Quaternion group

Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). For example, the cycle in red reflects the fact that i ² = −1, i ³ = −i and i ⁴ = 1. The red cycle also reflects the fact that (−i )² = −1, (−i )³ = i and (−i )⁴ = 1.

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

\mathrm{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.

Compared to dihedral group

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

Cayley graph		Cycle graphs
Q₈ The red arrows represent multiplication on the right by i, and the green arrows represent multiplication on the right by j.	Dih₄ Dihedral group	Q₈	Dih₄

The dihedral group D₄ arises in the split-quaternions in the same way that Q₈ lies in the quaternions.

Cayley table

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

Q×Q	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 the six imaginary units {±i, ±j, ±k} works like the cross product of unit vectors in three-dimensional Euclidean space.

\begin{alignat}{2} ij & = k, & 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 = x y.

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

Q. g. as a subgroup of SL(2,C)

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

\mathrm{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]

Q. g. as a subgroup of SL(2,3)

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

\mathrm{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 + 180 x^4 - 144 x^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 group is called a generalized quaternion group^[7] when its order is a power of 2 and it is dicyclic group.

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

It is a part of more general class of dicyclic groups.

Some authors define ^[4] generalized quaternion group to be the same as dicyclic group.

\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.^[8] 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^[9] x = e^iπ/n and y = j.

The generalized quaternion groups have the property that every abelian subgroup is cyclic.^[10] 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.^[11] 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.^[12] 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-subgroups are generalized quaternion cannot be simple.

Notes

↑ See also a table from Wolfram Alpha
↑ See Hall (1999), p. 190
↑ See Kurosh (1979), p. 67
1 2 3 Johnson 1980, pp. 44–45
↑ Artin 1991
↑ Dean, Richard (1981). "A Rational Polynomial whose Group is the Quaternions". The American Mathematical Monthly 88 (1): 42–45. JSTOR 2320711.
↑ Roman, Steven (2011). Fundamentals of Group Theory: An Advanced Approach. Springer. pp. 347–348. ISBN 9780817683016.
↑ 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 978-0-13-004763-2
Brown, Kenneth S. (1982), Cohomology of groups (3 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. and 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 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 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 (2 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.

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