\right\}$	$H_4=x^2{H}_4=\left\{1,x^2\right\}$	$H_4=H_4{x}^2=\left\{1,x^2\right\}$
	$x{H}_4=x^3{H}_4=\left\{x,x^3\right\}$	$H_{4}x=H_4{x}^3=\left\{x,x^3\right\}$
	$y{H}_4=x^{2}y{H}_4=\left\{y,x^{2}y\right\}$	$H_{4}y=H_4{x}^{2}y=\left\{y,x^{2}y\right\}$
	$xy{H}_4=x^{3}y{H}_4=\left\{xy,x^{3}y\right\}$	$H_{4}xy=H_4{x}^{3}y=\left\{xy,x^{3}y\right\}$
$H_5=\left\{1,y\right\}$	$H_5=y{H}_5=\left\{1,y\right\}$	$H_5=H_{5}y=\left\{1,y\right\}$
	$x{H}_5=xy{H}_5=\left\{x,xy\right\}$	$H_{5}x=H_5{x}^{3}y=\left\{x,x^{3}y\right\}$
	$x^2{H}_5=x^{2}y{H}_5=\left\{y,x^{2}y\right\}$	$H_5{x}^2=H_5{x}^{2}y=\left\{x^2,x^{2}y\right\}$
	$x^3{H}_5=x^{3}y{H}_5=\left\{x^3,x^{3}y\right\}$	$H_5{x}^3=H_{5}xy=\left\{x^3,xy\right\}$
$H_6=\left\{1,xy\right\}$	$H_6=xy{H}_6=\left\{1,xy\right\}$	$H_6=H_{6}xy=\left\{1,xy\right\}$
	$x{H}_6=x^{2}y{H}_6=\left\{x,x^{2}y\right\}$	$H_{6}x=H_6{x}^{2}y=\left\{x,y\right\}$
	$x^2{H}_6=x^{3}y{H}_6=\left\{x^2,x^{3}y\right\}$	$H_6{x}^2=H_6{x}^{3}y=\left\{x^2,x^{3}y\right\}$
	$x^3{H}_6=y{H}_6=\left\{x^3,y\right\}$	$H_6{x}^3=H_6{x}^{2}y=\left\{x^3,x^{2}y\right\}$
$H_7=\left\{1,x^{2}y\right\}$	$H_7=x^{2}y{H}_7=\left\{x,x^{2}y\right\}$	$H_7=H_7{x}^{2}y=\left\{1,x^{2}y\right\}$
	$x{H}_7=x^{3}y{H}_7=\left\{x,x^{3}y\right\}$	$H_{7}x=H_{7}xy=\left\{x,xy\right\}$
	$x^2{H}_7=y{H}_7=\left\{x^2,y\right\}$	$H_7{x}^2=H_{7}y=\left\{x^2,y\right\}$
	$x^3{H}_7=xy{H}_7=\left\{x^3,xy\right\}$	$H_7{x}^3=H_7{x}^{3}y=\left\{x^3,x^{3}y\right\}$
$H_8=\left\{1,x^{3}y\right\}$	$H_8=x^{3}y{H}_8=\left\{1,x^{3}y\right\}$	$H_8=H_8{x}^{3}y=\left\{1,x^{3}y\right\}$
	$x{H}_8=y{H}_8=\left\{x,y\right\}$	$H_{8}x=H_8{x}^{2}y=\left\{x^{2}y,y\right\}$
	$x^2{H}_8=xy{H}_8=\left\{x^2,xy\right\}$	$H_8{x}^2=H_{8}xy=\left\{x^2,xy\right\}$
	$x^3{H}_8=x^{2}y{H}_{\mathrm{<mn>8</mn>}}=\left\{x^3,x^{2}y\right\}$	$H_8{x}^3=H_{8}y=\left\{x^3,y\right\}$
$H_9=\left\{1\right\}$	$H_9=\left\{1\right\},x{H}_9=\left\{x\right\},$	$H_9=\left\{1\right\},H_{9}x=\left\{x\right\},$
	$x^2{H}_9=\left\{x^2\right\},x{H}_9=\left\{x\right\},$
	$y{H}_9=\left\{y\right\},xy{H}_9=\left\{xy\right\},$	$H_{9}y=\left\{y\right\},H_{9}xy=\left\{xy\right\},$
	$x^{2}y{H}_9=\left\{x^{2}y\right\},x^{3}y{H}_9=\left\{x^{3}y\right\},$	$H_9{x}^{2}y=\left\{x^{2}y\right\},H_9{x}^{3}y=\left\{x^{3}y\right\},$

Subgroups $H_i$	Normalizer $N_{H_i}$	Centralizer $Z_{H_i}$
$H_0=D_4$	$H_0=D_4$	$Z\left(D_4\right)=H_4=⟨x^2⟩$
$H_1=⟨x⟩$	$D_4$	$H_1=⟨x⟩$
$H_2=⟨x^2,y⟩$	$D_4$	$H_2=⟨x^2,y⟩$
$H_3=⟨x^2,xy⟩$	$D_4$	$H_3=⟨x^2,xy⟩$
$H_4=⟨x^2⟩$	$D_4$	$D_4$
$H_5=⟨y⟩$	$H_2=⟨x^2,y⟩$	$H_2=⟨x^2,y⟩$
$H_6=⟨xy⟩$	$H_3=⟨x^2,xy⟩$	$H_3=⟨x^2,xy⟩$
$H_7=⟨x^{2}y⟩$	$H_2=⟨x^2,y⟩$	$H_2=⟨x^2,y⟩$
$H_8=⟨x^{3}y⟩$	$H_3=⟨x^2,xy⟩$	$H_3=⟨x^2,xy⟩$
$H_9=⟨1⟩$	$D_4$	$D_4$

Homomorphism	Kernel	Image
$\begin{array}{rrcc}{\varphi }_0:& D_4& \to & ⟨1⟩\\ & s_1& \mapsto & 1\\ & s_2& \mapsto & 1\end{array}$	$ker\hspace{0.17em}{\varphi }_0=D_4$	$im\hspace{0.17em}{\varphi }_0=⟨1⟩$
$\begin{array}{rrcc}{\varphi }_1:& D_4& \to & {\mu }_2\\ & x& \mapsto & 1\\ & y& \mapsto & -1\end{array}$	$ker\hspace{0.17em}{\varphi }_1=H_1$	$im\hspace{0.17em}{\varphi }_1={\mu }_2$
$\begin{array}{rrcc}{\varphi }_2:& D_4& \to & {\mu }_2\\ & x& \mapsto & -1\\ & y& \mapsto & 1\end{array}$	$ker\hspace{0.17em}{\varphi }_2=\left\{1,x^2,y,x^{2}y\right\}=H_2$	$im\hspace{0.17em}{\varphi }_2={\mu }_2$
$\begin{array}{rrcc}{\varphi }_3:& D_4& \to & {\mu }_2\\ & x& \mapsto & -1\\ & y& \mapsto & -1\end{array}$	$ker\hspace{0.17em}{\varphi }_3=\left\{1,x^2,xy,x^{3}y\right\}=H_2$	$im\hspace{0.17em}{\varphi }_3={\mu }_2$
$\begin{array}{rrcc}{\varphi }_4:& D_4& \to & {\mu }_2\hspace{0.17em}\times \hspace{0.17em}{\mu }_2\\ & x& \mapsto & \left(\begin{array}{cc}-1& \\ & 1\end{array}\right)\\ & y& \mapsto & \left(\begin{array}{cc}1& \\ & -1\end{array}\right)\end{array}$	$ker\hspace{0.17em}{\varphi }_4=\left\{1,x^2\right\}=H_4$	$im\hspace{0.17em}{\varphi }_4={\mu }_2\hspace{0.17em}\times \hspace{0.17em}{\mu }_2$
$\begin{array}{rrcc}{\varphi }_9:& D_4& \to & D_4\\ & x& \mapsto & x\\ & y& \mapsto & y\end{array}$	$ker\hspace{0.17em}{\varphi }_3=\left\{1\right\}=H_9$	$im\hspace{0.17em}{\varphi }_9=D_4$

References

[CM] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980. MR0562913 (81a:20001)

[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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