Quaternion group

Cayley table

The Cayley table (multiplication table) for Q₈ is given by:

Properties

The elements i, j, and k all have order four in Q₈ and any two of them generate the entire group. Another presentation of Q₈ based in only two elements to skip this redundancy is:

${\displaystyle ⟨x,y\mid x^4=1,x^2=y^2,y^{-1}xy=x^{-1}⟩.}$

This presentation is still redundant, since the relation ${\displaystyle x^4=1}$ is implied by the other two relations, and may be dropped. The quaternion group has also a shorter presentation with two generators and two relations

${\displaystyle ⟨x,y\mid x^2=y^2,xyx=y⟩.}$

For instance, writing the group elements in lexicographically minimal normal forms, one may identify:

${\displaystyle \{e,\overline{e},i,\overline{i},j,\overline{j},k,\overline{k}\}\leftrightarrow \{e,x^2,x,x^3,y,x^{2}y,xy,x^{3}y\}.}$

The quaternion group has the unusual property of being Hamiltonian: Q₈ is non-abelian, but every subgroup is normal. Every Hamiltonian group contains a copy of Q₈.

The quaternion group Q₈ and the dihedral group D₄ are the two smallest examples of a nilpotent non-abelian group.

The center and the commutator subgroup of Q₈ is the subgroup ${\displaystyle \{e,\overline{e}\}}$. The inner automorphism group of Q₈ is given by the group modulo its center, i.e. the factor group ${\displaystyle {\mathrm{Q}}_8/\{e,\overline{e}\},}$ which is isomorphic to the Klein four-group V. The full automorphism group of Q₈ is isomorphic to S₄, the symmetric group on four letters (see Matrix representations below), and the outer automorphism group of Q₈ is thus S₄/V, which is isomorphic to S₃.

The quaternion group Q₈ has five conjugacy classes, ${\displaystyle \{e\},\{\overline{e}\},\{i,\overline{i}\},\{j,\overline{j}\},\{k,\overline{k}\},}$ and so five irreducible representations over the complex numbers, with dimensions 1, 1, 1, 1, 2:

Trivial representation.

Sign representations with i, j, k-kernel: Q₈ has three maximal normal subgroups: the cyclic subgroups generated by i, j, and k respectively. For each maximal normal subgroup N, we obtain a one-dimensional representation factoring through the 2-element quotient group G/N. The representation sends elements of N to 1, and elements outside N to −1.

2-dimensional representation: Described below in Matrix representations. It is not realizable over the real numbers, but is a complex representation: indeed, it is just the quaternions ${\displaystyle \mathbb{H}}$ considered as an algebra over ${\displaystyle \mathbb{C}}$, and the action is that of left multiplication by ${\displaystyle Q_8\subset \mathbb{H}}$.

The character table of Q₈ turns out to be the same as that of D₄:

Nevertheless, all the irreducible characters ${\displaystyle {\chi }_{\rho }}$ in the rows above have real values, this gives the decomposition of the real group algebra of ${\displaystyle G={\mathrm{Q}}_8}$ into minimal two-sided ideals:

${\displaystyle \mathbb{R}[{\mathrm{Q}}_8]=\underset{\rho }{⨁}(e_{\rho }),}$

where the idempotents ${\displaystyle e_{\rho }\in \mathbb{R}[{\mathrm{Q}}_8]}$ correspond to the irreducibles:

${\displaystyle e_{\rho }=\frac{\dim (\rho )}{|G|}\sum _{g\in G}{\chi }_{\rho }(g^{-1})g,}$

so that

${\displaystyle \begin{array}{cc}{e}_{\text{triv}}& =\frac{1}{8}(e+\overline{e}+i+\overline{i}+j+\overline{j}+k+\overline{k})\\ e_{i\text{-ker}}& =\frac{1}{8}(e+\overline{e}+i+\overline{i}-j-\overline{j}-k-\overline{k})\\ e_{j\text{-ker}}& =\frac{1}{8}(e+\overline{e}-i-\overline{i}+j+\overline{j}-k-\overline{k})\\ e_{k\text{-ker}}& =\frac{1}{8}(e+\overline{e}-i-\overline{i}-j-\overline{j}+k+\overline{k})\\ e_2& =\frac{2}{8}(2e-2\overline{e})=\frac{1}{2}(e-\overline{e})\end{array}}$

Each of these irreducible ideals is isomorphic to a real central simple algebra, the first four to the real field ${\displaystyle \mathbb{R}}$. The last ideal ${\displaystyle (e_2)}$ is isomorphic to the skew field of quaternions ${\displaystyle \mathbb{H}}$ by the correspondence:

${\displaystyle \begin{array}{cc}\frac{1}{2}(e-\overline{e})& ⟷1,\\ \frac{1}{2}(i-\overline{i})& ⟷i,\\ \frac{1}{2}(j-\overline{j})& ⟷j,\\ \frac{1}{2}(k-\overline{k})& ⟷k.\end{array}}$

Furthermore, the projection homomorphism ${\displaystyle \mathbb{R}[{\mathrm{Q}}_8]\to (e_2)\cong \mathbb{H}}$ given by ${\displaystyle r\mapsto r{e}_2}$ has kernel ideal generated by the idempotent:

${\displaystyle e_2^{\perp }=e_1+e_{i\text{-ker}}+e_{j\text{-ker}}+e_{k\text{-ker}}=\frac{1}{2}(e+\overline{e}),}$

so the quaternions can also be obtained as the quotient ring ${\displaystyle \mathbb{R}[{\mathrm{Q}}_8]/(e+\overline{e})\cong \mathbb{H}}$. Note that this is irreducible as a real representation of ${\displaystyle Q_8}$, but splits into two copies of the two-dimensional irreducible when extended to the complex numbers. Indeed, the complex group algebra is ${\displaystyle \mathbb{C}[{\mathrm{Q}}_8]\cong {\mathbb{C}}^{\oplus 4}\oplus M_2(\mathbb{C}),}$ where ${\displaystyle M_2(\mathbb{C})\cong \mathbb{H}{\otimes }_{\mathbb{R}}\mathbb{C}}$ is the algebra of biquaternions.

Matrix representations

The two-dimensional irreducible complex representation described above gives the quaternion group Q₈ as a subgroup of the general linear group ${\displaystyle \mathrm{GL}(2,\mathbb{C})}$. The quaternion group is a multiplicative subgroup of the quaternion algebra:

${\displaystyle \mathbb{H}=\mathbb{R}1+\mathbb{R}i+\mathbb{R}j+\mathbb{R}k=1\mathbb{C}+j\mathbb{C},}$

which has a regular representation ${\displaystyle \rho :\mathbb{H}\to \mathrm{M}(2,\mathbb{C})}$ by left multiplication on itself considered as a right complex vector space with basis ${\displaystyle \{1,j\},}$ so that ${\displaystyle z\in \mathbb{H}}$ corresponds to the ${\displaystyle \mathbb{C}}$-linear mapping ${\displaystyle {\rho }_z:a+jb\mapsto z\cdot (a+jb).}$ The resulting representation

${\displaystyle \{\begin{array}{l}\rho :{\mathrm{Q}}_8\to \mathrm{GL}(2,\mathbb{C})\\ g⟼{\rho }_g\end{array}}$

is given by:

${\displaystyle \begin{array}{cccc}e\mapsto \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)& i\mapsto \left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right)& j\mapsto \left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)& k\mapsto \left(\begin{array}{cc}0& -i\\ -i& 0\end{array}\right)\\ \overline{e}\mapsto \left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)& \overline{i}\mapsto \left(\begin{array}{cc}-i& 0\\ 0& i\end{array}\right)& \overline{j}\mapsto \left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)& \overline{k}\mapsto \left(\begin{array}{cc}0& i\\ i& 0\end{array}\right).\end{array}}$

Since all of the above matrices have unit determinant, this is a representation of Q₈ in the special linear group ${\displaystyle \mathrm{SL}(2,\mathbb{C})}$.

A variant gives a representation by unitary matrices (table at right). Let ${\displaystyle g\in {\mathrm{Q}}_8}$ correspond to the linear mapping ${\displaystyle {\rho }_g:a+bj\mapsto (a+bj)\cdot j{g}^{-1}{j}^{-1},}$ so that ${\displaystyle \rho :{\mathrm{Q}}_8\to \mathrm{SU}(2)}$ is given by:

${\displaystyle \begin{array}{cccc}e\mapsto \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)& i\mapsto \left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right)& j\mapsto \left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)& k\mapsto \left(\begin{array}{cc}0& i\\ i& 0\end{array}\right)\\ \overline{e}\mapsto \left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)& \overline{i}\mapsto \left(\begin{array}{cc}-i& 0\\ 0& i\end{array}\right)& \overline{j}\mapsto \left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)& \overline{k}\mapsto \left(\begin{array}{cc}0& -i\\ -i& 0\end{array}\right).\end{array}}$

It is worth noting that physicists exclusively use a different convention for the ${\displaystyle \mathrm{SU}(2)}$ matrix representation to make contact with the usual Pauli matrices:

${\displaystyle \begin{array}{ccccc}& e\mapsto \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)=1_{2\times 2}& i\mapsto \left(\begin{array}{cc}0& -i\\ -i& 0\end{array}\right)=-i{\sigma }_x& j\mapsto \left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)=-i{\sigma }_y& k\mapsto \left(\begin{array}{cc}-i& 0\\ 0& i\end{array}\right)=-i{\sigma }_z\\ & \overline{e}\mapsto \left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)=-1_{2\times 2}& \overline{i}\mapsto \left(\begin{array}{cc}0& i\\ i& 0\end{array}\right)=i{\sigma }_x& \overline{j}\mapsto \left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)=i{\sigma }_y& \overline{k}\mapsto \left(\begin{array}{cc}i& 0\\ 0& -i\end{array}\right)=i{\sigma }_z.\end{array}}$

This particular choice is convenient and elegant when one describes spin-1/2 states in the ${\displaystyle ({\overrightarrow{J}}^2,J_z)}$ basis and considers angular momentum ladder operators ${\displaystyle J_{\pm }=J_x\pm i{J}_y.}$

There is also an important action of Q₈ on the 2-dimensional vector space over the finite field ${\displaystyle {\mathbb{F}}_3=\{0,1,-1\}}$ (table at right). A modular representation ${\displaystyle \rho :{\mathrm{Q}}_8\to \mathrm{SL}(2,3)}$ is given by

${\displaystyle \begin{array}{cccc}e\mapsto \left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)& i\mapsto \left(\begin{array}{cc}1& 1\\ 1& -1\end{array}\right)& j\mapsto \left(\begin{array}{cc}-1& 1\\ 1& 1\end{array}\right)& k\mapsto \left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\\ \overline{e}\mapsto \left(\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right)& \overline{i}\mapsto \left(\begin{array}{cc}-1& -1\\ -1& 1\end{array}\right)& \overline{j}\mapsto \left(\begin{array}{cc}1& -1\\ -1& -1\end{array}\right)& \overline{k}\mapsto \left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right).\end{array}}$

This representation can be obtained from the extension field:

${\displaystyle {\mathbb{F}}_9={\mathbb{F}}_3[k]={\mathbb{F}}_{3}1+{\mathbb{F}}_{3}k,}$

where ${\displaystyle k^2=-1}$ and the multiplicative group ${\displaystyle {\mathbb{F}}_9^{\times }}$ has four generators, ${\displaystyle \pm (k\pm 1),}$ of order 8. For each ${\displaystyle z\in {\mathbb{F}}_9,}$ the two-dimensional ${\displaystyle {\mathbb{F}}_3}$-vector space ${\displaystyle {\mathbb{F}}_9}$ admits a linear mapping:

${\displaystyle \{\begin{array}{l}{\mu }_z:{\mathbb{F}}_9\to {\mathbb{F}}_9\\ {\mu }_z(a+bk)=z\cdot (a+bk)\end{array}}$

In addition we have the Frobenius automorphism ${\displaystyle \varphi (a+bk)=(a+bk{)}^3}$ satisfying ${\displaystyle {\varphi }^2={\mu }_1}$ and ${\displaystyle \varphi {\mu }_z={\mu }_{\varphi (z)}\varphi .}$ Then the above representation matrices are:

${\displaystyle \begin{array}{cc}\rho (\overline{e})& ={\mu }_{-1},\\ \rho (i)& ={\mu }_{k+1}\varphi ,\\ \rho (j)& ={\mu }_{k-1}\varphi ,\\ \rho (k)& ={\mu }_k.\end{array}}$

This representation realizes Q₈ as a normal subgroup of GL(2, 3). Thus, for each matrix ${\displaystyle m\in \mathrm{GL}(2,3)}$, we have a group automorphism

${\displaystyle \{\begin{array}{l}{\psi }_m:{\mathrm{Q}}_8\to {\mathrm{Q}}_8\\ {\psi }_m(g)=mg{m}^{-1}\end{array}}$

with ${\displaystyle {\psi }_I={\psi }_{-I}={\mathrm{i}\mathrm{d}}_{{\mathrm{Q}}_8}.}$ In fact, these give the full automorphism group as:

${\displaystyle \mathrm{Aut}({\mathrm{Q}}_8)\cong \mathrm{PGL}(2,3)=\mathrm{GL}(2,3)/\{\pm I\}\cong S_4.}$

This is isomorphic to the symmetric group S₄ since the linear mappings ${\displaystyle m:{\mathbb{F}}_3^2\to {\mathbb{F}}_3^2}$ permute the four one-dimensional subspaces of ${\displaystyle {\mathbb{F}}_3^2,}$ i.e., the four points of the projective space ${\displaystyle {\mathbb{P}}^1({\mathbb{F}}_3)=\mathrm{PG}(1,3).}$

Also, this representation permutes the eight non-zero vectors of ${\displaystyle {\mathbb{F}}_3^2,}$ giving an embedding of Q₈ in the symmetric group S₈, in addition to the embeddings given by the regular representations.

Galois group

Richard Dedekind considered the field ${\displaystyle \mathbb{Q}(\sqrt{2},\sqrt{3})}$ in attempting to relate the quaternion group to Galois theory. In 1936 Ernst Witt published his approach to the quaternion group through Galois theory.

In 1981, Richard Dean showed the quaternion group can be realized as the Galois group Gal(T/Q) where Q is the field of rational numbers and T is the splitting field of the polynomial

${\displaystyle x^8-72x^6+180x^4-144x^2+36}$.

The development uses the fundamental theorem of Galois theory in specifying four intermediate fields between Q and T and their Galois groups, as well as two theorems on cyclic extension of degree four over a field.

Generalized quaternion group

A generalized quaternion group Q_4n of order 4n is defined by the presentation

${\displaystyle ⟨x,y\mid x^{2n}=y^4=1,x^n=y^2,y^{-1}xy=x^{-1}⟩}$

for an integer n ≥ 2, with the usual quaternion group given by n = 2. Coxeter calls Q_4n the dicyclic group ${\displaystyle ⟨2,2,n⟩}$, a special case of the binary polyhedral group ${\displaystyle ⟨\ell ,m,n⟩}$ and related to the polyhedral group ${\displaystyle (p,q,r)}$ and the dihedral group ${\displaystyle (2,2,n)}$. The generalized quaternion group can be realized as the subgroup of ${\displaystyle {\mathrm{GL}}_2(\mathbb{C})}$ generated by

${\displaystyle \left(\begin{array}{cc}{\omega }_n& 0\\ 0& {\overline{\omega }}_n\end{array}\right)\text{ and }\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)}$

where ${\displaystyle {\omega }_n=e^{i\pi /n}}$. It can also be realized as the subgroup of unit quaternions generated by ${\displaystyle x=e^{i\pi /n}}$ and ${\displaystyle y=j}$.

The generalized quaternion groups have the property that every abelian subgroup is cyclic. It can be shown that a finite p-group with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above. Another characterization is that a finite p-group in which there is a unique subgroup of order p is either cyclic or a 2-group isomorphic to generalized quaternion group. In particular, for a finite field F with odd characteristic, the 2-Sylow subgroup of SL₂(F) is non-abelian and has only one subgroup of order 2, so this 2-Sylow subgroup must be a generalized quaternion group, (Gorenstein 1980, p. 42). Letting p^r be the size of F, where p is prime, the size of the 2-Sylow subgroup of SL₂(F) is 2ⁿ, where n = ord₂(p² − 1) + ord₂(r).

The Brauer–Suzuki theorem shows that the groups whose Sylow 2-subgroups are generalized quaternion cannot be simple.

Another terminology reserves the name "generalized quaternion group" for a dicyclic group of order a power of 2, which admits the presentation

${\displaystyle ⟨x,y\mid x^{2^m}=y^4=1,x^{2^{m-1}}=y^2,y^{-1}xy=x^{-1}⟩.}$