Maths - Group Theory

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:

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.

Symmetry

It is often possible to understand mathematical and physics structures by understanding their symmetries.

These symmetries are groups or, to put it the other way round, groups represent symmetries. This allows us to formalise symmetry see this page.

Group Family Type of Symmetry Example
Cyclic 'n' way rotation in finite steps in a plane. Square can be rotated to four positions
Alternating finite step rotational symmetry of the n-simplex in 'n' dimensions. Tetrahedron in 3 dimensions
Lie continuous symmetry
  over
reals
over
complex
over
quat
over
oct
An
special linear group
SL(n+1)      
Bn
special orthogonal group
SO(2n+1) SU    
Cn
symplectic group
Sp(2n)      
Dn
special orthogonal group
SO(2n)      
E6,E7,E8,F4,G2 exceptional cases      
Properties of physics are the same regardless of the orientation in 3D space of the coordinate system.
Sporadic ? ?

Axioms of a group

In order to be a group, a set of objects plus an operation, must obey the following axioms:

Note: that there is no requirement for commutivity, that is a*b need not in general be equal to b*a.

Classification of Finite Simple Groups

We will see (on these pages) that there is a sort of meta-algebra of groups, what I mean by that is that is that we can 'multiply' or 'divide' whole groups (rather than just elements of those groups) to form other whole groups. This analogy is not exact in that the rules for this type of 'meta multiplication and division' is different from the type of operations that we are used to when applied to elements of groups and rings. However it does mean that, in the same way that we can 'factorise' integers and derive the prime numbers we can 'factorise' groups into subgroups. If we can't reduce a group into subgroups (apart from the trivial case of itself and the identity element) then it is known as a 'simple group'. In the same way that the structure of prime numbers is very complicated then the structure of simple groups is also very complicated.

A subgroup of this type is known as a quotient group or factor group (a special case is the normal subgroup)

Any simple group belongs to one of the following families of groups:

Where:

History of Classification of Finite Simple Groups

As we have seen the structure of finite simple groups is very messy, the final sporadic group was not found until 1975. Then it was not until 1981 that the proof of the classification of finite simple groups was completed, that is the proof that the categories are complete and there are no more to be discovered. This is so complicated that even after 1981 gaps in the proof were still being found and filled in. Even now we can't be sure that there in not a hidden error.

This classification proof is perhaps the largest collaborative piece of pure mathematics ever attempted. It consists of over 10,000 pages, across about 500 journal articles, by over 100 different authors from around the world, it was without precedent.

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:

gm gn = gm+n

(gm)n = gmn

but in general:

(g h)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):

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

Coset

The coset is not a group because it does not necessarily have an identity element or inverse but it is a set with an associative binary operation (this is known as a semigroup).

We create a coset from:

Then we can create a left coset: gH
and a right coset : Hg

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 we were to choose the element 'g' to be an element of H then the coset would be H (since H is a group it is closed).

Normal Subgroup

If H is normal then:

Hg := gH.

multiplying left and right by g-1gives:

H := gHg-1

A coset is a subgroup if either:

Coset Normal Subgroup
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 | a∈A} set
a a∈A element of A
{a, b, c}   set containing a, b and c
  a H = {ah | h∈H} 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
&22C9;   Semidirect Product (left)
&22CA; {n, h}&22CA;φ {n',h'} Semidirect Product (right)
&22C8;   Bicrossed Product
also known as Knit or Zappa-Szep product
∈ there existsh = 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= intersection)
  or (U = union)
∈   is in (is an element of the set)
  if
<->   if and only if
for all   for all
there exists   there exists
isomorpic   is isomorphic to
orb(s)   Orbit - set of elements that can be reached
stab(s)   Stabilizer - set of elements that dont move s

More notation on this page


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see also:

 

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag Fearless Symmetry - This book approaches symmetry from the point of view of number theory. It may not be for you if you are only interested in the geometrical aspects of symmetry such as rotation groups but if you are interested in subjects like modulo n numbers, Galois theory, Fermats last theorem, to name a few topics the chances are you will find this book interesting. It is written in a friendly style for a general audience but I did not find it dumbed down. I found a lot of new concepts to learn. It certainly gives a flavor of the complexity of the subject and some areas where maths is still being discovered.

Terminology and Notation

Specific to this page here:

 

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