The Tetrahedral Group

## The Tetrahedral Group ${A}_{4}$

The group ${A}_{4}$ can be given in at least two natural ways. In the following tables we shall use one-line notation to represent the permutations in ${A}_{4}$

Set Operation
even permutations in ${S}_{4}$ composition of permutations
rotations preserving a tetrahedron compositions of rotations

Center Abelian Conjugacy classes
$Z\left({A}_{4}\right)=\left\{\left(1234\right)\right\}$ No ${𝒞}_{\left({1}^{4}\right)}=\left\{\left(1234\right)\right\}$
${𝒞}_{\left({2}^{2}\right)}=\left\{\left(2134\right),\left(3412\right),\left(4321\right)\right\}$
${𝒞}_{{\left(31\right)}^{+}}=\left\{\left(3124\right),\left(4213\right),\left(2431\right),\left(1342\right)\right\}$
${𝒞}_{{\left(31\right)}^{-}}=\left\{\left(2314\right),\left(3241\right),\left(4132\right),\left(1423\right)\right\}$

Subgroups
${H}_{0}={A}_{4}$
${H}_{1}=\left\{\left(1234\right),\left(2143\right),\left(3412\right),\left(4321\right)\right\}$
${H}_{2}=\left\{\left(1234\right),\left(3124\right),\left(2314\right)\right\}$
${H}_{3}=\left\{\left(1234\right),\left(4132\right),\left(2431\right)\right\}$
${H}_{4}=\left\{\left(1234\right),\left(4213\right),\left(3241\right)\right\}$
${H}_{5}=\left\{\left(1234\right),\left(1423\right),\left(1342\right)\right\}$
${H}_{6}=\left\{\left(1234\right),\left(3412\right)\right\}$
${H}_{7}=\left\{\left(1234\right),\left(2143\right)\right\}$
${H}_{8}=\left\{\left(1234\right),\left(4321\right)\right\}$
${H}_{8}=\left\{\left(1234\right)\right\}$

Element $g$ Order $ο\left(g\right)$ Centralizer ${Z}_{g}$ Conjugacy Class ${𝒞}_{g}$
$\left(1234\right)$ $1$ ${A}_{4}$ ${𝒞}_{\left({1}^{4}\right)}$
$\left(2143\right)$ $2$ ${H}_{1}$ ${𝒞}_{\left({2}^{2}\right)}$
$\left(3412\right)$ $2$ ${H}_{1}$ ${𝒞}_{\left({2}^{2}\right)}$
$\left(4321\right)$ $2$ ${H}_{1}$ ${𝒞}_{\left({2}^{2}\right)}$
$\left(3124\right)$ $3$ ${H}_{2}$ ${𝒞}_{{\left(31\right)}^{+}}$
$\left(4213\right)$ $3$ ${H}_{4}$ ${𝒞}_{{\left(31\right)}^{+}}$
$\left(2431\right)$ $3$ ${H}_{3}$ ${𝒞}_{{\left(31\right)}^{+}}$
$\left(1342\right)$ $3$ ${H}_{5}$ ${𝒞}_{{\left(31\right)}^{+}}$
$\left(2314\right)$ $3$ ${H}_{2}$ ${𝒞}_{{\left(31\right)}^{-}}$
$\left(3241\right)$ $3$ ${H}_{4}$ ${𝒞}_{{\left(31\right)}^{-}}$
$\left(4132\right)$ $3$ ${H}_{3}$ ${𝒞}_{{\left(31\right)}^{-}}$
$\left(1423\right)$ $3$ ${H}_{5}$ ${𝒞}_{{\left(31\right)}^{-}}$

Generators Relations Realization
$S,T$ ${S}^{3}={T}^{2}={\left(ST\right)}^{3}=1$ $S=\left(2314\right),T=\left(2143\right)$

Subgroups ${H}_{i}$ Structure Index Normal Quotient group
${H}_{0}={A}_{4}$ ${H}_{0}={A}_{4}$ $\left[{A}_{4}:{A}_{4}\right]=1$ Yes ${A}_{4}/{H}_{0}\cong ⟨1⟩$
${H}_{1}=\left\{\left(1234\right),\left(2143\right),\left(3412\right),\left(4321\right)\right\}$ ${H}_{1}\cong {\mu }_{2} × {\mu }_{2}$ $\left[{A}_{4}:{H}_{1}\right]=3$ Yes ${A}_{4}/{H}_{1}\cong {\mu }_{3}$
${H}_{2}=\left\{\left(1234\right),\left(3124\right),\left(2314\right)\right\}$ ${H}_{2}\cong {\mu }_{3}$ $\left[{A}_{4}:{H}_{2}\right]=4$ No
${H}_{3}=\left\{\left(1234\right),\left(4132\right),\left(2431\right)\right\}$ ${H}_{3}\cong {\mu }_{3}$ $\left\{{A}_{4}:{H}_{3}\right\}=4$ No
${H}_{1}=\left\{\left(1234\right),\left(4213\right),\left(3241\right)\right\}$ ${H}_{4}\cong {\mu }_{3}$ $\left[{A}_{4}:{H}_{4}\right]=4$ No
${H}_{5}=\left\{\left(1234\right),\left(1423\right),\left(1342\right)\right\}$ ${H}_{5}={\mu }_{3}$ $\left[{A}_{4}:⟨1⟩\right]=4$ No
${H}_{6}=\left\{\left(1234\right),\left(3412\right)\right\}$ ${H}_{6}={\mu }_{2}$ $\left[{A}_{4}:⟨1⟩\right]=6$ No
${H}_{7}=\left\{\left(1234\right),\left(2143\right)\right\}$ ${H}_{7}={\mu }_{2}$ $\left[{A}_{4}:⟨1⟩\right]=6$ No
${H}_{8}=\left\{\left(1234\right),\left(4321\right)\right\}$ ${H}_{8}={\mu }_{2}$ $\left[{A}_{4}:⟨1⟩\right]=6$ No
${H}_{9}=\left\{\left(1234\right)\right\}$ ${H}_{9}=⟨1⟩$ $\left[{A}_{4}:⟨1⟩\right]=12$ Yes ${A}_{4}/⟨1⟩\cong {A}_{4}$

Subgroup ${H}_{i}$ Normalizer ${N}_{{H}_{i}}$ Centralizer ${Z}_{{H}_{i}}$
${H}_{0}={A}_{4}$ ${A}_{4}$ ${H}_{9}=⟨1⟩$
$N$ ${A}_{4}$ $N$
${H}_{2}$ ${A}_{4}$ ${H}_{2}$
${H}_{3}$ ${A}_{4}$ ${H}_{3}$
${H}_{4}$ ${A}_{4}$ ${H}_{4}$
${H}_{5}$ ${A}_{4}$ ${H}_{5}$
${H}_{6}$ $N$ ${H}_{1}$
${H}_{7}$ $N$ ${H}_{1}$
${H}_{8}$ $N$ ${H}_{1}$
${H}_{9}=⟨1⟩$ ${A}_{4}$ ${A}_{4}$

Let $w$ be a primitive cube root of $1$ given by $w={e}^{2\pi i/3}$

Homomorphism Kernel
$\begin{array}{rrcc}{\varphi }_{0}:& {A}_{4}& \to & ⟨1⟩\\ & S& ↦& 1\\ & T& ↦& 1\end{array}$ $ker {\varphi }_{0}={A}_{4}$
$\begin{array}{rrcc}{\varphi }_{1}:& {A}_{4}& \to & {\mu }_{3}\\ & S& ↦& w\\ & T& ↦& 1\end{array}$ $ker {\varphi }_{1}={H}_{1}$
$\begin{array}{rrcc}{\varphi }_{2}:& {A}_{4}& \to & {\mu }_{3}\\ & S& ↦& {w}^{2}\\ & T& ↦& 1\end{array}$ $ker {\varphi }_{2}={H}_{1}$
$\begin{array}{rrcc}{\varphi }_{4}:& {A}_{4}& \to & {GL}_{3}\left(ℂ\right)\\ & S& ↦& \left(\begin{array}{ccc}1& 0& 0\\ 0& -1/2& -3/2\\ 0& 1/2& -1/2\end{array}\right)\\ & T& ↦& \left(\begin{array}{ccc}-1/3& -4/3& 0\\ -2/3& 1/3& 0\\ 0& 0& -1\end{array}\right)\end{array}$ $ker {\varphi }_{4}=⟨1⟩$

## 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)