This is a branch of mathematics called group theory. A group is any set of objects with an associated operation that combines pairs of objects in the set. In other words a group is defined as a set G together with a binary operation. We will use * to denote the operation although this does not imply that groups only apply to multiplication. An example of a group might be the set of all integers with the operation of addition.
Groups provide a level of abstraction apart from mathematical notations. For example, rotations might be modeled by matrices or by quaternions or by multivectors or by some other notation, however we may wish to study the properties of rotations without getting involved with the mechanics of matrices etc. Groups alone wont allow us to do the calculations but they do allow us to categorize the properties of rotations in this example.
(This page is about the mathematical concept of a group, other pages on this site uses the term group in a different way, as a type of node in a scenegraph which contains other nodes.)
There are many types of groups:
- Groups with a finite number of objects such as MOD(N).
- Groups with an infinite number of objects such as R(2)
Axioms of a group
In order to be a group, a set of objects plus an operation, must obey the following axioms:
- Closure law: The set of objects must be closed with regard to the operation, in other words, the result of an operation must always be an element of the set. If c = a * b then c must be an element of the set.
- Associatively: If we are combining 3 objects we can combine the first two first and then combine the result with the third or we can combine the second two and then combine the result with the first. (But we cant necessarily reverse them, see Abelian groups below). In other words; (a * b) * c = a * (b * c)
- Identity element: There is always an operation which does nothing, in other words, there is an element e in G such that e * a = a = a * e
- Inverse element: There is always a unique inverse operation, in other words, there is only one element a-1 for each a such that a * a-1 = e. There is an element b in G such that a * b = e = b * a
Abelion Groups
In general, for groups, there is no requirement for commutativity, so a * b is not necessarily equal to b * a. We can consider this as an optional property, if a group does have a commutativity property it is known as an Abelion Group.
Rotation Groups
There are 4 main Lie rotation groups (I have put the full story on this page):
- SO(n) - A group of rotations in 'n' real dimensions.
- SU(n) - A group of rotations in 'n' complex dimensions.
- SP(n) - A group of rotations in 'n' quaternion dimensions.
- G2,F4,E6,E7 and E8 - A group of rotations in 'n' octonion dimensions. (only works for 5 values of n)
How can we have rotations in 'n' complex/quaternion/octonion dimensions? I think this is shorthand for an isometry group of the projective plane over the complex/quaternions/octonions.
So, for instance, SU(2) might be thought of as a group of rotations in 2 complex dimensions. Two complex dimensions (represented by quaternions) contains 4 dimensions, but we use this to represent a double cover of rotation in 3D, so we project these 4 dimensions onto 3D space.
