Quaternion group

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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 ${\displaystyle \{1,i,j,k,-1,-i,-j,-k\}$ of the quaternions under multiplication. It is given by the group presentation

${\displaystyle \mathrm {Q} _{8}=\langle {\bar {e},i,j,k\mid {\bar {e}^{2}=e,\;i^{2}=j^{2}=k^{2}=ijk={\bar {e}\rangle ,}$

where e is the identity element and e commutes with the other elements of the group. These relations, discovered by W. R. Hamilton, also generate the quaternions as an algebra over the real numbers.

Another presentation of Q₈ is

${\displaystyle \mathrm {Q} _{8}=\langle a,b\mid a^{4}=e,a^{2}=b^{2},ba=a^{-1}b\rangle .}$

Like many other finite groups, it can be realized as the Galois group of a certain field of algebraic numbers.^[1]

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:

	Q8	D4
Cayley graph	Red arrows connect g→gi, green connect g→gj.
Cycle graph

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:^[2]

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^[3] based in only two elements to skip this redundancy is:

${\displaystyle \left\langle x,y\mid x^{4}=1,x^{2}=y^{2},y^{-1}xy=x^{-1}\right\rangle .}$

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

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

The quaternion group has the unusual property of being Hamiltonian: Q₈ is non-abelian, but every subgroup is normal.^[4] Every Hamiltonian group contains a copy of Q₈.^[5]

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 ${\displaystyle \{e,{\bar {e}\}$. The inner automorphism group of Q₈ is given by the group modulo its center, i.e. the factor group ${\displaystyle \mathrm {Q} _{8}/\{e,{\bar {e}\},}$ 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, ${\displaystyle \{e\},\{\bar {e}\},\{i,{\bar {i}\},\{j,{\bar {j}\},\{k,{\bar {k}\},}$ 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:

${\displaystyle \mathbb {R} [\mathrm {Q} _{8}]=\bigoplus _{\rho }(e_{\rho }),}$

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

${\displaystyle {\begin{aligned}e_{\text{triv}&={\tfrac {1}{8}(e+{\bar {e}+i+{\bar {i}+j+{\bar {j}+k+{\bar {k})\\e_{i{\text{-ker}&={\tfrac {1}{8}(e+{\bar {e}+i+{\bar {i}-j-{\bar {j}-k-{\bar {k})\\e_{j{\text{-ker}&={\tfrac {1}{8}(e+{\bar {e}-i-{\bar {i}+j+{\bar {j}-k-{\bar {k})\\e_{k{\text{-ker}&={\tfrac {1}{8}(e+{\bar {e}-i-{\bar {i}-j-{\bar {j}+k+{\bar {k})\\e_{2}&={\tfrac {2}{8}(2e-2{\bar {e})={\tfrac {1}{2}(e-{\bar {e})\end{aligned}$

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:

${\displaystyle {\begin{aligned}{\tfrac {1}{2}(e-{\bar {e})&\longleftrightarrow 1,\\{\tfrac {1}{2}(i-{\bar {i})&\longleftrightarrow i,\\{\tfrac {1}{2}(j-{\bar {j})&\longleftrightarrow j,\\{\tfrac {1}{2}(k-{\bar {k})&\longleftrightarrow k.\end{aligned}$

Furthermore, the projection homomorphism ${\displaystyle \mathbb {R} [\mathrm {Q} _{8}]\to (e_{2})\cong \mathbb {H} }$ given by ${\displaystyle r\mapsto re_{2}$ has kernel ideal generated by the idempotent:

${\displaystyle e_{2}^{\perp }=e_{1}+e_{i{\text{-ker}+e_{j{\text{-ker}+e_{k{\text{-ker}={\tfrac {1}{2}(e+{\bar {e}),}$

so the quaternions can also be obtained as the quotient ring ${\displaystyle \mathbb {R} [\mathrm {Q} _{8}]/(e+{\bar {e})\cong \mathbb {H} }$. 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 ${\displaystyle M_{2}(\mathbb {C} )\cong \mathbb {H} \otimes _{\mathbb {R} }\mathbb {C} \cong \mathbb {H} \oplus \mathbb {H} }$ 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 ${\displaystyle \operatorname {GL} (2,\mathbb {C} )}$. The quaternion group is a multiplicative subgroup of the quaternion algebra:

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

which has a regular representation ${\displaystyle \rho :\mathbb {H} \to \operatorname {M} (2,\mathbb {C} )}$ by left multiplication on itself considered as a 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

${\displaystyle {\begin{cases}\rho :\mathrm {Q} _{8}\to \operatorname {GL} (2,\mathbb {C} )\\g\longmapsto \rho _{g}\end{cases}$

is given by:

${\displaystyle {\begin{matrix}e\mapsto {\begin{pmatrix}1&0\\0&1\end{pmatrix}&i\mapsto {\begin{pmatrix}i&0\\0&\!\!\!\!-i\end{pmatrix}&j\mapsto {\begin{pmatrix}0&\!\!\!\!-1\\1&0\end{pmatrix}&k\mapsto {\begin{pmatrix}0&\!\!\!\!-i\\\!\!\!-i&0\end{pmatrix}\\{\overline {e}\mapsto {\begin{pmatrix}\!\!\!-1&0\\0&\!\!\!\!-1\end{pmatrix}&{\overline {i}\mapsto {\begin{pmatrix}\!\!\!-i&0\\0&i\end{pmatrix}&{\overline {j}\mapsto {\begin{pmatrix}0&1\\\!\!\!-1&0\end{pmatrix}&{\overline {k}\mapsto {\begin{pmatrix}0&i\\i&0\end{pmatrix}.\end{matrix}$

Since all of the above matrices have unit determinant, this is a representation of Q₈ in the special linear group ${\displaystyle \operatorname {SL} (2,\mathbb {C} )}$.^[6]

A variant gives a representation by unitary matrices (table at right). Let ${\displaystyle g\in \mathrm {Q} _{8}$ correspond to the linear mapping ${\displaystyle \rho _{g}:a+bj\mapsto (a+bj)\cdot jg^{-1}j^{-1},}$ so that ${\displaystyle \rho :\mathrm {Q} _{8}\to \operatorname {SU} (2)}$ is given by:

${\displaystyle {\begin{matrix}e\mapsto {\begin{pmatrix}1&0\\0&1\end{pmatrix}&i\mapsto {\begin{pmatrix}i&0\\0&\!\!\!\!-i\end{pmatrix}&j\mapsto {\begin{pmatrix}0&1\\\!\!\!-1&0\end{pmatrix}&k\mapsto {\begin{pmatrix}0&i\\i&0\end{pmatrix}\\{\overline {e}\mapsto {\begin{pmatrix}\!\!\!-1&0\\0&\!\!\!\!-1\end{pmatrix}&{\overline {i}\mapsto {\begin{pmatrix}\!\!\!-i&0\\0&i\end{pmatrix}&{\overline {j}\mapsto {\begin{pmatrix}0&\!\!\!\!-1\\1&0\end{pmatrix}&{\overline {k}\mapsto {\begin{pmatrix}0&\!\!\!\!-i\\\!\!\!-i&0\end{pmatrix}.\end{matrix}$

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

${\displaystyle {\begin{matrix}&e\mapsto {\begin{pmatrix}1&0\\0&1\end{pmatrix}=\quad \,1_{2\times 2}&i\mapsto {\begin{pmatrix}0&\!\!\!-i\!\\\!\!-i\!\!&0\end{pmatrix}=-i\sigma _{x}&j\mapsto {\begin{pmatrix}0&\!\!\!-1\!\\1&0\end{pmatrix}=-i\sigma _{y}&k\mapsto {\begin{pmatrix}\!\!-i\!\!&0\\0&i\end{pmatrix}=-i\sigma _{z}\\&{\overline {e}\mapsto {\begin{pmatrix}\!\!-1\!&0\\0&\!\!\!-1\!\end{pmatrix}=-1_{2\times 2}&{\overline {i}\mapsto {\begin{pmatrix}0&i\\i&0\end{pmatrix}=\,\,\,\,i\sigma _{x}&{\overline {j}\mapsto {\begin{pmatrix}0&1\\\!\!-1\!\!&0\end{pmatrix}=\,\,\,\,i\sigma _{y}&{\overline {k}\mapsto {\begin{pmatrix}i&0\\0&\!\!\!-i\!\end{pmatrix}=\,\,\,\,i\sigma _{z}.\end{matrix}$

This particular choice is convenient and elegant when one describes spin-1/2 states in the ${\displaystyle ({\vec {J}^{2},J_{z})}$ basis and considers angular momentum ladder operators ${\displaystyle J_{\pm }=J_{x}\pm iJ_{y}.}$

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 ${\displaystyle \rho :\mathrm {Q} _{8}\to \operatorname {SL} (2,3)}$ is given by

${\displaystyle {\begin{matrix}e\mapsto {\begin{pmatrix}1&0\\0&1\end{pmatrix}&i\mapsto {\begin{pmatrix}1&1\\1&\!\!\!\!-1\end{pmatrix}&j\mapsto {\begin{pmatrix}\!\!\!-1&1\\1&1\end{pmatrix}&k\mapsto {\begin{pmatrix}0&\!\!\!\!-1\\1&0\end{pmatrix}\\{\overline {e}\mapsto {\begin{pmatrix}\!\!\!-1&0\\0&\!\!\!\!-1\end{pmatrix}&{\overline {i}\mapsto {\begin{pmatrix}\!\!\!-1&\!\!\!\!-1\\\!\!\!-1&1\end{pmatrix}&{\overline {j}\mapsto {\begin{pmatrix}1&\!\!\!\!-1\\\!\!\!-1&\!\!\!\!-1\end{pmatrix}&{\overline {k}\mapsto {\begin{pmatrix}0&1\\\!\!\!-1&0\end{pmatrix}.\end{matrix}$

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:

${\displaystyle {\begin{cases}\mu _{z}:\mathbb {F} _{9}\to \mathbb {F} _{9}\\\mu _{z}(a+bk)=z\cdot (a+bk)\end{cases}$

In addition we have the Frobenius automorphism ${\displaystyle \phi (a+bk)=(a+bk)^{3}$ satisfying ${\displaystyle \phi ^{2}=\mu _{1}$ and ${\displaystyle \phi \mu _{z}=\mu _{\phi (z)}\phi .}$ Then the above representation matrices are:

${\displaystyle {\begin{aligned}\rho ({\bar {e})&=\mu _{-1},\\\rho (i)&=\mu _{k+1}\phi ,\\\rho (j)&=\mu _{k-1}\phi ,\\\rho (k)&=\mu _{k}.\end{aligned}$

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

${\displaystyle {\begin{cases}\psi _{m}:\mathrm {Q} _{8}\to \mathrm {Q} _{8}\\\psi _{m}(g)=mgm^{-1}\end{cases}$

with ${\displaystyle \psi _{I}=\psi _{-I}=\mathrm {id} _{\mathrm {Q} _{8}.}$ In fact, these give the full automorphism group as:

${\displaystyle \operatorname {Aut} (\mathrm {Q} _{8})\cong \operatorname {PGL} (2,3)=\operatorname {GL} (2,3)/\{\pm I\}\cong S_{4}.}$

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 ${\displaystyle \mathbb {P} ^{1}(\mathbb {F} _{3})=\operatorname {PG} (1,3).}$

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.^[7] In 1936 Ernst Witt published his approach to the quaternion group through Galois theory.^[8]

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.^[1]

Generalized quaternion group

A generalized quaternion group Q_4n of order 4n is defined by the presentation^[3]

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

for an integer n ≥ 2, with the usual quaternion group given by n = 2.^[9] Coxeter calls Q_4n the dicyclic group ${\displaystyle \langle 2,2,n\rangle }$, a special case of the binary polyhedral group ${\displaystyle \langle \ell ,m,n\rangle }$ 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 ${\displaystyle \operatorname {GL} _{2}(\mathbb {C} )}$ generated by

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

where ${\displaystyle \omega _{n}=e^{i\pi /n}$.^[3] It can also be realized as the subgroup of unit quaternions generated by^[10] ${\displaystyle x=e^{i\pi /n}$ and ${\displaystyle y=j}$.

The generalized quaternion groups have the property that every abelian subgroup is cyclic.^[11] 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.^[12] 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.^[13] 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,^[14] which admits the presentation

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

See also

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

Notes

References

• Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-13-004763-2
• Brown, Kenneth S. (1982), Cohomology of groups (3rd ed.), Springer-Verlag, ISBN 978-0-387-90688-1
• Cartan, Henri; Eilenberg, Samuel (1999), Homological Algebra, Princeton University Press, ISBN 978-0-691-04991-5
• Coxeter, H. S. M. & Moser, W. O. J. (1980). Generators and Relations for Discrete Groups. New York: Springer-Verlag. ISBN 0-387-09212-9.
• Dean, Richard A. (1981) "A rational polynomial whose group is the quaternions", American Mathematical Monthly 88:42–5.
• Gorenstein, D. (1980), Finite Groups, New York: Chelsea, ISBN 978-0-8284-0301-6, MR 0569209
• Johnson, David L. (1980), Topics in the theory of group presentations, Cambridge University Press, ISBN 978-0-521-23108-4, MR 0695161
• Rotman, Joseph J. (1995), An introduction to the theory of groups (4th ed.), Springer-Verlag, ISBN 978-0-387-94285-8
• P.R. Girard (1984) "The quaternion group and modern physics", European Journal of Physics 5:25–32.
• Hall, Marshall (1999), The theory of groups (2nd ed.), AMS Bookstore, ISBN 0-8218-1967-4
• Kurosh, Alexander G. (1979), Theory of Groups, AMS Bookstore, ISBN 0-8284-0107-1

External links

• Weisstein, Eric W. "Quaternion group". MathWorld.
• Quaternion groups on GroupNames
• Quaternion group on GroupProps
• Conrad, Keith. "Generalized Quaternions"