Quaternion 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.

The Cayley table (multiplication table) for Q₈ is given by:[1]

Note that 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.

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 SL₂(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:

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

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.

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]

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

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

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

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