The Dihedral Group of Order Eight

## The dihedral group ${D}_{4}$ of order eight

The groups ${D}_{4}$ is as in the following table.

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

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 $ο\left(g\right)$ 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 ${D}_{4}/{H}_{1}\cong {C}_{2}$
${H}_{2}=\left\{1,{x}^{2},y,{x}^{2}y\right\}$ ${H}_{2}\cong {C}_{2} × {C}_{2}$ $\left[{D}_{4}:{H}_{2}\right]=2$ Yes ${D}_{4}/{H}_{2}\cong {C}_{2}$
${H}_{3}=\left\{1,{x}^{2},xy,{x}^{3}y\right\}$ ${H}_{3}\cong {C}_{2} × {C}_{2}$ $\left\{{D}_{4}:{H}_{3}\right\}=2$ 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
${H}_{0}={D}_{4}$ ${D}_{4}=x{D}_{4}={x}^{3}{D}_{4}=y{D}_{4}$ ${D}_{4}={D}_{4}x={D}_{4}{x}^{2}={D}_{4}{x}^{3}={D}_{4}y$
$=xy{D}_{4}={x}^{2}y{D}_{4}={x}^{3}y{D}_{4}$ $={D}_{4}xy={D}_{4}{x}^{2}y={D}_{4}{x}^{3}y$
${H}_{1}=\left\{1,x,{x}^{2},{x}^{3}\right\}$ ${H}_{1}=x{H}_{1}={x}^{2}{H}_{1}={x}^{3}{H}_{1}$ ${H}_{1}={H}_{1}x={H}_{1}{x}^{2}={H}_{1}{x}^{3}$
$=\left\{1,x,{x}^{2},{x}^{3}\right\}$ $=\left\{1,x,{x}^{2},{x}^{3}\right\}$
$y{H}_{1}=xy{H}_{1}={x}^{2}y{H}_{1}={x}^{3}y{H}_{1}$ ${H}_{1}y={H}_{1}xy={H}_{1}{x}^{2}y={H}_{1}{x}^{3}y$
$=\left\{y,xy,{x}^{2}y,{x}^{3}y\right\}$ $=\left\{y,xy,{x}^{2}y,{x}^{3}y\right\}$
${H}_{2}=\left\{1,{x}^{2},y,{x}^{2}y\right\}$ ${H}_{2}={x}^{2}{H}_{2}=y{H}_{2}={x}^{2}y{H}_{2}$ ${H}_{2}={H}_{2}{x}^{2}={H}_{2}y={H}_{2}{x}^{2}y$
$=\left\{1,{x}^{2},y,{x}^{2}y\right\}$ $=\left\{1,{x}^{2},y,{x}^{2}y\right\}$
$x{H}_{2}={x}^{3}{H}_{2}=xy{H}_{2}={x}^{3}y{H}_{2}$ ${H}_{2}x={H}_{2}{x}^{3}={H}_{2}xy={H}_{2}{x}^{3}y$
$=\left\{x,{x}^{3},xy,{x}^{3}y\right\}$ $=\left\{x,{x}^{3},xy,{x}^{3}y\right\}$
${H}_{3}=\left\{1,{x}^{2},xy,{x}^{3}y\right\}$ ${H}_{3}={x}^{2}{H}_{3}=xy{H}_{3}={x}^{3}y{H}_{3}$ ${H}_{3}={H}_{3}{x}^{2}={H}_{3}xy={H}_{3}{x}^{3}y$
$=\left\{1,{x}^{2},xy,{x}^{3}y\right\}$ $=\left\{1,{x}^{2},xy,{x}^{3}y\right\}$
$x{H}_{3}={x}^{3}{H}_{3}=y{H}_{3}={x}^{2}y{H}_{3}$ ${H}_{3}x={H}_{3}{x}^{3}={H}_{3}y={H}_{3}{x}^{2}y$
$=\left\{x,{x}^{3},y,{x}^{2}y\right\}$ $=\left\{x,{x}^{3},y,{x}^{2}y\right\}$
${H}_{4}=\left\{1,{x}^{2}\right\}$ ${H}_{4}={x}^{2}{H}_{4}=\left\{1,{x}^{2}\right\}$ ${H}_{4}={H}_{4}{x}^{2}=\left\{1,{x}^{2}\right\}$
$x{H}_{4}={x}^{3}{H}_{4}=\left\{x,{x}^{3}\right\}$ ${H}_{4}x={H}_{4}{x}^{3}=\left\{x,{x}^{3}\right\}$
$y{H}_{4}={x}^{2}y{H}_{4}=\left\{y,{x}^{2}y\right\}$ ${H}_{4}y={H}_{4}{x}^{2}y=\left\{y,{x}^{2}y\right\}$
$xy{H}_{4}={x}^{3}y{H}_{4}=\left\{xy,{x}^{3}y\right\}$ ${H}_{4}xy={H}_{4}{x}^{3}y=\left\{xy,{x}^{3}y\right\}$
${H}_{5}=\left\{1,y\right\}$ ${H}_{5}=y{H}_{5}=\left\{1,y\right\}$ ${H}_{5}={H}_{5}y=\left\{1,y\right\}$
$x{H}_{5}=xy{H}_{5}=\left\{x,xy\right\}$ ${H}_{5}x={H}_{5}{x}^{3}y=\left\{x,{x}^{3}y\right\}$
${x}^{2}{H}_{5}={x}^{2}y{H}_{5}=\left\{y,{x}^{2}y\right\}$ ${H}_{5}{x}^{2}={H}_{5}{x}^{2}y=\left\{{x}^{2},{x}^{2}y\right\}$
${x}^{3}{H}_{5}={x}^{3}y{H}_{5}=\left\{{x}^{3},{x}^{3}y\right\}$ ${H}_{5}{x}^{3}={H}_{5}xy=\left\{{x}^{3},xy\right\}$
${H}_{6}=\left\{1,xy\right\}$ ${H}_{6}=xy{H}_{6}=\left\{1,xy\right\}$ ${H}_{6}={H}_{6}xy=\left\{1,xy\right\}$
$x{H}_{6}={x}^{2}y{H}_{6}=\left\{x,{x}^{2}y\right\}$ ${H}_{6}x={H}_{6}{x}^{2}y=\left\{x,y\right\}$
${x}^{2}{H}_{6}={x}^{3}y{H}_{6}=\left\{{x}^{2},{x}^{3}y\right\}$ ${H}_{6}{x}^{2}={H}_{6}{x}^{3}y=\left\{{x}^{2},{x}^{3}y\right\}$
${x}^{3}{H}_{6}=y{H}_{6}=\left\{{x}^{3},y\right\}$ ${H}_{6}{x}^{3}={H}_{6}{x}^{2}y=\left\{{x}^{3},{x}^{2}y\right\}$
${H}_{7}=\left\{1,{x}^{2}y\right\}$ ${H}_{7}={x}^{2}y{H}_{7}=\left\{x,{x}^{2}y\right\}$ ${H}_{7}={H}_{7}{x}^{2}y=\left\{1,{x}^{2}y\right\}$
$x{H}_{7}={x}^{3}y{H}_{7}=\left\{x,{x}^{3}y\right\}$ ${H}_{7}x={H}_{7}xy=\left\{x,xy\right\}$
${x}^{2}{H}_{7}=y{H}_{7}=\left\{{x}^{2},y\right\}$ ${H}_{7}{x}^{2}={H}_{7}y=\left\{{x}^{2},y\right\}$
${x}^{3}{H}_{7}=xy{H}_{7}=\left\{{x}^{3},xy\right\}$ ${H}_{7}{x}^{3}={H}_{7}{x}^{3}y=\left\{{x}^{3},{x}^{3}y\right\}$
${H}_{8}=\left\{1,{x}^{3}y\right\}$ ${H}_{8}={x}^{3}y{H}_{8}=\left\{1,{x}^{3}y\right\}$ ${H}_{8}={H}_{8}{x}^{3}y=\left\{1,{x}^{3}y\right\}$
$x{H}_{8}=y{H}_{8}=\left\{x,y\right\}$ ${H}_{8}x={H}_{8}{x}^{2}y=\left\{{x}^{2}y,y\right\}$
${x}^{2}{H}_{8}=xy{H}_{8}=\left\{{x}^{2},xy\right\}$ ${H}_{8}{x}^{2}={H}_{8}xy=\left\{{x}^{2},xy\right\}$
${x}^{3}{H}_{8}={x}^{2}y{H}_{8}=\left\{{x}^{3},{x}^{2}y\right\}$ ${H}_{8}{x}^{3}={H}_{8}y=\left\{{x}^{3},y\right\}$
${H}_{9}=\left\{1\right\}$ ${H}_{9}=\left\{1\right\},x{H}_{9}=\left\{x\right\},$ ${H}_{9}=\left\{1\right\},{H}_{9}x=\left\{x\right\},$
${x}^{2}{H}_{9}=\left\{{x}^{2}\right\},x{H}_{9}=\left\{x\right\},$ ${H}_{9}{x}^{2}=\left\{{x}^{2}\right\},{H}_{9}{x}^{3}=\left\{{x}^{3}\right\},$
$y{H}_{9}=\left\{y\right\},xy{H}_{9}=\left\{xy\right\},$ ${H}_{9}y=\left\{y\right\},{H}_{9}xy=\left\{xy\right\},$
${x}^{2}y{H}_{9}=\left\{{x}^{2}y\right\},{x}^{3}y{H}_{9}=\left\{{x}^{3}y\right\},$ ${H}_{9}{x}^{2}y=\left\{{x}^{2}y\right\},{H}_{9}{x}^{3}y=\left\{{x}^{3}y\right\},$

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
$\begin{array}{rrcc}{\varphi }_{0}:& {D}_{4}& \to & ⟨1⟩\\ & {s}_{1}& ↦& 1\\ & {s}_{2}& ↦& 1\end{array}$ $ker {\varphi }_{0}={D}_{4}$ $im {\varphi }_{0}=⟨1⟩$
$\begin{array}{rrcc}{\varphi }_{1}:& {D}_{4}& \to & {\mu }_{2}\\ & x& ↦& 1\\ & y& ↦& -1\end{array}$ $ker {\varphi }_{1}={H}_{1}$ $im {\varphi }_{1}={\mu }_{2}$
$\begin{array}{rrcc}{\varphi }_{2}:& {D}_{4}& \to & {\mu }_{2}\\ & x& ↦& -1\\ & y& ↦& 1\end{array}$ $ker {\varphi }_{2}=\left\{1,{x}^{2},y,{x}^{2}y\right\}={H}_{2}$ $im {\varphi }_{2}={\mu }_{2}$
$\begin{array}{rrcc}{\varphi }_{3}:& {D}_{4}& \to & {\mu }_{2}\\ & x& ↦& -1\\ & y& ↦& -1\end{array}$ $ker {\varphi }_{3}=\left\{1,{x}^{2},xy,{x}^{3}y\right\}={H}_{2}$ $im {\varphi }_{3}={\mu }_{2}$
$\begin{array}{rrcc}{\varphi }_{4}:& {D}_{4}& \to & {\mu }_{2} × {\mu }_{2}\\ & x& ↦& \left(\begin{array}{cc}-1& \\ & 1\end{array}\right)\\ & y& ↦& \left(\begin{array}{cc}1& \\ & -1\end{array}\right)\end{array}$ $ker {\varphi }_{4}=\left\{1,{x}^{2}\right\}={H}_{4}$ $im {\varphi }_{4}={\mu }_{2} × {\mu }_{2}$
$\begin{array}{rrcc}{\varphi }_{9}:& {D}_{4}& \to & {D}_{4}\\ & x& ↦& x\\ & y& ↦& y\end{array}$ $ker {\varphi }_{3}=\left\{1\right\}={H}_{9}$ $im {\varphi }_{9}={D}_{4}$

## 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)