Generalized Quaternion Group

In group theory, the quaternion group Q₈ (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset
$\$ of the quaternions under multiplication. It is given by the group presentation

:$\mathrm_8 = \langle \bar,i,j,k \mid \bar^2 = e, \;i^2 = j^2 = k^2 = ijk = \bar \rangle ,$

where ''e'' is the identity element and commutes with the other elements of the group.

Another presentation of Q₈ is
:$\mathrm_8 = \langle a,b \mid a^4 = e, a^2 = b^2, ba = a^b\rangle.$

Compared to dihedral group

The quaternion group Q₈ has the same order as the dihedral group D₄, but a different structure, as shown by their Cayley and cycle graphs:

In the diagrams for D₄, the group elements are marked with their action on a letter F in the defining representation R². The same cannot be done for Q₈, since it has no faithful representation in R² or R³. D₄ can be realized as a subset of the split-quaternions in the same way that Q₈ can be viewed as a subset of the quaternions.

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:

:$\left \langle x,y \mid x^4 = 1, x^2 = y^2, y^xy = x^ \right \rangle.$

One may take, for instance, $i = x, j = y,$ and $k = xy$.

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 $\$. The inner automorphism group of Q₈ is given by the group modulo its center, i.e. the factor group $\mathrm_8/\,$ 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, $\, \, \, \, \,$ 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''.

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

Since the irreducible characters $\chi_\rho$ in the rows above have real values, this gives the decomposition of the real group algebra of $G = \mathrm_8$ into minimal two-sided ideals:

:\R mathrm_8\bigoplus_\rho (e_\rho),

where the idempotents e_\rho\in \R mathrm_8/math> correspond to the irreducibles:

:$e_\rho = \frac\sum_ \chi_\rho(g^)g,$

so that

:$\begin e_ &= \tfrac 18(e + \bar e + i +\bar i+j+\bar j+k+\bar k) \\ e_ &= \tfrac 18(e + \bar e + i +\bar i-j-\bar j-k-\bar k) \\ e_ &= \tfrac 18(e + \bar e - i -\bar i+j+\bar j-k-\bar k) \\ e_ &= \tfrac 18(e + \bar e - i -\bar i-j-\bar j+k+\bar k) \\ e_ &= \tfrac 28(2e - 2\bar e) = \tfrac 12(e - \bar e) \end$

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

:$\begin \tfrac12(e-\bar e) &\longleftrightarrow 1, \\ \tfrac12(i-\bar i) &\longleftrightarrow i, \\ \tfrac12(j-\bar j) &\longleftrightarrow j, \\ \tfrac12(k-\bar k) &\longleftrightarrow k. \end$

Furthermore, the projection homomorphism \R mathrm_8to (e_2)\cong \mathbb given by $r\mapsto re_2$ has kernel ideal generated by the idempotent:

:$e_2^\perp = e_1+e_+e_+e_ = \frac 12(e+\bar e),$

so the quaternions can also be obtained as the quotient ring \R mathrm_8(e+\bar e)\cong \mathbb H.

The complex group algebra is thus \C mathrm_8\cong \C^ \oplus M_2(\C), where $M_2(\C) \cong \mathbb \otimes_ \C \cong \mathbb \oplus \mathbb$ 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 $\operatorname(2, \C)$. The quaternion group is a multiplicative subgroup of the quaternion algebra:

:$\H = \R 1 + \R i + \R j + \R k= \C 1+ \C j,$

which has a regular representation $\rho:\H \to \operatorname(2, \C)$ by left multiplication on itself considered as a complex vector space with basis $\,$ so that $z \in \H$ corresponds to the $\C$-linear mapping $\rho_z: a + jb \mapsto z\cdot(a + jb).$ The resulting representation

:$\begin \rho:\mathrm_8 \to \operatorname(2,\C)\\ g\longmapsto\rho_g \end$

is given by:

:$\begin e \mapsto \begin 1 & 0 \\ 0 & 1 \end & i \mapsto \begin i & 0 \\ 0 & \!\!\!\!-i \end& j \mapsto \begin 0 & \!\!\!\!-1 \\ 1 & 0 \end& k \mapsto \begin 0 & \!\!\!\!-i \\ \!\!\!-i & 0 \end \\ \overline \mapsto \begin \!\!\!-1 & 0 \\ 0 & \!\!\!\!-1 \end & \overline \mapsto \begin \!\!\!-i & 0 \\ 0 & i \end& \overline \mapsto \begin 0 & 1 \\ \!\!\!-1 & 0 \end& \overline \mapsto \begin 0 & i \\ i & 0 \end. \end$

Since all of the above matrices have unit determinant, this is a representation of Q₈ in the special linear group $\operatorname(2,\C)$.

A variant gives a representation by unitary matrices (table at right). Let $g\in \mathrm_8$ correspond to the linear mapping $\rho_g:a+bj\mapsto (a + bj)\cdot jg^j^,$ so that $\rho:\mathrm_8 \to \operatorname(2)$ is given by:

:$\begin e \mapsto \begin 1 & 0 \\ 0 & 1 \end & i \mapsto \begin i & 0 \\ 0 & \!\!\!\!-i \end& j \mapsto \begin 0 & 1 \\ \!\!\!-1 & 0 \end& k \mapsto \begin 0 & i \\ i & 0 \end \\ \overline \mapsto \begin \!\!\!-1 & 0 \\ 0 & \!\!\!\!-1 \end & \overline \mapsto \begin \!\!\!-i & 0 \\ 0 & i \end& \overline \mapsto \begin 0 & \!\!\!\!-1 \\ 1 & 0 \end& \overline \mapsto \begin 0 & \!\!\!\!-i \\ \!\!\!-i & 0 \end. \end$

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

:$\begin &e \mapsto \begin 1 & 0 \\ 0 & 1 \end = \quad\, 1_ &i \mapsto \begin 0 & \!\!\!-i\! \\ \!\!-i\!\! & 0 \end = -i \sigma_x &j \mapsto \begin 0 & \!\!\!-1\! \\ 1 & 0 \end = -i \sigma_y &k \mapsto \begin \!\!-i\!\! & 0 \\ 0 & i \end = -i \sigma_z\\ &\overline \mapsto \begin \!\!-1\! & 0 \\ 0 & \!\!\!-1\! \end = -1_ &\overline \mapsto \begin 0 & i \\ i & 0 \end = \,\,\,\, i \sigma_x &\overline \mapsto \begin 0 & 1 \\ \!\!-1\!\! & 0 \end = \,\,\,\, i \sigma_y &\overline \mapsto \begin i & 0 \\ 0 & \!\!\!-i\! \end = \,\,\,\, i \sigma_z. \end$

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

There is also an important action of Q₈ on the 2-dimensional vector space over the finite field $\mathbb_3 =\$ (table at right). A modular representation $\rho: \mathrm_8 \to \operatorname(2, 3)$ is given by

:$\begin e \mapsto \begin 1 & 0 \\ 0 & 1 \end & i \mapsto \begin 1 & 1 \\ 1 & \!\!\!\!-1 \end & j \mapsto \begin \!\!\!-1 & 1 \\ 1 & 1 \end & k \mapsto \begin 0 & \!\!\!\!-1 \\ 1 & 0 \end \\ \overline \mapsto \begin \!\!\!-1 & 0 \\ 0 & \!\!\!\!-1 \end & \overline \mapsto \begin \!\!\!-1 & \!\!\!\!-1 \\ \!\!\!-1 & 1 \end & \overline \mapsto \begin 1 & \!\!\!\!-1 \\ \!\!\!-1 & \!\!\!\!-1 \end & \overline \mapsto \begin 0 & 1 \\ \!\!\!-1 & 0 \end. \end$

This representation can be obtained from the extension field:

: \mathbb_9 = \mathbb_3 = \mathbb_3 1 + \mathbb_3 k,

where $k^2=-1$ and the multiplicative group $\mathbb_9^$ has four generators, $\pm(k\pm1),$ of order 8. For each $z \in \mathbb_9,$ the two-dimensional $\mathbb_3$-vector space $\mathbb_9$ admits a linear mapping:

:$\begin \mu_z: \mathbb_9 \to \mathbb_9 \\ \mu_z(a+bk)=z\cdot(a+bk) \end$

In addition we have the Frobenius automorphism $\phi(a+bk)=(a+bk)^3$ satisfying $\phi^2 = \mu_1$ and $\phi\mu_z = \mu_\phi.$ Then the above representation matrices are:

:$\begin \rho(\bar e) &=\mu_, \\ \rho(i) &=\mu_\phi, \\ \rho(j)&=\mu_ \phi, \\ \rho(k)&=\mu_. \end$

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

:$\begin \psi_m:\mathrm_8\to\mathrm_8 \\ \psi_m(g)=mgm^ \end$

with $\psi_I =\psi_=\mathrm_.$ In fact, these give the full automorphism group as:

:$\operatorname(\mathrm_8) \cong \operatorname(2, 3) = \operatorname(2,3)/\\cong S_4.$

This is isomorphic to the symmetric group S₄ since the linear mappings $m:\mathbb_3^2 \to \mathbb_3^2$ permute the four one-dimensional subspaces of $\mathbb_3^2,$ i.e., the four points of the projective space $\mathbb^1 (\mathbb_3) = \operatorname(1,3).$

Also, this representation permutes the eight non-zero vectors of $\mathbb_3^2,$ giving an embedding of Q₈ in the symmetric group S₈, in addition to the embeddings given by the regular representations.

Galois group

As Richard Dean showed in 1981, the quaternion group can be presented as the Galois group Gal(T/Q) where Q is the field of rational numbers and T is the splitting field over Q of the polynomial
:$x^8 - 72 x^6 + 180 x^4 - 144 x^2 + 36$.
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_4''n'' of order 4''n'' is defined by the presentation
:$\langle x,y \mid x^ = y^4 = 1, x^n = y^2, y^xy = x^\rangle$
for an integer , with the usual quaternion group given by ''n'' = 2. Coxeter calls Q_4''n'' the dicyclic group $\langle 2, 2, n\rangle$, a special case of the binary polyhedral group $\langle \ell, m, n\rangle$ and related to the polyhedral group $(p,q,r)$ and the dihedral group $(2,2,n)$. The generalized quaternion group can be realized as the subgroup of $\operatorname_2(\Complex)$ generated by

:$\left(\begin \omega_n & 0 \\ 0 & \overline_n \end \right) \mbox \left(\begin 0 & -1 \\ 1 & 0 \end \right)$

where $\omega_n = e^$. It can also be realized as the subgroup of unit quaternions generated by $x=e^$ and $y=j$.

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., Theorem 4.3, p. 99 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, . Letting ''p^r'' be the size of ''F'', where ''p'' is prime, the size of the 2-Sylow subgroup of SL₂(''F'') is 2^''n'', where .

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
:$\langle x,y \mid x^ = y^4 = 1, x^ = y^2, y^xy = x^\rangle.$

See also

*16-cell
*Binary tetrahedral group
*Clifford algebra
* Dicyclic group
* Hurwitz integral quaternion
*List of small groups

Notes

References

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* Dean, Richard A. (1981) "A rational polynomial whose group is the quaternions", American Mathematical Monthly 88:42–5.
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* P.R. Girard (1984) "The quaternion group and modern physics", European Journal of Physics 5:25–32.
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External links

* {{MathWorld , urlname = QuaternionGroup , title = Quaternion group

Quaternion groups on GroupNames
* Quaternion group o
GroupProps
* Conrad, Keith
"Generalized Quaternions"

Group theory
Finite groups
Quaternions