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“I consider nature a vast chemical laboratory in which all kinds of composition and decompositions are formed.” – Antoine Lavoisier

 
Here we leverage work covered in Chapter 8 and Chapter 9 to demonstrate how to decompose Groups into more fundamental elements. The process is somewhat similar to the one we used to decompose Natural Numbers into multiples of Prime Numbers back in Chapter 8. Often one part of Mathematics has echoes of other parts. Such echoes can form the basis of entire research programmes and, while we understand some of these types of relationships between ostensibly different fields well, others are the subject of ongoing enquiry.
 
 
A Simple Algorithm

To return to our work on the decomposition of a Natural Number into primes, we had an algorithm which essentially did the following:

Let our number to be decomposed be n ∈ ℕ

1. If is n is prime we are done
    
2. Otherwise:

   n = p₁n₁

   for some prime p₁ and n₁ ∈ ℕ
    

3. If n₁ is prime, we are done
    
4. Otherwise:

   n₁ = p₂n₂

   for some prime p₂ and n₂ ∈ ℕ

   Which also means that:

   n = p₁p₂n₂
    

5. Basically keep going until we reach an n_m which is equal to some prime p_m
    
6. Then:

   n = p₁p₂p₃…p_m

   for some primes p₁, p₂, … , p_m

Can we find another algorithm which does the same thing for Groups? Well our analogue of prime numbers in Group Theory world is Simple Groups. Primes cannot be divided by any whole number other than 1 and themselves. Simple Groups have no Normal Subgroups except for the trivial Group consisting of the Identity Element and the whole Group itself.

We have found that we can use a Normal Subgroup H to “divide” a Group G, creating a Quotient Group, G/H. What would be nice is if we could find a way to make one of H or G/H a Simple Group. There is such a method, which relies upon picking the Maximal Normal Subgroup of G, let’s call this H_max.

Maximal has a particular meaning in Group Theory, it means the largest Normal Subgroup which is not contained by another Normal Subgroup (not necessarily the one of largest order).

My assertion is then that G/H_max is always a Simple Group. If you want to accept this at face value, then the proof in the box below can be skipped.

The Quotient Group of a Maximal Normal Subgroup There is a result in Group Theory called the Correspondence Theorem. This says that for a Normal Subgroup F ◁ G we can establish a relationship between the members of the Quotient Group G/F and the set of Subgroups of G which contain F. That is the set of all all Subgroups J of G such that: F ⊂ J ⊂ G If we use GF to refer to the set of all Subgroups of G that contain F, then the relationship is set up via a function [1] mapping GF to G/F as follows: f: GF ↦ G/F, J ↦ J/F i.e. we form the Quotient Group for each Subgroup J containing F and map J to this. I’m not going to prove the Correspondence Theorem, but the above relationship (much like the isomorphisms that we have met before) preserves essential features in the mapping. This is to the extent that the structures of the two sets GF and G/F can be viewed as essentially the same. In particular, for a given Subgroup J containing F, The Correspondence Theorem states that: J ◁ G iff [2] J/F ◁ G/F. Considering this and noting that, by definition: F ⊂ J ⊂ G This means that if we replace F with Hmax we have: Hmax ⊂ J ⊂ G But the definition of Hmax is that no other Normal Subgroup contains it. So we conclude that either J = Hmax, or J = G. In the first instance: J/Hmax = Hmax/Hmax = 1 In the second instance: J/Hmax = G/Hmax So any Normal Subgroup of G/Hmax is either the Identity Element or the whole of G/Hmax. This is the definition of G/Hmax being Simple.

So with the result from the box under our belts, we can design our algorithm as follows:

1. Start with a Group G and set a counter n = 1
    
2. If G is Simple, write down A_n = G in a list and we are finished
    
3. Otherwise determine the Maximal Normal Subgroup of G, let’s call this H
    
4. Form the Quotient Group G/H, which we have shown to be Simple and write this down
    
5. Write down A_n = H in a list
    
6. Go back to 1. But replace G with H and increment n by 1 (n = n+1)

It may be seen that the above will generate a list of nested Normal Subgroups ^[3] as follows:

e ◁ A_m ◁ A_m-1 ◁ … ◁ A₂ ◁ A₁ ◁ G

Such a list is called a Composition Series. We can also form the related list of Quotient Groups, each of which is Simple:

A_m-1/A_m, … , A₁/A₂, G/A₁

So we have achieved our Group Theory analogue of the first part of the Fundamental Theorem of Arithmetic, namely that such a decomposition exists. The Jordan-Holder Theorem (which again I won’t prove here, but which proceeds very much along the same lines as the analogous proof for unique prime decomposition) guarantees not precisely the uniqueness of this, but at least that if we rearrange any such decomposition, we will be able to establish isomorphism between the various Groups in any two Composition Series.

This fundamental result which means that we can – in a certain sense – study all finite Groups by forming an understanding of just the finite Simple Groups, which are the building blocks for all larger structures. Again our connection with quarks comes to mind, not for the last time.

Many years of work led to a definitive Cataloguing of all Finite Simple Groups (a theorem which at the time was the longest Mathematical proof in history). This quest was motivated in no small part by the result we have walked through above.
 
 
Multiplication Redux

So we have worked out how to take Groups apart (at least in some sense). How do we put them back together? This is the province of Group Extensions, which in some ways can be viewed as the opposite of the Composition Series work we have covered above. Group Extensions remain a field of active Mathematical study and the work here rapidly gets to a level of complexity that I probably want to avoid.

However, rather luckily, the complex elements of Group Extensions are not needed on our journey to better understand the Mathematics underpinning the Standard Model of Partice Physics. Instead the object we have been working our way towards considering…

SU(3) × SU(2) × U(1)

points us to the most simple way of putting Groups together to create larger ones. The clue is in the × signs. Once more Mathematicians have triumphed by using the humble × to mean something else, a direct product of Groups.

Given two Groups, G and H, we define the set forming their direct product, written G × H as follows:

G × H = {(g,h) such that g ∈ G and h ∈ H}

However, in order to have a Group, we also need a binary operator. If the binary operator for G is given by * and that for H by ○, then for elements g₁, g₂∈ G and h₁, h₁ ∈ H, we define a binary operator ◊ on G × H as follows:

(g₁, h₁) ◊ (g₂, h₂) = (g₁ * g₂, h₁ ○ h₂)

The set element of the above is the equivalent of forming the Cartesian Product of two sets.

Let’s check that these definitions do indeed give us a Group.

1. Closure

   Combining any two elements in G yields an element of g. The same goes for any two elements of H. Thus:

   g₁ * g₂ ∈ G

   h₁ ○ h₂ ∈ H

   and so

   (g₁ * g₂, h₁ ○ h₂) ∈ G × H

   So we have Closure.
    

2. Identity

   If e_G is the identity element in G and e_H is the one in H, then it is straightforward to see that (e_G, e_H) is the identity element in G × H.
    

3. Inverses

   Simmilar to the last point, for any g ∈ G and h ∈ H, there must exist inverses, g^-1 and h^-1 respectively. It is clear that then:

   (g, h)^-1 = (g^-1, h^-1)

   So we have Inverses.
    

4. Associativity

   Following the normal trend we get a bit hand wavy when it comes to associativity. The reader can work it out long hand, but as we have defined our new binary operator ◊ in terms of two other binary operators which we know are associative on their specific sets, it is pretty clear that ◊ is also associative on G × H.

As I mention above, there are other ways to glue Groups together and in many ways the direct product is perhaps the ugliest. However it is what we need to understand SU(3) × SU(2) × U(1) so I won’t delve any further into this area.
 
 
At this point, our discussions about Simple Groups are complete. In the next two Chapters, Root of the Problem and Mont Évariste, we will go back to the origins of Group Theory. In the succeeding Chapters we will start to combine several of the concepts that we have assembled in earlier Chapters to offer our first sight of the Unitary Groups.
 
 

Concepts Introduced in this Chapter Maximal Normal Subgroup A Normal Subgroup that is not contained by any other Subgroups. Quotient of a Maximal Normal Subgroup G/F is always Simple if F is a Maximal Normal Subgroup. Correspondence Theorem For any Normal Subgroup F of a Group G, a one-to-one mapping exists between the set of all Subgroups J of G which contain F and the Quotient Groups J/F. Composition Series A set of Maximal Normal Subgroups nested within each other. So if A1 is the Maximal Normal Subgroup of some Group G, A2 is the Maximal Normal Subgroup of A1 and so on until a Simple Group (one with no Normal Subgroups) is reached. A Composition Series also generates a set of Quotient Groups, G/A1, A1/A2… each of which is Simple. Group Decomposition Algorithm A programmatic approach to creating a Composition Series and related set of Quotient Groups for any Group. Analogous to the algorithm for factorising any Natural Number uniquely into a product of prime numbers. Jordan-Holder Theorem Any Composition Series of a Group is unique up to isomorphism of the Subgroups generated. Finite Simple Groups A long Mathematical proof cataloguing every single Finite Simple Group; the building blocks for all other Finite Groups. Group Extensions A collective term for ways to combine Groups to make more Groups. This complex area is highly related to Composition Series and is a field of active study in the Mathematical community. Direct Product The simplest way to create Group Extensions. Analogous to gluing together two Groups. The direct product of two Groups G and H, written G × H is the Cartesian product of the two sets with the binary operators of G and H acting independently on the two elements of the Cartesian product, I.e. Using multiplicative terminology: (g,h)(g’,h’) = (gg’,hh’) where g, g’ ∈ G and h, h’ ∈ H

Groups Discovered in this Chapter G × H The Direct Product Group of two Groups G and H.

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Chapter 10 – Notes

[1]	We have managed to avoid functions to this point; or rather we have used lots of them and not made this explicit. A function is a mapping, denoted by a letter, sometimes “f” for “function”, sometimes by a greek letter, which connects two set A and B such that each element of A becomes an element of B. So for all a ∈ A we have a corresponding f(a) ∈ B. The most familiar of functions will be things like: f(x) = ax + c f(x) = x2 f(x) = log(x) + √x and so on. Each of the above will map the Real Numbers to the Real Numbers (or some subset of them) and will indeed do the same with the Complex Numbers. We write: f: ℝ ↦ ℝ, x ↦ x2 or f: ℂ ↦ ℂ, x ↦ log(x) + √x A bit like elements of a Group, functions can have inverses (though not all of them) and we would write this as f-1, such that: f-1(f(x)) = x It should also be noted that: If f: A ↦ B Then f-1: B ↦ A All of the binary operators acting on a Group G that we have met are functions. If the binary operator is denoted by “*” then we can write a function: f(x, y) = x * y where x, y ∈ G Here the mapping is from two copies of G (which we show as G × G) to G itself. We write: f: G × G ↦ G (x, y) ↦ x * y The above is the explanation for the rather recherché definition of a binary operator that I initially provided in Chapter 2. There are many flavours of functions. The one we are going to look at is a one-to-one correspondence (i.e. each element a of A is mapped to a unique f(a) in B) where all of A is mapped to all of B (i.e. f(A) = B). This is called a bijective function, or just a bijection.
[2]	If and only if once more. I.e. each statement implies the other one.
[3]	At least it will do if G is finite, no such result holds for infinite Groups.

Text: © Peter James Thomas 2016-18. Images: © Peter James Thomas 2016-17, unless stated otherwise. Published under a Creative Commons Attribution 4.0 International License.