Maths - Group Theory - Martin Baker

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 * (or sometimes 'o') 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.

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)
Infinite Groups that are continous (Lie groups)

We might think of groups as a set (finite or infinite) plus extra structure which is an operation that takes two elements of the set to give another element of the set. This 'composition law' turns a set into a group

Group theory is related to symmetry and allows us to formalise symmetry.

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

Properties of Multiplicative Groups

If multiplication is the group operation, then applying it multiple times is represented by a superscript operator. For instance: g³ is the result of applying g in succession 3 times.

This is like the exponent for groups but there are many differences. For instance, the exponent can only be a whole integer, not a fractional value.

g⁰ = e = the identity element

g^-1 = the inverse of g

g^-n = (g^-1)ⁿ

It is true that:

g^m gⁿ = g^m+n

(g^m)ⁿ = g^mn

but in general:

(gh)ⁿ ≠ gⁿhⁿ

where:

m and n are integers which may be +ve, 0 or -ve.

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. Here the group operation is usually represented as '+' .

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.

Coset and Normal Subgroup

If G is a group, H is a subgroup of G, and g is an element of G, then

gH = {gh : h an element of H} is a left coset of H in G, and
Hg = {hg : h an element of H} is a right coset of H in G.

If H is normal then:

Hg := gH.

multiplying left and right by g^-1gives:

H := gHg^-1

A coset is a subgroup if either:

A subgroup where the left and right cosets are the same, or,
A subgroup where H is preserved under conjugation.

Coset	Normal Subgroup

Conjugate

The conjugate is: aba^-1

Not to be confused with conjugate of a complex number or quaternion.

If multiplication commutes then:

aba^-1= baa^-1= b

So the conjugate is a measure of how much multiplication commutes.

cojugate set

Notation

notation	example	meaning
A	A = {a \| aA}	set
a	aA	element of A
{a, b, c}		set containing a, b and c
	a H = {ah \| hH}	left coset of a
(a,b)		cycle notation
<a,b>		all the elements of group generated by a and b
	<r,f \| r³=1, f²=1, frf=r^-1>	generators with constrains

\| A \|		grade (size) = number of elements in A
\| G : H \|		number of left cosets of H in G
×		Direct Product
		Semidirect Product (left)
	{n, h}_φ {n',h'}	Semidirect Product (right)
		Bicrossed Product also known as Knit or Zappa-Szep product
	h = h'H such that a h' = h a	Normal subgroup a H = H a for all a in G
Ø		empty set
¬		not
∧		and (= intersection)
∨		or (U = union)
		is in (is an element of the set)
→		if
<->		if and only if
		for all
		there exists
		is isomorphic to
orb(s)		Orbit - set of elements that can be reached
stab(s)		Stabilizer - set of elements that dont move s