Quaternion group

In group theory , the quaternion group is a non-Abelian group of order . It is often referred to with the symbol . It gets its name from the fact that it consists of the eight elements in the oblique body of the Hamiltonian quaternions . ${\displaystyle 8}$ ${\displaystyle Q_{8}}$ ${\displaystyle \pm 1,\pm \mathrm {i} ,\pm \mathrm {j} ,\pm \mathrm {k} }$

definition

The quaternion group is the eight-element set with the link , which, in addition to the usual sign rules , fulfills the following relations: ${\displaystyle Q_{8}=\{\pm 1,\pm \mathrm {i} ,\pm \mathrm {j} ,\pm \mathrm {k} \}}$ ${\displaystyle \cdot \colon Q_{8}\times Q_{8}\to Q_{8}}$

${\displaystyle \mathrm {i} \cdot \mathrm {i} =\mathrm {j} \cdot \mathrm {j} =\mathrm {k} \cdot \mathrm {k} =\mathrm {i} \cdot \mathrm {j} \cdot \mathrm {k} =-1}$.

These rules were found by William Rowan Hamilton . This results in the following link table :

${\displaystyle \cdot }$	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

properties

The quaternion group is not Abelian , for example . Except for isomorphism, they and the dihedral group are the only two non-Abelian groups with eight elements. ${\displaystyle Q_{8}}$ ${\displaystyle \mathrm {i} \cdot \mathrm {j} =\mathrm {k} \neq \mathrm {j} \cdot \mathrm {i} =-\mathrm {k} }$ ${\displaystyle D_{4}}$

The group is also a Hamiltonian group : it is non-Abelian, but every subgroup is a normal divisor . Every Hamiltonian group has a subgroup that is too isomorphic. ${\displaystyle Q_{8}}$ ${\displaystyle Q_{8}}$

The oblique field of the Hamiltonian quaternions consists of the real vector space with a base and the multiplication, which continues the above multiplication table bilinearly. Conversely, starting from the oblique body, the quaternion group can be defined as the subgroup formed by the elements . ${\displaystyle \mathbb {H} }$ ${\displaystyle \{1,\mathrm {i} ,\mathrm {j} ,\mathrm {k} \}}$ ${\displaystyle \mathbb {H} }$ ${\displaystyle \pm 1,\pm \mathrm {i} ,\pm \mathrm {j} ,\pm \mathrm {k} }$

One can also represent a subgroup of the general linear group by the matrices and and . ${\displaystyle Q_{8}}$ ${\displaystyle \operatorname {GL} _{2}(\mathbb {C} )}$ ${\displaystyle \mathrm {i} =\left({\begin{smallmatrix}{\sqrt {-1}}&0\\0&-{\sqrt {-1}}\end{smallmatrix}}\right)}$ ${\displaystyle \mathrm {j} =\left({\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}}\right)}$ ${\displaystyle \mathrm {k} =\left({\begin{smallmatrix}0&{\sqrt {-1}}\\{\sqrt {-1}}&0\end{smallmatrix}}\right)}$

The quaternion group can be used in synthetic geometry . There quasi-bodies serve as coordinate areas of an affine or projective plane and it turns out that one of the smallest quasi-bodies, which is not a sloping body and above which there are non-Desargue planes, has an isomorphic multiplicative group. → see ternary body . ${\displaystyle Q_{8}}$

Automorphisms

An automorphism (here of ) is a bijective mapping in which the multiplication is treated homomorphically , i.e. H. ${\displaystyle Q_{8}}$ ${\displaystyle \phi \colon Q_{8}\to Q_{8}}$

${\displaystyle \phi (x\cdot y)=\phi (x)\cdot \phi (y)\,}$.

Since the order of group elements is retained, the only elements with order 1 or 2 must remain fixed. On the other hand, the 3 imaginary units can each be converted into another. More precisely: the first, let's say , has all 6 corners of this octahedron to choose from, the negative of this value must then be assigned to the “antipode” . For the second, say , 4 corners remain. Then the remaining assignments are determined: Antipode as well as because (this orientation prohibits reflections see below) and its antipode . So there are 6 · 4 = 24 automorphisms which are in one-to-one correspondence with the rotations of the said octahedron. Thus the automorphism group is isomorphic to the rotation group of the octahedron , which in turn is isomorphic to the symmetric group S ₄ . ${\displaystyle \pm 1}$ ${\displaystyle \mathrm {i} ,\mathrm {j} ,\mathrm {k} }$ ${\displaystyle \mathrm {i} }$ ${\displaystyle \pm \mathrm {i} ,\pm \mathrm {j} ,\pm \mathrm {k} }$ ${\displaystyle -\mathrm {i} \,}$ ${\displaystyle \mathrm {j} }$ ${\displaystyle -\mathrm {j} }$ ${\displaystyle \mathrm {k} }$ ${\displaystyle \mathrm {k} =\mathrm {i} \cdot \mathrm {j} }$ ${\displaystyle -\mathrm {k} }$ ${\displaystyle \operatorname {Aut} (Q_{8})}$

An elegant realization of in the context of the quaternions can be found in Hurwitzquaternions . ${\displaystyle \operatorname {Aut} (Q_{8})}$

The inner automorphisms of are conveyed to the center by modulo . They form the group isomorphic to that the small group of four rule V is isomorphic. ${\displaystyle Q_{8}}$ ${\displaystyle q\in Q_{8}}$ ${\displaystyle Z=\left\{\pm 1\right\}}$ ${\displaystyle x\mapsto q^{-1}\cdot x\cdot q}$ ${\displaystyle \operatorname {Inn} (Q_{8})}$ ${\displaystyle Q_{8}/Z}$

The conjugation as a reflection on the real axis, which also represents the inversion image here, is antihomomorphic, that is

${\displaystyle {\overline {x\cdot y}}={\bar {y}}\cdot {\bar {x}}}$     and also     , ${\displaystyle {\bar {\mathrm {k} }}={\bar {\mathrm {j} }}\cdot {\bar {\mathrm {i} }}\;\neq \;{\bar {\mathrm {i} }}\cdot {\bar {\mathrm {j} }}}$

and is therefore called involutive anti-automorphism .

Character board

The quaternion group has the following character table :

${\displaystyle G}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 2}$	${\displaystyle 2}$	${\displaystyle 2}$
	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle {\rm {i}}}$	${\displaystyle {\rm {j}}}$	${\displaystyle {\rm {k}}}$
${\displaystyle \chi _{1}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$
${\displaystyle \chi _{2}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle -1}$
${\displaystyle \chi _{3}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle 1}$	${\displaystyle -1}$
${\displaystyle \chi _{4}}$	${\displaystyle 1}$	${\displaystyle 1}$	${\displaystyle -1}$	${\displaystyle -1}$	${\displaystyle 1}$
${\displaystyle \chi _{5}}$	${\displaystyle 2}$	${\displaystyle -2}$	${\displaystyle 0}$	${\displaystyle 0}$	${\displaystyle 0}$

The dihedral group D ₄ has the same character table without being isomorphic to the quaternion group. The quaternion group is an example of how a group cannot be reconstructed from its character table.

Dicyclic groups and generalized quaternion groups

The quaternion group can be presented as follows using generators and relations: ${\displaystyle Q_{8}}$

${\displaystyle \left\langle x,y\mid x^{4}=1,x^{2}=y^{2},yxy^{-1}=x^{-1}\right\rangle }$.

In the above notation applies and . ${\displaystyle x=\mathrm {i} }$ ${\displaystyle y=\mathrm {j} }$

The quaternion group is therefore a so-called dicyclic group . The dicyclic group of the order for is obtained from the following presentation about generators and relations: ${\displaystyle 4n}$ ${\displaystyle n\geq 2}$

${\displaystyle \left\langle x,y\mid x^{2n}=1,x^{n}=y^{2},yxy^{-1}=x^{-1}\right\rangle }$.

The dicyclic groups, whose order is a power of two, are called generalized quaternion groups.

See also

• Lipschitz quaternions
• Hurwitz quaternions

Web links

• Platonic polychora

Individual evidence

1. ↑ William Rowan Hamilton : Incision in a stone of the Broom (also: Brougham) Bridge . Dublin 1843.
2. ↑ Hans-Dieter Ebbinghaus et al .: Numbers . In: Basic Knowledge of Mathematics , Volume 1. Springer-Verlag, Berlin / Heidelberg / New York / Tokyo 1983, ISBN 3-540-12666-X , pp. 138–154.
3. ↑ Eric W. Weisstein : Antihomomorphism . In: MathWorld (English).
4. ^ JL Alperin, RB Bell: Groups and Representations . Springer-Verlag, (1995), ISBN 0-387-94525-3 , chap. 6, example 8
5. ↑ Steven Roman: Fundamentals of group theory . Birkhäuser, Basel 2012, Chapter 12, pp. 347/348.
6. ↑ Thomas Keilen: Finite Groups . (PDF) Example 9.11, p. 37.
7. ↑ Bertram Huppert: Finite Groups I . Springer, Berlin 1967, Chapter I, § 14, Sentence 14.9, p. 91.