Quaternion group

Algebraic structure → Group theory Group theory
Basic notions Subgroup Normal subgroup Quotient group (Semi-)direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable action Glossary of group theory List of group theory topics
Finite groups Classification of finite simple groups cyclic alternating Lie type sporadic Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Symmetric group Sn Klein four-group V Dihedral group Dn Quaternion group Q Dicyclic group Dicn
Discrete groups Lattices Integers (${\displaystyle \mathbb {Z} }$) Free group Modular groups PSL(2, ${\displaystyle \mathbb {Z} }$) SL(2, ${\displaystyle \mathbb {Z} }$) Arithmetic group Lattice Hyperbolic group
Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve
v t

Cycle diagram of Q₈. Each color specifies a series of powers of any element connected to the identity element e = 1. For example, the cycle in red reflects the fact that i² = e, i³ = i and i⁴ = e. The red cycle also reflects that i² = e, i³ = i and i⁴ = e.

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.

Another presentation of Q₈ is

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

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:^[1]

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^[2] based in only two elements to skip this redundancy is:

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

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

The quaternion group has the unusual property of being Hamiltonian: Q₈ is non-abelian, but every subgroup is normal.^[3] Every Hamiltonian group contains a copy of Q₈.^[4]

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 Q₈/{e,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, { e }, { e }, { i, i }, { j, j }, { k, 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.

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

Since the irreducible characters ${\displaystyle \chi _{\rho }}$ in the rows above have real values, this gives the decomposition of the real group algebra of ${\displaystyle G=Q_{8}}$ into minimal two-sided ideals: ${\displaystyle \textstyle \mathbb {R} [Q_{8}]\ =\ \bigoplus _{\rho }(e_{\rho })}$, where the idempotents ${\displaystyle e_{\rho }\in \mathbb {R} [Q_{8}]}$ correspond to the irreducibles: ${\displaystyle \textstyle e_{\rho }={\frac {\dim(\rho )}{|G|}}\sum _{g\in G}\chi _{\rho }(g^{-1})g}$, so that

${\displaystyle e_{\text{triv}}={\tfrac {1}{8}}(e+{\bar {e}}+i+{\bar {i}}+j+{\bar {j}}+k+{\bar {k}})}$

${\displaystyle e_{i{\text{-ker}}}={\tfrac {1}{8}}(e+{\bar {e}}+i+{\bar {i}}-j-{\bar {j}}-k-{\bar {k}})}$

${\displaystyle e_{j{\text{-ker}}}={\tfrac {1}{8}}(e+{\bar {e}}-i-{\bar {i}}+j+{\bar {j}}-k-{\bar {k}})}$

${\displaystyle e_{k{\text{-ker}}}={\tfrac {1}{8}}(e+{\bar {e}}-i-{\bar {i}}-j-{\bar {j}}+k+{\bar {k}})}$

${\displaystyle e_{2}={\tfrac {2}{8}}(2e-2{\bar {e}})={\tfrac {1}{2}}(e-{\bar {e}})}$.

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:

${\displaystyle \textstyle {\frac {1}{2}}(e-{\bar {e}})\longleftrightarrow 1,\ \ {\frac {1}{2}}(i-{\bar {i}})\longleftrightarrow i,\ \ {\frac {1}{2}}(j-{\bar {j}})\longleftrightarrow j,\ \ {\frac {1}{2}}(k-{\bar {k}})\longleftrightarrow k.}$

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

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

so the quaternions can also be obtained as the quotient ring ${\displaystyle \mathbb {R} [Q_{8}]/(e+{\bar {e}})\cong \mathbb {H} }$.

The complex group algebra is thus ${\displaystyle \mathbb {C} [Q_{8}]\cong \mathbb {C} ^{\oplus 4}\oplus M_{2}(\mathbb {C} )}$, 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[]

Multiplication table of quaternion group as a subgroup of SL(2,C). The entries are represented by sectors corresponding to their arguments: 1 (green), i (blue), -1 (red), -i (yellow).

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 ${\displaystyle \rho :\mathbb {H} \to \mathrm {M} _{2}(\mathbb {C} )}$ 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 C-linear mapping ${\displaystyle \rho _{z}:a{+}jb\mapsto z\cdot (a{+}jb)}$. The resulting representation ${\displaystyle \rho :\mathrm {Q} _{8}\to \mathrm {GL} (2,\mathbb {C} ),\ g\mapsto \rho _{g},}$ 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 SL(2,C).^[5]

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

Multiplication table of the quaternion group as a subgroup of SL(2,3). The field elements are denoted 0,+,−.

There is also an important action of Q₈ on the 2-dimensional vector space over the finite field F₃ = {0,1,−1} (table at right). A modular representation ${\displaystyle \rho :\mathrm {Q} _{8}\to \mathrm {SL} (2,3)}$ 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 F₉ = F₃[k] = F₃1 + F₃k, where k² = −1 and the multiplicative group (F₉)^× has generators ±(k+1), ±(k-1) of order 8. The two-dimensional F₃-vector space F₉ admits the linear mappings ${\displaystyle \mu _{z}(a+bk)=z\cdot (a+bk)}$ for z in F₉, as well as 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 ${\displaystyle \rho ({\bar {e}})=\mu _{-1}}$, ${\displaystyle \rho (i)=\mu _{k+1}\phi }$, ${\displaystyle \rho (j)=\mu _{k-1}\phi }$, and ${\displaystyle \rho (k)=\mu _{k}}$.

The above 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 ${\displaystyle \psi _{m}:Q_{8}\to Q_{8}}$ defined by ${\displaystyle \psi _{m}(g)=mgm^{-1}}$, with ${\displaystyle \psi _{I}=\psi _{-I}=\mathrm {id} _{Q_{8}}}$. In fact, these give the full automorphism group as:

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

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

Also, this representation permutes the eight non-zero vectors of (F₃)², giving an embedding of Q₈ in the symmetric group S_8, 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

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

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.^[6]

Generalized quaternion group[]

A generalized quaternion group Q_4n of order 4n is defined by the presentation^[2]

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

for an integer n ≥ 2, with the usual quaternion group given by n = 2.^[7] 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 ${\displaystyle \omega _{n}=e^{i\pi /n}}$.^[2] It can also be realized as the subgroup of unit quaternions generated by^[8] ${\displaystyle x=e^{i\pi /n}}$ and ${\displaystyle y=j}$.

The generalized quaternion groups have the property that every abelian subgroup is cyclic.^[9] 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.^[10] 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.^[11] 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,^[12] 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"