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
\{1,i,j,k,-1,-i,-j,-k\}
Q8=\langle\bar{e},i,j,k\mid\bar{e}2=e, i2=j2=k2=ijk=\bar{e}\rangle,
where e is the identity element and 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
Q8=\langlea,b\mida4=e,a2=b2,ba=a-1b\rangle.
Like many other finite groups, it can be realized as the Galois group of a certain field of algebraic numbers.
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.
The Cayley table (multiplication table) for Q8 is given by:[1]
× | e | i | j | k | |||||
---|---|---|---|---|---|---|---|---|---|
e | e | i | j | k | |||||
e | i | j | k | ||||||
i | i | e | k | j | |||||
i | e | k | j | ||||||
j | j | k | e | i | |||||
j | k | e | i | ||||||
k | k | j | i | e | |||||
k | j | i | e |
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\langlex,y\midx4=1,x2=y2,y-1xy=x-1\right\rangle.
For instance, writing the group elements in lexicographically minimal normal forms, one may identify:
The quaternion group has the unusual property of being Hamiltonian: Q8 is non-abelian, but every subgroup is normal.[2] Every Hamiltonian group contains a copy of Q8.[3]\{e,\bare,i,\bar{i},j,\bar{j},k,\bar{k}\} \leftrightarrow\{e,x2,x,x3,y,x2y,xy,x3y\}.
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
\{e,\bar{e}\}
Q8/\{e,\bar{e}\},
The quaternion group Q8 has five conjugacy classes,
\{e\},\{\bar{e}\},\{i,\bar{i}\},\{j,\bar{j}\},\{k,\bar{k}\},
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
H
C
Q8\subsetH
The character table of Q8 turns out to be the same as that of D4:
Representation(ρ)/Conjugacy class | ||||||
---|---|---|---|---|---|---|
Trivial representation | 1 | 1 | 1 | 1 | 1 | |
Sign representation with i-kernel | 1 | 1 | 1 | −1 | −1 | |
Sign representation with j-kernel | 1 | 1 | −1 | 1 | −1 | |
Sign representation with k-kernel | 1 | 1 | −1 | −1 | 1 | |
2-dimensional representation | 2 | −2 | 0 | 0 | 0 |
Nevertheless, all the irreducible characters
\chi\rho
G=Q8
\R[Q8]=oplus\rho(e\rho),
e\rho\in\R[Q8]
e\rho=
\dim(\rho) | |
|G| |
\sumg\in
-1 | |
\chi | |
\rho(g |
)g,
so that
\begin{align} etriv&=\tfrac18(e+\bare+i+\bari+j+\barj+k+\bark)\\ ei-ker&=\tfrac18(e+\bare+i+\bari-j-\barj-k-\bark)\\ ej-ker&=\tfrac18(e+\bare-i-\bari+j+\barj-k-\bark)\\ ek-ker&=\tfrac18(e+\bare-i-\bari-j-\barj+k+\bark)\\ e2&=\tfrac28(2e-2\bare)=\tfrac12(e-\bare) \end{align}
Each of these irreducible ideals is isomorphic to a real central simple algebra, the first four to the real field
\R
(e2)
H
\begin{align} \tfrac12(e-\bare)&\longleftrightarrow1,\\ \tfrac12(i-\bari)&\longleftrightarrowi,\\ \tfrac12(j-\barj)&\longleftrightarrowj,\ \tfrac12(k-\bark)&\longleftrightarrowk. \end{align}
Furthermore, the projection homomorphism
\R[Q8]\to(e2)\congH
r\mapstore2
\perp | |
e | |
2 |
=e1+ei-ker+ej-ker+ek-ker=\tfrac{1}{2}(e+\bare),
\R[Q8]/(e+\bare)\congH
Q8
\C[Q8]\cong\C ⊕ ⊕ M2(\C),
M2(\C)\congH ⊗ \R\C\congH ⊕ H
\operatorname{GL}(2,\C)
\H=\R1+\Ri+\Rj+\Rk=\C1+\Cj,
\rho:\H\to\operatorname{M}(2,\C)
\{1,j\},
z\in\H
\C
\rhoz:a+jb\mapstoz ⋅ (a+jb).
\begin{cases}\rho:Q8\to\operatorname{GL}(2,\C)\ g\longmapsto\rhog\end{cases}
is given by:
\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}
\operatorname{SL}(2,\C)
A variant gives a representation by unitary matrices (table at right). Let
g\inQ8
\rhog:a+bj\mapsto(a+bj) ⋅ jg-1j-1,
\rho:Q8\to\operatorname{SU}(2)
\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
\operatorname{SU}(2)
\begin{matrix} &e\mapsto\begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}= 12 x 2&i\mapsto\begin{pmatrix} 0&-i\\ -i&0 \end{pmatrix}=-i\sigmax &j\mapsto\begin{pmatrix} 0&-1\\ 1&0 \end{pmatrix}=-i\sigmay &k\mapsto\begin{pmatrix} -i&0\\ 0&i \end{pmatrix}=-i\sigmaz\\ &\overline{e}\mapsto\begin{pmatrix} -1&0\\ 0&-1 \end{pmatrix}=-12 x 2&\overline{i}\mapsto\begin{pmatrix} 0&i\\ i&0 \end{pmatrix}=i\sigmax &\overline{j}\mapsto\begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}=i\sigmay &\overline{k}\mapsto\begin{pmatrix} i&0\\ 0&-i \end{pmatrix}=i\sigmaz. \end{matrix}
This particular choice is convenient and elegant when one describes spin-1/2 states in the
(\vec{J}2,Jz)
J\pm=Jx\pmiJy.
F3=\{0,1,-1\}
\rho:Q8\to\operatorname{SL}(2,3)
\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
x | |
F | |
9 |
\pm(k\pm1),
z\inF9,
F3
F9
\begin{cases}\muz:F9\toF9\ \muz(a+bk)=z ⋅ (a+bk)\end{cases}
\phi(a+bk)=(a+bk)3
\phi2=\mu1
\phi\muz=\mu\phi(z)\phi.
\begin{align} \rho(\bare)&=\mu-1,\\ \rho(i)&=\muk+1\phi,\\ \rho(j)&=\muk-1\phi,\\ \rho(k)&=\muk. \end{align}
This representation realizes Q8 as a normal subgroup of . Thus, for each matrix
m\in\operatorname{GL}(2,3)
\begin{cases}\psim:Q8\toQ8
-1 | |
\ \psi | |
m(g)=mgm |
\end{cases}
with
\psiI=\psi-I
=id | |
Q8 |
.
\operatorname{Aut}(Q8)\cong\operatorname{PGL}(2,3)=\operatorname{GL}(2,3)/\{\pmI\}\congS4.
This is isomorphic to the symmetric group S4 since the linear mappings
2 | |
m:F | |
3 |
\to
2 | |
F | |
3 |
2, | |
F | |
3 |
P1(F3)=\operatorname{PG}(1,3).
Also, this representation permutes the eight non-zero vectors of
2, | |
F | |
3 |
Richard Dedekind considered the field
Q[\sqrt{2},\sqrt{3}]
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
x8-72x6+180x4-144x2+36
A generalized quaternion group Q4n of order 4n is defined by the presentation
\langlex,y\midx2n=y4=1,xn=y2,y-1xy=x-1\rangle
\langle2,2,n\rangle
\langle\ell,m,n\rangle
(p,q,r)
(2,2,n)
\operatorname{GL}2(\Complex)
\left(\begin{array}{cc} \omegan&0\\ 0&\overline{\omega}n \end{array} \right) and \left(\begin{array}{cc} 0&-1\\ 1&0 \end{array} \right)
where
\omegan=ei\pi/n
x=ei\pi/n
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, . Letting pr be the size of F, where p is prime, the size of the 2-Sylow subgroup of SL2(F) is 2n, 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,[12] which admits the presentation
\langlex,y\mid
2m | |
x |
=y4=1,
2m-1 | |
x |
=y2,y-1xy=x-1\rangle.