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 .
definition
The quaternion group is the eight-element set with the link , which, in addition to the usual sign rules , fulfills the following relations:
- .
These rules were found by William Rowan Hamilton . This results in the following link table :
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.
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.
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 .
One can also represent a subgroup of the general linear group by the matrices and and .
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 .
Automorphisms
An automorphism (here of ) is a bijective mapping in which the multiplication is treated homomorphically , i.e. H.
- .
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 .
An elegant realization of in the context of the quaternions can be found in Hurwitzquaternions .
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.
The conjugation as a reflection on the real axis, which also represents the inversion image here, is antihomomorphic, that is
- and also ,
and is therefore called involutive anti-automorphism .
Character board
The quaternion group has the following character table :
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:
- .
In the above notation applies and .
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:
- .
The dicyclic groups, whose order is a power of two, are called generalized quaternion groups.
See also
Web links
Individual evidence
- ↑ William Rowan Hamilton : Incision in a stone of the Broom (also: Brougham) Bridge . Dublin 1843.
- ↑ 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.
- ↑ Eric W. Weisstein : Antihomomorphism . In: MathWorld (English).
- ^ JL Alperin, RB Bell: Groups and Representations . Springer-Verlag, (1995), ISBN 0-387-94525-3 , chap. 6, example 8
- ↑ Steven Roman: Fundamentals of group theory . Birkhäuser, Basel 2012, Chapter 12, pp. 347/348.
- ↑ Thomas Keilen: Finite Groups . (PDF) Example 9.11, p. 37.
- ↑ Bertram Huppert: Finite Groups I . Springer, Berlin 1967, Chapter I, § 14, Sentence 14.9, p. 91.