The Nonabelian Group of Order Six

## The nonabelian group of order six

Let $1= 1 0 0 0 1 0 0 0 1 , 12= 0 1 0 1 0 0 0 0 1 , 23= 1 0 0 0 0 1 0 1 0 , 13= 0 0 1 0 1 0 1 0 0 , 132= 0 1 0 0 0 1 1 0 0 , 123= 0 0 1 1 0 0 0 1 0 .$

The groups ${S}_{3}$ and ${D}_{3}$ are as in the following table.

Set Operation
${S}_{3}=\left\{1,\left(12\right),\left(23\right),\left(13\right),\left(132\right),\left(123\right)\right\}$ Ordinary matrix multiplication
${D}_{3}=\left\{1,x,{x}^{2},y,xy,{x}^{2}y\right\}$ ${x}^{i}{y}^{j}{x}^{k}{y}^{l}={x}^{\left(i-k\right)mod 3}{y}^{\left(j+l\right)mod 2}$

The complete multiplication tables for these groups are as follows.

Multiplication tables
 ${S}_{3}$ $1$ $\left(12\right)$ $\left(23\right)$ $\left(13\right)$ $\left(132\right)$ $\left(123\right)$ $1$ $1$ $\left(12\right)$ $\left(23\right)$ $\left(13\right)$ $\left(132\right)$ $\left(123\right)$ $\left(12\right)$ $\left(12\right)$ $1$ $\left(123\right)$ $\left(132\right)$ $\left(13\right)$ $\left(12\right)$ $\left(23\right)$ $\left(23\right)$ $\left(132\right)$ $1$ $\left(123\right)$ $\left(12\right)$ $\left(13\right)$ $\left(13\right)$ $\left(13\right)$ $\left(123\right)$ $\left(132\right)$ $1$ $\left(23\right)$ $\left(12\right)$ $\left(132\right)$ $\left(132\right)$ $\left(23\right)$ $\left(13\right)$ $\left(12\right)$ $\left(123\right)$ $1$ $\left(123\right)$ $\left(123\right)$ $\left(13\right)$ $\left(12\right)$ $\left(23\right)$ $1$ $\left(132\right)$
 ${D}_{3}$ $1$ $y$ ${x}^{2}y$ $xy$ ${x}^{2}$ $x$ $1$ $1$ $y$ ${x}^{2}y$ $xy$ ${x}^{2}$ $x$ $y$ $y$ $1$ $x$ ${x}^{2}$ $xy$ ${x}^{2}y$ ${x}^{2}y$ ${x}^{2}y$ ${x}^{2}$ $1$ $x$ $y$ $xy$ $xy$ $xy$ $x$ ${x}^{2}$ $1$ ${x}^{2}y$ $y$ ${x}^{2}$ ${x}^{2}$ ${x}^{2}y$ $xy$ $y$ $x$ $1$ $x$ $x$ $xy$ $y$ ${x}^{2}y$ $1$ ${x}^{2}$

HW: Prove that the group homomorphism given as in the following table is an isomorphism.

Isomorphism
$\begin{array}{cccc}\Phi :& {D}_{3}& \to & {S}_{3}\\ & x& ↦& \left(123\right)\\ & y& ↦& \left(12\right)\end{array}$

Center Abelian Conjugacy classes Subgroups
$Z\left({S}_{3}\right)=⟨1⟩$ No ${𝒞}_{\left({1}^{3}\right)}=\left\{1\right\}$ ${H}_{0}={S}_{3}$
${𝒞}_{\left(21\right)}=\left\{\left(12\right),\left(23\right),\left(13\right)\right\}$ ${H}_{1}=\left\{1,\left(132\right),\left(123\right)\right\}$
${𝒞}_{\left(3\right)}=\left\{\left(123\right),\left(132\right)\right\}$ ${H}_{2}=\left\{1,\left(12\right)\right\}$
${H}_{3}=\left\{1,\left(13\right)\right\}$
${H}_{4}=\left\{1,\left(23\right)\right\}$
${H}_{5}=\left\{1\right\}$

Element $g$ Order $ο\left(g\right)$ Centralizer ${Z}_{g}$ Conjugacy Class ${𝒞}_{g}$
$\left(1,1\right)$ $1$ ${S}_{3}$ ${𝒞}_{\left({1}^{3}\right)}$
$\left(12\right)$ $2$ ${H}_{2}$ ${𝒞}_{\left(21\right)}$
$\left(23\right)$ $2$ ${H}_{4}$ ${𝒞}_{\left(21\right)}$
$\left(13\right)$ $2$ ${H}_{3}$ ${𝒞}_{\left(21\right)}$
$\left(132\right)$ $3$ ${H}_{1}$ ${𝒞}_{\left(3\right)}$
$\left(123\right)$ $3$ ${H}_{1}$ ${𝒞}_{\left(3\right)}$

Generators Relations Realization
${D}_{3}$ $x,y$ ${x}^{3}={y}^{2}=1$ $x=\left(123\right)$
${\left(xy\right)}^{2}=1$ $y=\left(12\right)$
${S}_{3}$ ${s}_{1},{s}_{2}$ ${s}_{1}^{2}={s}_{2}^{2}=1$ ${s}_{1}=y=\left(12\right)$
${s}_{1}{s}_{2}{s}_{1}={s}_{2}{s}_{1}{s}_{2}$ ${s}_{2}={x}^{2}y=\left(23\right)$

Subgroups ${H}_{i}$ Structure Order $\left|{H}_{i}\right|$ Index Normal Quotient group
${H}_{0}={S}_{3}$ ${H}_{0}={S}_{3}$ $6$ $\left[{S}_{3}:{S}_{3}\right]=1$ Yes ${S}_{3}/{H}_{0}\cong ⟨1⟩$
${H}_{1}=\left\{1,\left(132\right),\left(123\right)\right\}$ ${H}_{1}\cong {C}_{3}\cong {A}_{3}$ $3$ $\left[{S}_{3}:{H}_{1}\right]=2$ Yes ${S}_{3}/{H}_{1}\cong {C}_{2}$
${H}_{2}=\left\{1,\left(12\right)\right\}$ ${H}_{2}\cong {C}_{2}$ $2$ $\left[{S}_{3}:{H}_{2}\right]=3$ No
${H}_{3}=\left\{1,\left(13\right)\right\}$ ${H}_{3}\cong {C}_{2}$ $2$ $\left\{{S}_{3}:{H}_{3}\right\}=3$ No
${H}_{4}=\left\{1,\left(23\right)\right\}$ ${H}_{4}\cong {C}_{2}$ $2$ $\left[{S}_{3}:{H}_{4}\right]=3$ No
${H}_{5}=\left\{1\right\}$ ${H}_{5}=⟨1⟩$ $1$ $\left[{S}_{3}:{H}_{5}\right]=6$ Yes ${S}_{3}/⟨1⟩\cong {S}_{3}$

Subgroups ${H}_{i}$ Left Cosets Right Cosets
${H}_{0}={S}_{3}$ ${S}_{3}$ ${S}_{3}$
${H}_{1}=\left\{1,\left(132\right),\left(123\right)\right\}$ ${H}_{1}=\left\{1,\left(132\right),\left(123\right)\right\}$ ${H}_{1}=\left\{1,\left(132\right),\left(123\right)\right\}$
$\left(12\right){H}_{1}=\left\{\left(12\right),\left(13\right),\left(23\right)\right\}$ ${H}_{1}\left(12\right)=\left\{\left(12\right),\left(13\right),\left(23\right)\right\}$
${H}_{2}=\left\{1,\left(12\right)\right\}$ ${H}_{2}=\left\{1,\left(12\right)\right\}$ ${H}_{2}=\left\{1,\left(12\right)\right\}$
$\left(23\right){H}_{2}=\left\{\left(23\right),\left(132\right)\right\}$ ${H}_{2}\left(23\right)=\left\{\left(23\right),\left(123\right)\right\}$
$\left(13\right){H}_{2}=\left\{\left(13\right),\left(123\right)\right\}$ ${H}_{2}\left(13\right)=\left\{\left(13\right),\left(132\right)\right\}$
${H}_{3}=\left\{1,\left(13\right)\right\}$ ${H}_{3}=\left\{1,\left(13\right)\right\}$ ${H}_{3}=\left\{1,\left(13\right)\right\}$
$\left(23\right){H}_{3}=\left\{\left(23\right),\left(123\right)\right\}$ ${H}_{3}\left(23\right)=\left\{\left(23\right),\left(132\right)\right\}$
$\left(12\right){H}_{3}=\left\{\left(12\right),\left(132\right)\right\}$ ${H}_{3}\left(12\right)=\left\{\left(12\right),\left(123\right)\right\}$
${H}_{4}=\left\{1,\left(23\right)\right\}$ ${H}_{4}=\left\{1,\left(23\right)\right\}$ ${H}_{4}=\left\{1,\left(23\right)\right\}$
$\left(12\right){H}_{4}=\left\{\left(12\right),\left(123\right)\right\}$ ${H}_{4}\left(12\right)=\left\{\left(12\right),\left(132\right)\right\}$
$\left(13\right){H}_{4}=\left\{\left(13\right),\left(132\right)\right\}$ ${H}_{4}\left(13\right)=\left\{\left(13\right),\left(123\right)\right\}$
${H}_{5}=\left\{1\right\}$ ${H}_{5}=\left\{1\right\}$ ${H}_{5}=\left\{1\right\}$
$\left(12\right){H}_{5}=\left\{\left(12\right)\right\}$ ${H}_{5}\left(12\right)=\left\{\left(12\right)\right\}$
$\left(23\right){H}_{5}=\left\{\left(23\right)\right\}$ ${H}_{5}\left(23\right)=\left\{\left(23\right)\right\}$
$\left(13\right){H}_{5}=\left\{\left(13\right)\right\}$ ${H}_{5}\left(13\right)=\left\{\left(13\right)\right\}$
$\left(132\right){H}_{5}=\left\{\left(132\right)\right\}$ ${H}_{5}\left(132\right)=\left\{\left(132\right)\right\}$
$\left(123\right){H}_{5}=\left\{\left(123\right)\right\}$ ${H}_{5}\left(123\right)=\left\{\left(123\right)\right\}$

Subgroups ${H}_{i}$ Normalizer ${N}_{{H}_{i}}$ Centralizer ${Z}_{{H}_{i}}$
${H}_{0}={S}_{3}$ ${H}_{0}={S}_{3}$ ${H}_{5}=\left\{1\right\}$
${H}_{1}=\left\{1,\left(132\right),\left(123\right)\right\}$ ${H}_{0}={S}_{3}$ ${H}_{1}=\left\{1,\left(123\right),\left(132\right)\right\}$
${H}_{2}=\left\{1,\left(12\right)\right\}$ ${H}_{2}=\left\{1,\left(12\right)\right\}$ ${H}_{2}=\left\{1,\left(12\right)\right\}$
${H}_{3}=\left\{1,\left(13\right)\right\}$ ${H}_{3}=\left\{1,\left(13\right)\right\}$ ${H}_{3}=\left\{1,\left(13\right)\right\}$
${H}_{4}=\left\{1,\left(23\right)\right\}$ ${H}_{4}=\left\{1,\left(23\right)\right\}$ ${H}_{4}=\left\{1,\left(23\right)\right\}$
${H}_{5}=\left\{1\right\}$ ${H}_{0}={S}_{3}$ ${H}_{0}={S}_{3}$

Homomorphism Kernel Image
$\begin{array}{rrcc}{\phi }_{0}:& {S}_{3}& \to & ⟨1⟩\\ & {s}_{1}& ↦& 1\\ & {s}_{2}& ↦& 1\end{array}$ $ker {\phi }_{0}={S}_{3}$ $im {\phi }_{0}=⟨1⟩$
$\begin{array}{rrcc}\epsilon :& {S}_{3}& \to & {\mu }_{2}\\ & {s}_{1}& ↦& -1\\ & {s}_{2}& ↦& -1\end{array}$ $ker \epsilon ={A}_{3}$ $im {\phi }_{0}={\mu }_{2}$
$\begin{array}{rrcc}{\phi }_{2}:& {S}_{3}& \to & O\left(3\right)\\ & \left(12\right)& ↦& \left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right)\\ & \left(23\right)& ↦& \left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right)\end{array}$ $ker {\phi }_{2}=⟨1⟩$ $im {\phi }_{2}=\left\{\begin{array}{cc}\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),& \left(\begin{array}{ccc}0& 1& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right),\\ \left(\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right),& \left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right),\\ \left(\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right),& \left(\begin{array}{ccc}0& 0& 1\\ 1& 0& 0\\ 0& 1& 0\end{array}\right)\end{array}\right\}$
$\begin{array}{rrcc}{\phi }_{3}:& {S}_{3}& \to & O\left(2\right)\\ & \left(12\right)& ↦& \left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right)\\ & \left(23\right)& ↦& \left(\begin{array}{cc}\frac{1}{2}& \frac{1}{2}\\ \frac{3}{2}& \frac{-1}{2}\end{array}\right)\end{array}$ $ker {\phi }_{3}=⟨1⟩$ $im {\phi }_{3}=\left\{\begin{array}{cc}\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right),& \left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right),\\ \left(\begin{array}{cc}\frac{1}{2}& \frac{1}{2}\\ \frac{3}{2}& \frac{-1}{2}\end{array}\right),& \left(\begin{array}{cc}\frac{-1}{2}& \frac{-1}{2}\\ \frac{3}{2}& \frac{-1}{2}\end{array}\right),\\ \left(\begin{array}{cc}\frac{-1}{2}& \frac{1}{2}\\ \frac{-3}{2}& \frac{-1}{2}\end{array}\right),& \left(\begin{array}{cc}\frac{-1}{2}& \frac{-1}{2}\\ \frac{-3}{2}& \frac{-1}{2}\end{array}\right)\end{array}\right\}$
$\begin{array}{rrcc}{\phi }_{4}:& {S}_{3}& \to & {D}_{3}\\ & \left(12\right)& ↦& y\\ & \left(132\right)& ↦& x\end{array}$ $ker {\phi }_{4}=⟨1⟩$ $im {\phi }_{4}={D}_{3}$

## 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)