Quaternion group

In group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication. It is often denoted by Q or Q₈, and is given by the group presentation

$Q = \langle -1,i,j,k \mid (-1)^2 = 1, \;i^2 = j^2 = k^2 = ijk = -1 \rangle, \,\!$

where 1 is the identity element and −1 commutes with the other elements of the group.

1 Cayley graph
2 Cayley table
3 Properties
4 Matrix representations
5 Galois group
6 Generalized quaternion group
7 See also
8 Notes
9 References
10 External links

Cayley graph

The Q₈ group has the same order as the Dihedral group, D₄, but a different structure, as shown by their Cayley graphs:

Cayley graph
Q₈ The red arrows represent multiplication on the right by i, and the green arrows represent multiplication on the right by j.	D₄ Dihedral group

Cayley table

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

	1	−1	i	−i	j	−j	k	−k
1	1	−1	i	−i	j	−j	k	−k
−1	−1	1	−i	i	−j	j	−k	k
i	i	−i	−1	1	k	−k	−j	j
−i	−i	i	1	−1	−k	k	j	−j
j	j	−j	−k	k	−1	1	i	−i
−j	−j	j	k	−k	1	−1	−i	i
k	k	−k	j	−j	−i	i	−1	1
−k	−k	k	−j	j	i	−i	1	−1

The multiplication of pairs of elements from the subset {±i, ±j, ±k} works like the cross product of unit vectors in three-dimensional Euclidean space.

$\begin{alignat}{2} ij & = k, & \qquad ji & = -k, \\ jk & = i, & kj & = -i, \\ ki & = j, & ik & = -j. \end{alignat}$

Properties

The quaternion group has the unusual property of being Hamiltonian: every subgroup of Q is a normal subgroup, but the group is non-abelian.^[2] Every Hamiltonian group contains a copy of Q.^[3]

In abstract algebra, one can construct a real four-dimensional vector space with basis {1, i, j, k} and turn it into an associative algebra by using the above multiplication table and distributivity. The result is a skew field called the quaternions. Note that this is not quite the same as the group algebra on Q (which would be eight-dimensional). Conversely, one can start with the quaternions and define the quaternion group as the multiplicative subgroup consisting of the eight elements {1, −1, i, −i, j, −j, k, −k}. The complex four-dimensional vector space on the same basis is called the algebra of biquaternions.

Note that i, j, and k all have order four in Q and any two of them generate the entire group. Another presentation of Q^[4] demonstrating this is:

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

One may take, for instance, i = x, j = y and k = xy.

The center and the commutator subgroup of Q is the subgroup {±1}. The factor group Q/{±1} is isomorphic to the Klein four-group V. The inner automorphism group of Q is isomorphic to Q modulo its center, and is therefore also isomorphic to the Klein four-group. The full automorphism group of Q is isomorphic to S₄, the symmetric group on four letters. The outer automorphism group of Q is then S₄/V which is isomorphic to S₃.

Matrix representations

The quaternion group can be represented as a subgroup of the general linear group GL₂(C). A representation

$Q = \{\pm 1, \pm i, \pm j, \pm k\} \to \mathrm{GL}_{2}(\mathbf{C})$

is given by

$1 \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

$i \mapsto \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}$

$j \mapsto \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$

$k \mapsto \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$

Since all of the above matrices have unit determinant, this is a representation of Q in the special linear group SL₂(C). The standard identities for quaternion multiplication can be verified using the usual laws of matrix multiplication in GL₂(C).^[5]

There is also an important action of Q on the eight nonzero elements of the 2-dimensional vector space over the finite field F₃. A representation

$Q = \{\pm 1, \pm i, \pm j, \pm k\} \to \mathrm{GL}(2,3)$

is given by

$1 \mapsto \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

$i \mapsto \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$

$j \mapsto \begin{pmatrix} -1 & 1 \\ 1 & 1 \end{pmatrix}$

$k \mapsto \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$

where {−1,0,1} are the three elements of F₃. Since all of the above matrices have unit determinant over F₃, this is a representation of Q in the special linear group SL(2, 3). Indeed, the group SL(2, 3) has order 24, and Q is a normal subgroup of SL(2, 3) of index 3.

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

$x^8 - 72 x^6 %2B 180 x^4 - 144 x^2 %2B 36$ .

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.

Generalized quaternion group

A group is called a generalized quaternion group or dicyclic group if it has a presentation^[4]

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

for some integer n ≥ 2. This group is denoted Q_4n and has order 4n.^[6] Coxeter labels these dicyclic groups <2,2,n>, being a special case of the binary polyhedral group <l,m,n> and related to the polyhedral groups (p,q,r), and dihedral group (2,2,n). The usual quaternion group corresponds to the case n = 2. The generalized quaternion group can be realized as the subgroup of GL₂(C) generated by

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

where ω_n = e^iπ/n.^[4] It can also be realized as the subgroup of unit quaternions generated by^[7] x = e^iπ/n and y = j.

The generalized quaternion groups have the property that every abelian subgroup is cyclic.^[8] 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.^[9] Another characterization is that a finite p-group in which there is a unique subgroup of order p is either cyclic or generalized quaternion (of order a power of 2).^[10] 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 groups whose Sylow 2-subgroup is generalized quaternion cannot be simple.

Notes

^ See also a table from Wolfram Alpha
^ See Hall (1999), p. 190
^ See Kurosh (1979), p. 67
^ ^a ^b ^c Johnson 1980, pp. 44–45
^ Artin 1991
^ Some authors (e.g., Rotman 1995, pp. 87, 351) refer to this group as the dicyclic group, reserving the name generalized quaternion group to the case where n is a power of 2.
^ Brown 1982, p. 98
^ Brown 1982, p. 101, exercise 1
^ Cartan & Eilenberg 1999, Theorem 11.6, p. 262
^ Brown 1982, Theorem 4.3, p. 99

References

Artin, Michael (1991), Algebra, Prentice Hall, ISBN 978-0-130-04763-2
Brown, Kenneth S. (1982), Cohomology of groups (3 ed.), Springer-Verlag, ISBN 9780387906881
Cartan, Henri; Eilenberg, Samuel (1999), Homological Algebra, Princeton University Press, ISBN 9780691049915
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 81b:20002
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 (4 ed.), Springer-Verlag, ISBN 9780387942858
P.R. Girard (1984) "The quaternion group and modern physics", European Journal of Physics 5:25–32.
Hall, Marshall (1999), The theory of groups (2 ed.), AMS Bookstore, ISBN 0821819674
Kurosh, Alexander G. (1979), Theory of Groups, AMS Bookstore, ISBN 0828401071

External links

Weisstein, Eric W., "Quaternion group" from MathWorld.