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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
 If is n is prime we are done
 Otherwise:
n = p_{1}n_{1}
for some prime p_{1} and n_{1} ∈ ℕ
 If n_{1} is prime, we are done
 Otherwise:
n_{1} = p_{2}n_{2}
for some prime p_{2} and n_{2} ∈ ℕ
Which also means that:
n = p_{1}p_{2}n_{2}
 Basically keep going until we reach an n_{m} which is equal to some prime p_{m}
 Then:
n = p_{1}p_{2}p_{3}…p_{m}
for some primes p_{1}, p_{2}, … , 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:
If we use G_{F} to refer to the set of all Subgroups of G that contain F, then the relationship is set up via a function ^{[1]} mapping G_{F} to G/F as follows:
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 G_{F} and G/F can be viewed as essentially the same. In particular, for a given Subgroup J containing F, The Correspondence Theorem states that:
Considering this and noting that, by definition:
This means that if we replace F with H_{max} we have:
But the definition of H_{max} is that no other Normal Subgroup contains it. So we conclude that either J = H_{max}, or J = G. In the first instance:
In the second instance:
So any Normal Subgroup of G/H_{max} is either the Identity Element or the whole of G/H_{max}. This is the definition of G/H_{max} being Simple. 
So with the result from the box under our belts, we can design our algorithm as follows:
 Start with a Group G and set a counter n = 1
 If G is Simple, write down A_{n} = G in a list and we are finished
 Otherwise determine the Maximal Normal Subgroup of G, let’s call this H
 Form the Quotient Group G/H, which we have shown to be Simple and write this down
 Write down A_{n} = H in a list
 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_{m1} ◁ … ◁ A_{2} ◁ A_{1} ◁ 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_{m1}/A_{m}, … , A_{1}/A_{2}, G/A_{1}
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 JordanHolder 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 g_{1}, g_{2}∈ G and h_{1}, h_{1} ∈ H, we define a binary operator ◊ on G × H as follows:
(g_{1}, h_{1}) ◊ (g_{2}, h_{2}) = (g_{1} * g_{2}, h_{1} ○ h_{2})
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.
 Closure
Combining any two elements in G yields an element of g. The same goes for any two elements of H. Thus:
g_{1} * g_{2} ∈ G
h_{1} ○ h_{2} ∈ H
and so
(g_{1} * g_{2}, h_{1} ○ h_{2}) ∈ G × H
So we have Closure.
 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.
 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.
 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.


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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:
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:
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:
It should also be noted that:
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:
Here the mapping is from two copies of G (which we show as G × G) to G itself. We write:
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 onetoone 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 201618. 
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