The Quaternion Group Q

## 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 ${𝒞}_{1}=\left\{1\right\}$ ${H}_{0}=Q$
${𝒞}_{-1}=\left\{-1\right\}$ ${H}_{1}=\left\{±1,±i\right\}$
${𝒞}_{i}=\left\{±i\right\}$ ${H}_{2}=\left\{±1,±j\right\}$
${𝒞}_{j}=\left\{±j\right\}$ ${H}_{3}=\left\{±1,±k\right\}$
${𝒞}_{k}=\left\{±k\right\}$ ${H}_{4}=\left\{±1\right\}$
${H}_{5}=\left\{1\right\}$

Element $g$ Order $ο\left(g\right)$ 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 $Q/{H}_{0}\cong ⟨1⟩$
${H}_{1}=\left\{±1,±i\right\}$ ${H}_{1}\cong {\mu }_{4}$ $\left[Q:{H}_{1}\right]=2$ Yes $Q/{H}_{1}\cong {\mu }_{2}$
${H}_{2}=\left\{±1,±j\right\}$ ${H}_{2}\cong {\mu }_{4}$ $\left[Q:{H}_{2}\right]=2$ Yes $Q/{H}_{2}\cong {\mu }_{2}$
${H}_{3}=\left\{±1,±k\right\}$ ${H}_{3}\cong {\mu }_{4}$ $\left\{Q:{H}_{3}\right\}=2$ Yes $Q/{H}_{3}\cong {\mu }_{2}$
${H}_{4}=\left\{±1\right\}$ ${H}_{4}\cong {\mu }_{2}$ $\left[Q:{H}_{4}\right]=4$ Yes $Q/{H}_{4}\cong {\mu }_{2} × {\mu }_{2}$
${H}_{9}=\left\{1\right\}$ ${H}_{9}=⟨1⟩$ $\left[Q:⟨1⟩\right]=8$ Yes $Q/{H}_{9}\cong Q$

Subgroups ${H}_{i}$ Left Cosets Right Cosets
${H}_{0}=Q$ $Q$ $Q$
${H}_{1}=\left\{±1,±i\right\}$ ${H}_{1}=\left\{±1,±i\right\}$ ${H}_{1}=\left\{±1,±i\right\}$
$j{H}_{1}=\left\{±j,±k\right\}$ ${H}_{1}j=\left\{±j,±k\right\}$
${H}_{2}=\left\{±1,±j\right\}$ ${H}_{2}=\left\{±1,±j\right\}$ ${H}_{2}=\left\{±1,±j\right\}$
$i{H}_{2}=\left\{±i,±k\right\}$ ${H}_{2}j=\left\{±i,±k\right\}$
${H}_{3}=\left\{±1,±k\right\}$ ${H}_{3}=\left\{±1,±k\right\}$ ${H}_{3}=\left\{±1,±k\right\}$
$i{H}_{3}=\left\{±i,±j\right\}$ ${H}_{3}i=\left\{±i,±j\right\}$
${H}_{4}=\left\{±1\right\}$ ${H}_{4}=\left\{±1\right\}$ ${H}_{4}=\left\{±1\right\}$
$i{H}_{4}=\left\{±i\right\}$ ${H}_{4}i=\left\{±i\right\}$
$j{H}_{4}=\left\{±j\right\}$ ${H}_{4}j=\left\{±j\right\}$
$k{H}_{4}=\left\{±k\right\}$ ${H}_{4}k=\left\{±k\right\}$
${H}_{5}=\left\{1\right\}$ ${H}_{5}=\left\{1\right\}$ ${H}_{5}=\left\{1\right\}$
$\left(-1\right){H}_{5}=\left\{-1\right\}$ ${H}_{5}\left(-1\right)=\left\{-1\right\}$
$i{H}_{5}=\left\{i\right\}$ ${H}_{5}i=\left\{i\right\}$
$-i{H}_{5}=\left\{-i\right\}$ ${H}_{5}\left(-i\right)=\left\{-i\right\}$
$j{H}_{5}=\left\{j\right\}$ ${H}_{5}j=\left\{j\right\}$
$-j{H}_{5}=\left\{-j\right\}$ ${H}_{5}\left(-j\right)=\left\{-j\right\}$
$k{H}_{5}=\left\{k\right\}$ ${H}_{5}k=\left\{k\right\}$
$-k{H}_{5}=\left\{k\right\}$ ${H}_{5}\left(-k\right)=\left\{-k\right\}$

Subgroups ${H}_{i}$ Normalizer ${N}_{{H}_{i}}$ Centralizer ${Z}_{{H}_{i}}$
${H}_{0}=Q$ $Q$ $Z\left(Q\right)={H}_{4}=\left\{±1\right\}$
${H}_{1}=⟨i⟩$ $Q$ ${H}_{1}=⟨i⟩$
${H}_{2}=⟨j⟩$ $Q$ ${H}_{2}=⟨j⟩$
${H}_{3}=⟨k⟩$ $Q$ ${H}_{3}=⟨k⟩$
${H}_{4}=⟨±1⟩$ $Q$ $Q$
${H}_{5}=⟨1⟩$ $Q$ $Q$

Homomorphism Kernel Image
$\begin{array}{rrcc}{\varphi }_{0}:& Q& \to & ⟨1⟩\\ & i& ↦& 1\\ & j& ↦& 1\end{array}$ $ker {\varphi }_{0}=Q$ $im {\varphi }_{0}=⟨1⟩$
$\begin{array}{rrcc}{\varphi }_{1}:& Q& \to & {\mu }_{2}\\ & i& ↦& 1\\ & j& ↦& -1\end{array}$ $ker {\varphi }_{1}={H}_{1}=\left\{±1,±i\right\}$ $im {\varphi }_{1}={\mu }_{2}$
$\begin{array}{rrcc}{\varphi }_{2}:& Q& \to & {\mu }_{2}\\ & i& ↦& -1\\ & j& ↦& 1\end{array}$ $ker {\varphi }_{2}={H}_{2}=\left\{±1,±j\right\}$ $im {\varphi }_{2}={\mu }_{2}$
$\begin{array}{rrcc}{\varphi }_{3}:& Q& \to & {\mu }_{2}\\ & i& ↦& -1\\ & j& ↦& -1\end{array}$ $ker {\varphi }_{3}={H}_{2}=\left\{±1,±k\right\}$ $im {\varphi }_{3}={\mu }_{2}$
$\begin{array}{rrcc}{\varphi }_{4}:& Q& \to & {GL}_{2}\left(ℂ\right)\\ & i& ↦& \left(\begin{array}{cc}i& 0\\ 0& i\end{array}\right)\\ & j& ↦& \left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)\\ & k& ↦& \left(\begin{array}{cc}0& i\\ i& 0\end{array}\right)\end{array}$ $ker {\varphi }_{4}={H}_{5}=⟨1⟩$ $im {\varphi }_{4}=\left\{\begin{array}{cc}\left(\begin{array}{cc}±1& 0\\ 0& ±1\end{array}\right),& \left(\begin{array}{cc}±i& 0\\ 0& \mp i\end{array}\right),\\ \left(\begin{array}{cc}0& ±1\\ \mp 1& 0\end{array}\right),& \left(\begin{array}{cc}0& ±i\\ \mp i& 0\end{array}\right)\end{array}\right\}$
$\begin{array}{rrcc}{\varphi }_{5}:& Q& \to & {\mu }_{2} × {\mu }_{2}\\ & i& ↦& \left(-1,1\right)\\ & j& ↦& \left(1,-1\right)\end{array}$ $ker {\varphi }_{5}={H}_{4}=\left\{±1\right\}$ $im {\varphi }_{5}={\mu }_{2} × {\mu }_{2}$

## 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)