The Nonabelian Group of Order Six

$S_{3} ≅ D_{3}$ : The Nonabelian Group of Order Six

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

Last updates: 14 December 2010

The nonabelian group of order six

Let $\begin{array}{rrr} 1 = (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}), & (12) = (\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}), & (23) = (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}), \\ (13) = (\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}), & (132) = (\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}), & (123) = (\begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}) . \end{array}$

The groups $S_{3}$ and $D_{3}$ are as in the following table.

Set	Operation
$S_{3} = \{1, (12), (23), (13), (132), (123)\}$	Ordinary matrix multiplication
$D_{3} = \{1, x, x^{2}, y, x y, x^{2} y\}$	$x^{i} y^{j} x^{k} y^{l} = x^{(i - k) mod 3} y^{(j + l) mod 2}$

The complete multiplication tables for these groups are as follows.

Multiplication tables

$S_{3}$	$1$	$(12)$	$(23)$	$(13)$	$(132)$	$(123)$
$1$	$1$	$(12)$	$(23)$	$(13)$	$(132)$	$(123)$
$(12)$	$(12)$	$1$	$(123)$	$(132)$	$(13)$	$(12)$
$(23)$	$(23)$	$(132)$	$1$	$(123)$	$(12)$	$(13)$
$(13)$	$(13)$	$(123)$	$(132)$	$1$	$(23)$	$(12)$
$(132)$	$(132)$	$(23)$	$(13)$	$(12)$	$(123)$	$1$
$(123)$	$(123)$	$(13)$	$(12)$	$(23)$	$1$	$(132)$

$D_{3}$	$1$	$y$	$x^{2} y$	$x y$	$x^{2}$	$x$
$1$	$1$	$y$	$x^{2} y$	$x y$	$x^{2}$	$x$
$y$	$y$	$1$	$x$	$x^{2}$	$x y$	$x^{2} y$
$x^{2} y$	$x^{2} y$	$x^{2}$	$1$	$x$	$y$	$x y$
$x y$	$x y$	$x$	$x^{2}$	$1$	$x^{2} y$	$y$
$x^{2}$	$x^{2}$	$x^{2} y$	$x y$	$y$	$x$	$1$
$x$	$x$	$x y$	$y$	$x^{2} y$	$1$	$x^{2}$

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

Isomorphism
$\begin{matrix} Φ : & D_{3} & \to & S_{3} \\ x & \mapsto & (123) \\ y & \mapsto & (12) \end{matrix}$

Center	Abelian	Conjugacy classes	Subgroups
$Z (S_{3}) = ⟨ 1 ⟩$	No	$𝒞_{(1^{3})} = \{1\}$	$H_{0} = S_{3}$
		$𝒞_{(21)} = \{(12), (23), (13)\}$	$H_{1} = \{1, (132), (123)\}$
		$𝒞_{(3)} = \{(123), (132)\}$	$H_{2} = \{1, (12)\}$
			$H_{3} = \{1, (13)\}$
			$H_{4} = \{1, (23)\}$
			$H_{5} = \{1\}$

Element $g$	Order $ο (g)$	Centralizer $Z_{g}$	Conjugacy Class $𝒞_{g}$
$(1, 1)$	$1$	$S_{3}$	$𝒞_{(1^{3})}$
$(12)$	$2$	$H_{2}$	$𝒞_{(21)}$
$(23)$	$2$	$H_{4}$	$𝒞_{(21)}$
$(13)$	$2$	$H_{3}$	$𝒞_{(21)}$
$(132)$	$3$	$H_{1}$	$𝒞_{(3)}$
$(123)$	$3$	$H_{1}$	$𝒞_{(3)}$

	Generators	Relations	Realization
$D_{3}$	$x, y$	$x^{3} = y^{2} = 1$	$x = (123)$
		${(x y)}^{2} = 1$	$y = (12)$
$S_{3}$	$s_{1}, s_{2}$	$s_{1}^{2} = s_{2}^{2} = 1$	$s_{1} = y = (12)$
		$s_{1} s_{2} s_{1} = s_{2} s_{1} s_{2}$	$s_{2} = x^{2} y = (23)$

Subgroups $H_{i}$	Structure	Order $\|H_{i}\|$	Index	Normal	Quotient group
$H_{0} = S_{3}$	$H_{0} = S_{3}$	$6$	$[S_{3} : S_{3}] = 1$	Yes	$S_{3} / H_{0} ≅ ⟨ 1 ⟩$
$H_{1} = \{1, (132), (123)\}$	$H_{1} ≅ C_{3} ≅ A_{3}$	$3$	$[S_{3} : H_{1}] = 2$	Yes	$S_{3} / H_{1} ≅ C_{2}$
$H_{2} = \{1, (12)\}$	$H_{2} ≅ C_{2}$	$2$	$[S_{3} : H_{2}] = 3$	No
$H_{3} = \{1, (13)\}$	$H_{3} ≅ C_{2}$	$2$	$\{S_{3} : H_{3}\} = 3$	No
$H_{4} = \{1, (23)\}$	$H_{4} ≅ C_{2}$	$2$	$[S_{3} : H_{4}] = 3$	No
$H_{5} = \{1\}$	$H_{5} = ⟨ 1 ⟩$	$1$	$[S_{3} : H_{5}] = 6$	Yes	$S_{3} / ⟨ 1 ⟩ ≅ S_{3}$

Subgroups $H_{i}$	Left Cosets	Right Cosets
$H_{0} = S_{3}$	$S_{3}$	$S_{3}$
$H_{1} = \{1, (132), (123)\}$	$H_{1} = \{1, (132), (123)\}$	$H_{1} = \{1, (132), (123)\}$
	$(12) H_{1} = \{(12), (13), (23)\}$	$H_{1} (12) = \{(12), (13), (23)\}$
$H_{2} = \{1, (12)\}$	$H_{2} = \{1, (12)\}$	$H_{2} = \{1, (12)\}$
	$(23) H_{2} = \{(23), (132)\}$	$H_{2} (23) = \{(23), (123)\}$
	$(13) H_{2} = \{(13), (123)\}$	$H_{2} (13) = \{(13), (132)\}$
$H_{3} = \{1, (13)\}$	$H_{3} = \{1, (13)\}$	$H_{3} = \{1, (13)\}$
	$(23) H_{3} = \{(23), (123)\}$	$H_{3} (23) = \{(23), (132)\}$
	$(12) H_{3} = \{(12), (132)\}$	$H_{3} (12) = \{(12), (123)\}$
$H_{4} = \{1, (23)\}$	$H_{4} = \{1, (23)\}$	$H_{4} = \{1, (23)\}$
	$(12) H_{4} = \{(12), (123)\}$	$H_{4} (12) = \{(12), (132)\}$
	$(13) H_{4} = \{(13), (132)\}$	$H_{4} (13) = \{(13), (123)\}$
$H_{5} = \{1\}$	$H_{5} = \{1\}$	$H_{5} = \{1\}$
	$(12) H_{5} = \{(12)\}$	$H_{5} (12) = \{(12)\}$
	$(23) H_{5} = \{(23)\}$	$H_{5} (23) = \{(23)\}$
	$(13) H_{5} = \{(13)\}$	$H_{5} (13) = \{(13)\}$
	$(132) H_{5} = \{(132)\}$	$H_{5} (132) = \{(132)\}$
	$(123) H_{5} = \{(123)\}$	$H_{5} (123) = \{(123)\}$

Subgroups $H_{i}$	Normalizer $N_{H_{i}}$	Centralizer $Z_{H_{i}}$
$H_{0} = S_{3}$	$H_{0} = S_{3}$	$H_{5} = \{1\}$
$H_{1} = \{1, (132), (123)\}$	$H_{0} = S_{3}$	$H_{1} = \{1, (123), (132)\}$
$H_{2} = \{1, (12)\}$	$H_{2} = \{1, (12)\}$	$H_{2} = \{1, (12)\}$
$H_{3} = \{1, (13)\}$	$H_{3} = \{1, (13)\}$	$H_{3} = \{1, (13)\}$
$H_{4} = \{1, (23)\}$	$H_{4} = \{1, (23)\}$	$H_{4} = \{1, (23)\}$
$H_{5} = \{1\}$	$H_{0} = S_{3}$	$H_{0} = S_{3}$

Homomorphism	Kernel	Image
$\begin{array}{rrcc} φ_{0} : & S_{3} & \to & ⟨ 1 ⟩ \\ s_{1} & \mapsto & 1 \\ s_{2} & \mapsto & 1 \end{array}$	$ker φ_{0} = S_{3}$	$im φ_{0} = ⟨ 1 ⟩$
$\begin{array}{rrcc} ε : & S_{3} & \to & μ_{2} \\ s_{1} & \mapsto & - 1 \\ s_{2} & \mapsto & - 1 \end{array}$	$ker ε = A_{3}$	$im φ_{0} = μ_{2}$
$\begin{array}{rrcc} φ_{2} : & S_{3} & \to & O (3) \\ (12) & \mapsto & (\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}) \\ (23) & \mapsto & (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}) \end{array}$	$ker φ_{2} = ⟨ 1 ⟩$	$im φ_{2} = \{\begin{matrix} (\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}), & (\begin{matrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}), \\ (\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}), & (\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{matrix}), \\ (\begin{matrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix}), & (\begin{matrix} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}) \end{matrix}\}$
$\begin{array}{rrcc} φ_{3} : & S_{3} & \to & O (2) \\ (12) & \mapsto & (\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix}) \\ (23) & \mapsto & (\begin{matrix} \frac{1}{2} & \frac{1}{2} \\ \frac{3}{2} & \frac{- 1}{2} \end{matrix}) \end{array}$	$ker φ_{3} = ⟨ 1 ⟩$	$im φ_{3} = \{\begin{matrix} (\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}), & (\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix}), \\ (\begin{matrix} \frac{1}{2} & \frac{1}{2} \\ \frac{3}{2} & \frac{- 1}{2} \end{matrix}), & (\begin{matrix} \frac{- 1}{2} & \frac{- 1}{2} \\ \frac{3}{2} & \frac{- 1}{2} \end{matrix}), \\ (\begin{matrix} \frac{- 1}{2} & \frac{1}{2} \\ \frac{- 3}{2} & \frac{- 1}{2} \end{matrix}), & (\begin{matrix} \frac{- 1}{2} & \frac{- 1}{2} \\ \frac{- 3}{2} & \frac{- 1}{2} \end{matrix}) \end{matrix}\}$
$\begin{array}{rrcc} φ_{4} : & S_{3} & \to & D_{3} \\ (12) & \mapsto & y \\ (132) & \mapsto & x \end{array}$	$ker φ_{4} = ⟨ 1 ⟩$	$im φ_{4} = D_{3}$

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)

page history

S3≅D3: The Nonabelian Group of Order Six