Quaternion group

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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.

Cayley graph of Q. The red arrows represent multiplication on the right by i, and the green arrows represent multiplication on the right by j.

In group theory, the quaternion group is a non-abelian group of order 8, isomorphic to a certain eight-element subset of the quaternions under multiplication. It is often denoted by Q or Q₈, and has the following eight elements:

$Q = \{\pm 1,\, \pm i, \,\pm j, \,\pm k\}.\,\!$

Here 1 is the identity element. The element −1 squares to 1, and the other elements square to −1:

$(-1)^2 = 1\quad\text{and}\quad i^2 = j^2 = k^2 = -1\!\,$

The element −1 commutes with the other elements of the group:

$(-1)i = i(-1) = -i.\!\,$

In addition,

$ij = k,\quad jk = i,\quad ki = j,\,\!$

and

$ji = -k,\quad kj=-i,\quad ik = -j.\,\!$

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

This information can be summarized by the following group presentation:

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

The entire Cayley table (multiplication table) for Q is given by:

	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

[edit] 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. Every Hamiltonian group contains a copy of Q.

In abstract algebra, one can construct a real 4-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 group algebra on Q (which would be 8-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}.

Note that i, j, and k all have order 4 in Q and any two of them generate the entire group. Q has the presentation

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

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.

The quaternion group Q may be regarded as acting on the eight nonzero elements of the 2-dimensional vector space over the finite field GF(3).

[edit] Matrix representation of the quaternion group

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

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

is given by

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

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

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

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

Note: the $i$ 's inside the matrices represent the imaginary number $i$ . In fact, 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).^[1]

[edit] Generalized quaternion group

A group is called a generalized quaternion group if it has a presentation

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

for some integer n ≥ 3. The order of this group is 2ⁿ. The ordinary quaternion group corresponds to the case n = 3. The generalized quaternion group can be realized as the subgroup of unit quaternions generated by

$\begin{align} x & = e^{2\pi i/2^{n-1}} \\ y & = j. \end{align}$

The generalized quaternion groups are members of the still larger family of dicyclic groups. The generalized quaternion groups have the property that every abelian subgroup is cyclic. 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. Another characterization is that a finite p-group in which there is a unique subgroup of order p is either cyclic or generalized quaternion. 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 q = 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).

[edit] See also

[edit] Notes

^ Michael Artin (1991). Algebra. Prentice Hall. ISBN 9780130047632.

[0] Michael Artin (1991). Algebra. Prentice Hall. ISBN 9780130047632.

[1]

Quaternion group

From Wikipedia, the free encyclopedia

Contents

[edit] Properties

[edit] Matrix representation of the quaternion group

[edit] Generalized quaternion group

[edit] See also

[edit] Notes

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