The Tetrahedral Group $A_4$

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

Last updates: 02 February 2011

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 $o\left(g\right)$	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	$A_4/H_0\cong ⟨1⟩$
$H_1=\left\{\left(1234\right),\left(2143\right),\left(3412\right),\left(4321\right)\right\}$		$\left[A_4:H_1\right]=3$	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
$\begin{array}{rrcc}{\varphi }_0:& A_4& \to & ⟨1⟩\\ & S& \mapsto & 1\\ & T& \mapsto & 1\end{array}$	$ker\hspace{0.17em}{\varphi }_0=A_4$
$\begin{array}{rrcc}{\varphi }_1:& A_4& \to & {\mu }_3\\ & S& \mapsto & w\\ & T& \mapsto & 1\end{array}$	$ker\hspace{0.17em}{\varphi }_1=H_1$
$\begin{array}{rrcc}{\varphi }_2:& A_4& \to & {\mu }_3\\ & S& \mapsto & w^2\\ & T& \mapsto & 1\end{array}$	$ker\hspace{0.17em}{\varphi }_2=H_1$
$\begin{array}{rrcc}{\varphi }_4:& A_4& \to & {GL}_3\left(ℂ\right)\\ & S& \mapsto & \left(\begin{array}{ccc}1& 0& 0\\ 0& -1/2& -3/2\\ 0& 1/2& -1/2\end{array}\right)\\ & T& \mapsto & \left(\begin{array}{ccc}-1/3& -4/3& 0\\ -2/3& 1/3& 0\\ 0& 0& -1\end{array}\right)\end{array}$	$ker\hspace{0.17em}{\varphi }_4=⟨1⟩$

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)

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