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Ingroup theory, thequaternion group Q₈ (sometimes just denoted by Q) is anon-abeliangroup oforder eight, isomorphic to the eight-element subset${\displaystyle \{1,i,j,k,-1,-i,-j,-k\}}$ of thequaternions under multiplication. It is given by thegroup presentation

${\displaystyle {\mathrm{Q}}_8=⟨\overline{e},i,j,k\mid {\overline{e}}^2=e,i^2=j^2=k^2=ijk=\overline{e}⟩,}$

wheree is the identity element andecommutes with the other elements of the group. These relations, discovered byW. R. Hamilton, also generate the quaternions as an algebra over the real numbers.

Another presentation of Q₈ is

${\displaystyle {\mathrm{Q}}_8=⟨a,b\mid a^4=e,a^2=b^2,ba=a^{-1}b⟩.}$

Like many other finite groups, itcan be realized as theGalois group of a certain field ofalgebraic numbers.^[1]

The quaternion group Q₈ has the same order as thedihedral groupD₄, but a different structure, as shown by their Cayley and cycle graphs:

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

In the diagrams for D₄, the group elements are marked with their action on a letter F in the defining representationR². The same cannot be done for Q₈, since it has no faithful representation inR² orR³. D₄ can be realized as a subset of thesplit-quaternions in the same way that Q₈ can be viewed as a subset of the quaternions.

TheCayley table (multiplication table) for Q₈ is given by:^[2]

The elementsi,j, andk all haveorder four in Q₈ and any two of them generate the entire group. Anotherpresentation of Q₈^[3] based in only two elements to skip this redundancy is:

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

For instance, writing the group elements inlexicographically 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 beingHamiltonian: Q₈ is non-abelian, but everysubgroup isnormal.^[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 anilpotent non-abelian group.

Thecenter and thecommutator subgroup of Q₈ is the subgroup${\displaystyle \{e,\overline{e}\}}$. Theinner automorphism group of Q₈ is given by the group modulo its center, i.e. thefactor group${\displaystyle {\mathrm{Q}}_8/\{e,\overline{e}\},}$ which isisomorphic to theKlein four-group V. The fullautomorphism group of Q₈ isisomorphic to S₄, thesymmetric group on four letters (seeMatrix representations below), and theouter 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 fiveirreducible 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 subgroupN, we obtain a one-dimensional representation factoring through the 2-elementquotient groupG/N. The representation sends elements ofN to 1, and elements outsideN to −1.

2-dimensional representation: Described below inMatrix representations. It is notrealizable 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${\displaystyle Q_8\subset \mathbb{H}}$.

Thecharacter 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 thedecomposition of the realgroup algebra of${\displaystyle G={\mathrm{Q}}_8}$ into minimal two-sidedideals:

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

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

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

so that

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

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 thequotient 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 ofbiquaternions.

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

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

which has aregular 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${\displaystyle \mathbb{C}}$-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:

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

A variant gives a representation byunitary 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:

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

This particular choice is convenient and elegant when one describesspin-1/2 states in the${\displaystyle ({\overrightarrow{J}}^2,J_z)}$ basis and considersangular 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 thefinite field${\displaystyle {\mathbb{F}}_3=\{0,1,-1\}}$ (table at right). Amodular representation${\displaystyle \rho :{\mathrm{Q}}_8\to \mathrm{SL}(2,3)}$ is given by

This representation can be obtained from theextension 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 theFrobenius 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:

This representation realizes Q₈ as anormal subgroup ofGL(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 theprojective 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 thesymmetric group S₈, in addition to the embeddings given by the regular representations.

Richard Dedekind considered the field${\displaystyle \mathbb{Q}[\sqrt{2},\sqrt{3}]}$ in attempting to relate the quaternion group toGalois theory.^[7] In 1936Ernst Witt published his approach to the quaternion group through Galois theory.^[8]

In 1981, Richard Dean showed the quaternion group can be realized as theGalois group Gal(T/Q) whereQ is the field ofrational numbers and T is thesplitting field of the polynomial

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

The development uses thefundamental theorem of Galois theory in specifying four intermediate fields betweenQ and T and their Galois groups, as well as two theorems on cyclic extension of degree four over a field.^[1]

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

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

for an integern ≥ 2, with the usual quaternion group given byn = 2.^[9]Coxeter calls Q_4n thedicyclic group${\displaystyle ⟨2,2,n⟩}$, a special case of thebinary polyhedral group${\displaystyle ⟨\ell ,m,n⟩}$ and related to thepolyhedral group${\displaystyle (p,q,r)}$ and thedihedral group${\displaystyle (2,2,n)}$. The generalized quaternion group can be realized as the subgroup of${\displaystyle {\mathrm{GL}}_2(\mathbb{C})}$ generated by

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 everyabelian subgroup is cyclic.^[11] It can be shown that a finitep-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 finitep-group in which there is a unique subgroup of orderp is either cyclic or a 2-group isomorphic to generalized quaternion group.^[13] In particular, for a finite fieldF 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). Lettingp^r be the size ofF, wherep is prime, the size of the 2-Sylow subgroup of SL₂(F) is 2ⁿ, wheren = ord₂(p² − 1) + ord₂(r).

TheBrauer–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

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

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

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• 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

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

• Group theory
• Finite groups
• Quaternions

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