Quaternion group

In group theory, the quaternion group is a non- abelian group of order. It is often referred to with the symbol. Your name it receives from the fact that it consists of the eight elements in the skew field of Hamilton's quaternions.

Definition

The quaternion group is the achtelementige amount with the link, in addition to the usual sign rules satisfies the following relations:

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

Properties

The quaternion group is not abelian, since, for example, apply. Together with the dihedral group, this is up to isomorphism the only two non- abelian groups with eight elements.

The Group is also a Hamiltonian group: it is indeed non- abelian, but each subgroup is a normal subgroup. Every Hamiltonian group has a subgroup isomorphic to.

The skew field of Hamilton's quaternions consists of the real vector space with basis and multiplication, which continues the above multiplication table bilinear. Conversely, we can, starting from the quaternion skew field defined as the subgroup formed by the elements.

You can also as a subgroup of the general linear group represented by the matrices and and.

An application of the quaternion group results in synthetic geometry. There Quasi body as the coordinates of areas of an affine or projective plane and it turns out that one of the smallest quasi body which is not a division ring and above the levels therefore nichtdesarguesche revealed a multiplicative group isomorphic to have. → see Ternärkörper.

Automorphisms

As automorphism ( here of ) applies a bijective mapping, in which the multiplication is treated homomorphic, i.e.

Since the order of group elements here remains, must remain fixed as the only elements with order 1 and 2 respectively. In contrast, the imaginary three units can be transferred in each case to another. More precisely: the first, say, has all six corners of the octahedron to choose from, the negative of this value must be the " antipodes " are allocated. Stay for the second, say, another 4 corners. After that, the remaining assignments are defined: antipode as well as because of ( this orientation prohibits the reflections see below ) and its antipode. So there are 6.4 = 24 automorphisms, which are in one to one correspondence to the rotations of said octahedron. Thus, the automorphism group is isomorphic to the rotation group of the octahedron, which in turn is isomorphic to the symmetric group S4.

An elegant implementation of the context of quaternions can be found in Hurwitzquaternionen.

The inner automorphisms of are mediated by the modulo the center virtue. They form the group isomorphic to, which is isomorphic to the group of four small between V.

The conjugation as a reflection in the real axis, which also represents the inversion illustration here is antihomomorph, ie

And is therefore called involutional Antiautomorphismus.

Generalized quaternion

The quaternion group can be presented as follows by generators and relations:

In the above notation applies for and.

The generalized quaternion group of order for, we obtain the following presentation on generators and relations:

The generalized Quaternionengruppen belong to the even larger family of dicyclic groups.