The Quaternion Group $Q$

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

Last updates: 27 January 2011

The quaternion group $Q$

The quaternion group $Q$ is as in the following table. The element $-1$ acts like $-1$ in the complex numbers, it takes everything to its negative, and the negative of a negative is a positive.

Set	Operation
$Q=\left\{1,-1,i,-i,j,-j,k,-k\right\}$	$i^2=j^2=k^2=ijk=-1$

The complete multiplication table for $Q$ is as follows.

Multiplication table
$Q$ $1$ $-1$ $i$ $-i$ $j$ $-j$ $k$ $-k$ $1$ $1$ $-1$ $i$ $-i$ $j$ $-j$ $k$ $-k$ $-1$ $-1$ $1$ $-i$ $i$ $-j$ $j$ $-k$ $k$ $i$ $i$ $-i$ $-1$ $1$ $k$ $-k$ $-j$ $j$ $-i$ $-i$ $i$ $1$ $-1$ $-k$ $k$ $j$ $-j$ $j$ $j$ $-j$ $-k$ $k$ $-1$ $1$ $i$ $-i$ $-j$ $-j$ $j$ $k$ $-k$ $1$ $-1$ $-i$ $i$ $k$ $k$ $-k$ $j$ $-j$ $-i$ $i$ $-1$ $1$ $-k$ $-k$ $k$ $-j$ $j$ $i$ $-i$ $1$ $-1$

Center	Abelian	Conjugacy classes	Subgroups
$Z\left(Q\right)=\left\{1,-1\right\}$	No	${𝒞}_1=\left\{1\right\}$	$H_0=Q$
		${𝒞}_{-1}=\left\{-1\right\}$	$H_1=\left\{\pm 1,\pm i\right\}$
		${𝒞}_i=\left\{\pm i\right\}$	$H_2=\left\{\pm 1,\pm j\right\}$
		${𝒞}_j=\left\{\pm j\right\}$	$H_3=\left\{\pm 1,\pm k\right\}$
		${𝒞}_k=\left\{\pm k\right\}$	$H_4=\left\{\pm 1\right\}$
			$H_5=\left\{1\right\}$

Element $g$	Order $o\left(g\right)$	Centralizer $Z_g$	Conjugacy Class ${𝒞}_g$
$1$	$1$	$Q$	${𝒞}_1$
$-1$	$2$	$Q$	${𝒞}_{-1}$
$i$	$4$	$H_1$	${𝒞}_i$
$-i$	$4$	$H_1$	${𝒞}_i$
$j$	$4$	$H_2$	${𝒞}_j$
$-j$	$4$	$H_2$	${𝒞}_j$
$k$	$4$	$H_3$	${𝒞}_k$
$-k$	$4$	$H_3$	${𝒞}_k$

Generators	Relations	Realization
$S,T$	$S^2=T^2={\left(ST\right)}^2$	$S=i,T=j,ST=k$

Subgroups $H_i$	Structure	Index	Normal	Quotient group
$H_0=Q$	$H_0=Q$	$\left[Q:Q\right]=1$	Yes	$Q/H_0\cong ⟨1⟩$
$H_1=\left\{\pm 1,\pm i\right\}$	$H_1\cong {\mu }_4$	$\left[Q:H_1\right]=2$	Yes	$Q/H_1\cong {\mu }_2$
$H_2=\left\{\pm 1,\pm j\right\}$	$H_2\cong {\mu }_4$	$\left[Q:H_2\right]=2$	Yes	$Q/H_2\cong {\mu }_2$
$H_3=\left\{\pm 1,\pm k\right\}$	$H_3\cong {\mu }_4$	$\left\{Q:H_3\right\}=2$	Yes	$Q/H_3\cong {\mu }_2$
$H_4=\left\{\pm 1\right\}$	$H_4\cong {\mu }_2$	$\left[Q:H_4\right]=4$	Yes	$Q/H_4\cong {\mu }_2\hspace{0.17em}\times \hspace{0.17em}{\mu }_2$
$H_9=\left\{1\right\}$	$H_9=⟨1⟩$	$\left[Q:⟨1⟩\right]=8$	Yes	$Q/H_9\cong Q$

Subgroups $H_i$	Left Cosets	Right Cosets
$H_0=Q$	$Q$	$Q$
$H_1=\left\{\pm 1,\pm i\right\}$	$H_1=\left\{\pm 1,\pm i\right\}$	$H_1=\left\{\pm 1,\pm i\right\}$
	$j{H}_1=\left\{\pm j,\pm k\right\}$	$H_{1}j=\left\{\pm j,\pm k\right\}$
$H_2=\left\{\pm 1,\pm j\right\}$	$H_2=\left\{\pm 1,\pm j\right\}$	$H_2=\left\{\pm 1,\pm j\right\}$
	$i{H}_2=\left\{\pm i,\pm k\right\}$	$H_{2}j=\left\{\pm i,\pm k\right\}$
$H_3=\left\{\pm 1,\pm k\right\}$	$H_3=\left\{\pm 1,\pm k\right\}$	$H_3=\left\{\pm 1,\pm k\right\}$
	$i{H}_3=\left\{\pm i,\pm j\right\}$	$H_{3}i=\left\{\pm i,\pm j\right\}$
$H_4=\left\{\pm 1\right\}$	$H_4=\left\{\pm 1\right\}$	$H_4=\left\{\pm 1\right\}$
	$i{H}_4=\left\{\pm i\right\}$	$H_{4}i=\left\{\pm i\right\}$
	$j{H}_4=\left\{\pm j\right\}$	$H_{4}j=\left\{\pm j\right\}$
	$k{H}_4=\left\{\pm k\right\}$	$H_{4}k=\left\{\pm k\right\}$
$H_5=\left\{1\right\}$	$H_5=\left\{1\right\}$	$H_5=\left\{1\right\}$
	$\left(-1\right)H_5=\left\{-1\right\}$	$H_5\left(-1\right)=\left\{-1\right\}$
	$i{H}_5=\left\{i\right\}$	$H_{5}i=\left\{i\right\}$
	$-i{H}_5=\left\{-i\right\}$	$H_5\left(-i\right)=\left\{-i\right\}$
	$j{H}_5=\left\{j\right\}$	$H_{5}j=\left\{j\right\}$
	$-j{H}_5=\left\{-j\right\}$	$H_5\left(-j\right)=\left\{-j\right\}$
	$k{H}_5=\left\{k\right\}$	$H_{5}k=\left\{k\right\}$
	$-k{H}_5=\left\{k\right\}$	$H_5\left(-k\right)=\left\{-k\right\}$

Subgroups $H_i$	Normalizer $N_{H_i}$	Centralizer $Z_{H_i}$
$H_0=Q$	$Q$	$Z\left(Q\right)=H_4=\left\{\pm 1\right\}$
$H_1=⟨i⟩$	$Q$	$H_1=⟨i⟩$
$H_2=⟨j⟩$	$Q$	$H_2=⟨j⟩$
$H_3=⟨k⟩$	$Q$	$H_3=⟨k⟩$
$H_4=⟨\pm 1⟩$	$Q$	$Q$
$H_5=⟨1⟩$	$Q$	$Q$

Homomorphism	Kernel	Image
$\begin{array}{rrcc}{\varphi }_0:& Q& \to & ⟨1⟩\\ & i& \mapsto & 1\\ & j& \mapsto & 1\end{array}$	$ker\hspace{0.17em}{\varphi }_0=Q$	$im\hspace{0.17em}{\varphi }_0=⟨1⟩$
$\begin{array}{rrcc}{\varphi }_1:& Q& \to & {\mu }_2\\ & i& \mapsto & 1\\ & j& \mapsto & -1\end{array}$	$ker\hspace{0.17em}{\varphi }_1=H_1=\left\{\pm 1,\pm i\right\}$	$im\hspace{0.17em}{\varphi }_1={\mu }_2$
$\begin{array}{rrcc}{\varphi }_2:& Q& \to & {\mu }_2\\ & i& \mapsto & -1\\ & j& \mapsto & 1\end{array}$	$ker\hspace{0.17em}{\varphi }_2=H_2=\left\{\pm 1,\pm j\right\}$	$im\hspace{0.17em}{\varphi }_2={\mu }_2$
$\begin{array}{rrcc}{\varphi }_3:& Q& \to & {\mu }_2\\ & i& \mapsto & -1\\ & j& \mapsto & -1\end{array}$	$ker\hspace{0.17em}{\varphi }_3=H_2=\left\{\pm 1,\pm k\right\}$	$im\hspace{0.17em}{\varphi }_3={\mu }_2$
$\begin{array}{rrcc}{\varphi }_4:& Q& \to & {GL}_2\left(ℂ\right)\\ & i& \mapsto & \left(\begin{array}{cc}i& 0\\ 0& i\end{array}\right)\\ & j& \mapsto & \left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\\ & k& \mapsto & \left(\begin{array}{cc}0& i\\ i& 0\end{array}\right)\end{array}$	$ker\hspace{0.17em}{\varphi }_4=H_5=⟨1⟩$	$im\hspace{0.17em}{\varphi }_4=\left\{\begin{array}{cc}\left(\begin{array}{cc}\pm 1& 0\\ 0& \pm 1\end{array}\right),& \left(\begin{array}{cc}\pm i& 0\\ 0& \mp i\end{array}\right),\\ \left(\begin{array}{cc}0& \pm 1\\ \mp 1& 0\end{array}\right),& \left(\begin{array}{cc}0& \pm i\\ \mp i& 0\end{array}\right)\end{array}\right\}$
	$ker\hspace{0.17em}{\varphi }_5=H_4=\left\{\pm 1\right\}$

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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