2 – What is a Group?

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Group Theory [see Acknowledgements for Image Credit]

  “What immortal hand or eye, could frame thy fearful symmetry?”

– William Blake

 

 
A Collective Noun

If you look at the Wikipedia entry for Special Unitary Groups you quickly end up disappearing down a rabbit-hole [1]. The initial definition is as follows:

“The special unitary group of degree n, denoted SU(n), is the Lie group of n×n unitary matrices with determinant 1”

A number of questions naturally arise:

  1. What is a Group?
  2. What does “unitary” mean?
  3. What is a Lie Group?
  4. What are matrices?
  5. What is a determinant?

With its helpful structure of internal hyperlinks, one might assume that an answer to each of these questions is but a click away. However, the democratic way in which Wikipedia is constructed means that, while some of the articles you find could be understood quite quickly by the non-specialist, others [perhaps the majority] require a degree of subject background that is unlikely to exist outside of people who are already students of Mathematics. Indeed sometimes – in an effort to find meaning – you can find yourself executing loops round Wikipedia pages, returning to the place whence you started [2]. In this book, I am going to attempt to be your guide through this maze. I’ll look to eschew those elements which are likely only to further muddy the waters and instead concentrate on the essentials.

With respect to the questions above, I will look to answer them, but with a tighter focus on the task in hand. So in Chapter 4, I will introduce several different concepts of number, each of which has had a role to play in the story and whose development has to some extent proceeded hand-in-hand with that of Group Theory. In Chapter 5 and Chapter 6, I’ll cover the concept of matrices and how they fit into the overall picture. In Chapter 7, I will lay some further groundwork necessary for explaining what “unitary” means, before discussing Groups more generally in a series of Chapters, beginning with Chapter 8. After this, Chapter 11 and Chapter 12 focus on the early development of Group Theory, explaining how it’s initial considerations have later circled back to be pertinent to modern Physics. In Chapter 13 and Chapter 14, I will offer a first sight of Unitary Groups and, in later Chapters, I’ll discuss how some of the Mathematical concepts that I have introduced are used somewhat differently in Physics to Mathematics. The intention is of course to bring all of these disparate threads together in the final Chapters.

However, in this chapter and the next, I’m going to start the journey by looking to answer the very first question: What is a Group?

Readers will have already picked up that there is a strong relationship between Groups and symmetry. While this is undoubtedly true, we need to start with a formal definition. To do this we need to introduce two concepts, one probably very familiar the other perhaps a tweak on a more everyday concept.
 
 
Setting a Good Example

Elephants ∩ Vegetables = ∅

The quotidian concept is that of a set. Without getting into all of the formalism of Set Theory, this is an idea with which we should all be comfortable. A set is just a collection of things, the things being pretty malleable. All elephants in Africa is a set, all elephants in India is a second, which does not overlap with the first. All vegetables is a set and all root vegetables is another, the latter being a subset (a set wholly contained within another set) of the former. If a set X is a subset of a set Y, we denote this using a special symbol as follows:

X ⊂ Y

More mathematically, the whole numbers greater than zero (normally called the Natural Numbers and denoted by ℕ) form a set as do all positive even numbers (multiples of 2); again the latter is a subset of the former. Unlike the number of elephants in any given continent, these sets are of infinite size, there is no final element to them.

At this point I am also going to introduce some mathematical shorthand, which is going to save us some time and effort later on. This is the symbol ∈ which is often spoken as one of: “is a member of”, “belongs to” or “is in”. The mathematical phrase x ∈ S reads “x is an element of set S”.

Another piece of notation that needs mentioning is curly brackets; or braces. These are typically used to denote a set. So if a finite set called S has elements a, b and c we write:

S = {a, b, c}

Infinite sets, like the Natural Numbers mentioned above, can be shown by reference to their initial members, where the extension to any member is fairly obvious:

ℕ = {1, 2, 3, 4, …}

where “…” means that the numbers continue indefinitely

What appears within the “{}” can also be a set of rules for defining the members of the set. To take our example of all positive even numbers we can write:

even = {a ∈ ℕ, where there exists n ∈ ℕ such that a = 2n}

We can add sets to each other and also do subtraction. In the latter case, if A ⊂ B, then the set consisting of all elements of B that are not elements of A is written:

B\A

However, rather than employing the natural description of subtraction, this is referred to as the relative complement of A in B.

As an example if B = {1, 2, 3, 4, 5} and A = {2, 4} then:

B\A = {1, 3, 5}

Finally in this section, a set does not have to include anything, it can be empty. We have a special symbol to denote the empty set:

∅ = {}

 
Smooth Operators

The slightly tweaked concept is that of a binary operator. While once more the Wikipedia definition:

“A binary operation on a set S is a map which sends elements of the Cartesian product S × S to S”

may conceal more than it reveals, there is a simpler (if less rigorous) way of looking at this. A binary operator is a machine which, when given two similar things as input, reliably produces a third, which is also of the same type.

Black Box

While we are talking in more general terms, a specific example is that of addition. What is addition but a machine that takes two numbers and gives us a third as a result:

1 + 2 = 3

A binary operator is a generalisation of things like addition and multiplication, recognising that other things can be like these, without being precisely the same [3]. To avoid confusion with things like addition, generic binary operators are usually denoted by things like “○” or “*”.

Having introduced these two concepts, we can combine them, together with some rules to define a Group.
 
 
The Formal Answer to “What is a Group?”

A Group is a set G combined with a binary operator ○ which has the following behaviour:

  1. It is closed.
     
    Which means for any two elements of G, say a and b, then a ○ b is still an element of G.

    In mathematical terms:

    For all a and b ∈ G then a ○ b ∈ G.

  2. There exists a special element of G, called the identity element, denoted by e [4]
     
    The identity element is such that, for any other element, a, belonging to G, combining this with the identity leads to no change.

    Again in mathematical terminology:

    For all a ∈ G then a ○ e = a.

  3. Related to the last rule, every element has an inverse.
     
    This means that for any element a in G we can find another element, normally written as a-1, which when combined with a gives us the identity element.

    We write this as follows:

    For all a ∈ G, there exists a-1 ∈ G, such that a ○ a-1 = e.

  4. The binary operation is associative.
     
    This means that if we want to use the binary operator across three elements of G, it doesn’t matter if we combine the first two and then consider the last, or if we combine the first with the result of combining the second and third.

    Here the mathematical notation is probably much clearer than the prose:

    For all a, b and c ∈ G, a ○ (b ○ c) = (a ○ b) ○ c.

We sometimes capture the fact that a Group consists of a set and a binary operator by writing (G,○), though just using G to denote the Group is also very common.
 
 
In Addition it May be Noted…

I have nothing to add... [see Acknowledgements for Image Credit]

The above may seem rather arcane, but the rules actually describe a whole range of familiar entities. Let’s start by looking at the same binary operator which we used to explain what something like a ○ b might mean, namely addition.

If we look at our rules we see the following:

  1. Closure
     
    If we think about equations like 1 + 2 = 3 we can see that adding two numbers always results in a third number, so tick.
     
  2. Identity
     
    The obvious candidate here is 0 because for any number a, a + 0 = a.
     
  3. Inverse
     
    Again this is fairly straightforward if we think of things like 3 – 3 = 0, this is the same as saying 3 + (-3) = 0, so the inverse of 3 is -3 [5].
     
  4. Associativity
     
    Once more addition plays nicely and it is fairly clear that for any numbers a, b and c, a + (b + c) = (a + b) + c.
     

Before we get too carried away, it is worth providing an immediate counterexample. If instead of addition we used the equally familiar binary operator of subtraction, we don’t get a group. The associative rule does not hold as:

3 – (2 – 1) = 3 – 1 = 2

Whereas

(3 – 2) – 1 = 1 – 1 = 0.

The other point to note here is that I have been pretty loose about what I mean when I say number [6]. To an extent I have been focussing on the properties of the binary operator without thinking about the set on which this is acting.

Take the closure attribute. If, instead of loosely saying “numbers” (and actually probably meaning integers [7]), we focus on the actual sets of numbers that we are operating on, then we can pretty quickly think of a couple of simple sets where addition does not lead to a Group:

  1. G = the Natural Numbers (the positive integers 1, 2, 3, 4, …).

    Here there is both no identity as 0 is not part of the set and, even if we extend the definition to include 0, no inverses as -1 is not a Natural Number.

  2. G = {0, 1, 2, 3, 4} a finite set.

    Here as well as the inverse problem we found above, the set is not closed under addition as 1 + 4 = 5 and 5 is not a member of the set.

 
Modular Arithmetic

As an aside we can turn the second set into a Group under addition by modifying our binary operator. Instead of normal addition, we can define our binary operator a ○ b to be the remainder when a + b is divided by 5 – something that Mathematicians call addition Modulo 5. We write this as follows:

|n|5 = the remainder when n is divided by 5

So, for example:

|16|5 = 1, or alternatively 16 = 1 (mod 5), because 16 = (5 × 3) + 1.

If we now consider the equation that broke our Group definition before then:

1 ○ 4 = |1 + 4|5 = |5|5 = 0, so our result is 0 which is part of the set

Equally

4 ○ 3 = |1 + 3|5 = |7|5 = 2, which is again a member of the set.

So we have closure. 0 is the obvious identity. What about inverses? Well there aren’t so many set elements, so we can just list them:

  • 1 ○ 4 = |1 + 4|5 = |5|5 = 0, so 1-1 = 4
     
  • 2 ○ 3 = |2 + 3|5 |5|5 = 0, so 2-1 = 3
     
  • 3 ○ 2 = |2 + 3|5 = |5|5 = 0, so 3-1 = 2
     
  • 4 ○ 1 = |4 + 1|5 = |5|5 = 0, so 4-1 = 1
     
  • The identity element (as in all Groups) is its own inverse.

What about associativity? The proof of this equality involves the type of mathematical sleight of hand that I am aiming to avoid at this early to stage of the book. Addition modulo 5 (or indeed modulo n) can be shown to be associative, but for an explanation, see the notes section [8].

The moral of this aside is that it is a only via combination of the attributes of the set and the attributes of the binary operator that we can decide whether or not we have a Group.

 
The Symmetry Angle

A sixfold way? [see Acknowledgements for Image Credit]

In both the Foreword and Introduction, I mentioned the link between Groups and symmetry. This is a topic which I’ll look at more directly in the next chapter. However, while it may not have been entirely evident at the time, two symmetries have already emerged from the areas we have considered above. Before looking at these, it’s worth pausing to explain what is meant by symmetry in real life and in Mathematics.

Well the two concepts are pretty related. We talk about an object being symmetrical if the structure of one part is similar to the structure of another. The human body (and the bodies of all chordates [9]) is broadly symmetrical if you take a line from the top of the head down and out though the base of the spine and between the legs. Our left hand and right hand sides are close to mirror images of each other, something called bilateral symmetry.

At this point the word “mirror” has been introduced; this is related to reflectional symmetry, one type of symmetry. In the next chapter we will think about reflectional symmetry as it applies to some geometric shapes and how this links to Group Theory. It is however worth pausing to better define symmetry. It tends to relate to things that are invariant when you transform an object. In the case of reflectional symmetry, this means that the shape (or me and you) is invariant when reflected. A further type of symmetry is rotational symmetry. If you think about a square, then it remains the same if you rotate it by 90 degrees. A pentagon remains the same if you rotate it by 72 degrees and a hexagon if you rotate it by 60 degrees. As you get into higher dimensional shapes, other types of symmetries can emerge, e.g. you could rotate a cube (the three dimensional analogue of a square) through any two diagonally opposite vertices and leave the shape the same.

Returning to mirrors let’s think about our Group of the integers under addition; here we can consider one of the two symmetries we have already touched on. One way that we can think about symmetry within this group is that if we place a mathematical mirror across our number line at 0, then the inverse of any positive integer is its “reflection” in the negative numbers (and vice versa of course). In general, the existence of inverses is one of the ways that the idea of symmetry is embodied in Group Theory.

Clock arithmetic

The second symmetry relates to Modular Arithmetic (or Clock Arithmetic as it is sometimes known [10]). In our Modulo 5 example, adding 5 to any number gets you back to where you started. Equally, even adding 1 could be thought of as our clock hand rotating by 72 degrees (like the pentagon above) leaving the clock face the same as it was before. This link between Modular Arithmetic and rotational symmetry is something we will come back to later.
 
 
I introduced a few examples of Groups based on numbers and addition above. In Chapter 3, I’ll be getting more physical and the symmetric properties of Groups will be more on the surface of what we are doing.
 

Concepts Introduced in this Chapter
Set A collection of elements or members, the elements can be anything. In our examples the members of sets will often be numbers.
Subset A set wholly contained in another set. E.g. all left-handed people is a subset of all people. If A is a subset of B, we write: A ⊂ B.
Relative Complement If A is a subset of B then the relative complement of A in B, written B/A, is all members of B which are not members of A.
Empty Set The set with no elements, denoted by {} or ∅.
A Mathematical symbol used to denote an element being a member of a set. So x ∈ S may be read as “x belongs to S”.
Natural Numbers Whole numbers greater than zero, sometimes called the Counting Numbers. Denoted by ℕ = {1, 2, 3, 4, …}.
Integers All whole numbers, including both zero and negative numbers like -48. Denoted by ℤ = {… , -3, -2, -1, 0, 1, 2, 3, …}.
Binary Operator Something which combines two member of a set to generate a third. Addition and multiplication are binary operators, but there are more esoteric examples.
Group A set, together with a binary operator acting on its elements in such a way that:

  1. When you combine two elements, the result is still a member of the Group: Closure
  2. The set has a special element, such that when you combine this with any member of the set using the binary operator the member is unchanged: Identity Element
  3. For any member of the set, we can find another member such that when these are combined the result is the identity element: Inverses
  4. If you combine three elements of the set using the binary operator, it doesn’t matter if you combine the first two and then the third, or combine the first with the result of combining the second and third: Associative
Modular Arithmetic A variation on normal addition where the result of a + b is given by the remainder when dividing by a number n; this defines addition Modulo n. We use vertical bars to denote this so: |3 + 4|4 = 3, because the remainder when dividing 7 by 4 is 3. This is often called Clock Arithmetic because when we get to n, it is the same as retuning to 0; the same way that clock hands return to the beginning after completing a cycle. The Group defined by additional Modulo n is written as ℤn.
Groups Discovered in this Chapter
(ℤ,+) The integers under normal addition. An infinite Group.
(ℤn,+) The integers under addition Modulo n. A finite Group of size n.
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Chapter 2 – Notes

 

 
[1]
 
Something I am intimately familiar with as per Curiouser and Curiouser – The Limits of Brexit Voting Analysis.
 
[2]
 
While the jury is still out (though perhaps leaning to the case for the defence) on the subject of whether or not space-time is finite, curving back on itself, it seems that – to borrow from Douglas Adams – routes through Wikipedia-space are not merely curved, but in fact totally bent.
 
[3]
 
A lot of Mathematics proceeds this way by making generalisations of familiar things to encompass the less familiar, see also Chapter 4 and Chapter 5.
 
[4]
 
Generally the use of e to denote the identity element is thought to be because of the German einheit, meaning “unit”.
 
[5]
 
In this case, symmetrically we see that the inverse of -3 is just 3.
 
[6]
 
Though as it happens, for addition in most cases at least, my complacency was relatively justified.
 
[7]
 
The integers are the positive and negative whole numbers plus zero, so the infinite set {… , -4, -3, -2, -1, 0, 1, 2, 3, 4, …}, they are normally denoted by the symbol ℤ.
 
[8]
 
Mathematicians (myself included) have a bad habit of skipping steps in proofs, or saying “it is therefore trivial to see…” when it is anything but trivial to follow their logic. I’ll try to avoid this problem in the following.

We want to show that:

| |a + b|5 + c |5 = | a + |b + c|5 |5    …(1)

This is the definition of the operator (addition Moduluo 5) being associative on the set in question.

For each side of the equation, consider the terms between the innermost vertical lines. We can write:

a + b = 5x + r, so |a + b|5 = r    …(2)

b + c = 5y + s, so |b + c|5 = s    …(3)

Where x and y are some integers and both r and s are < 5, so then, looking at the left hand side of the associativity equation (1) which we want to prove, we can say that – using (2) above:

|a + b|5 + c = r + c    …(4)

but, if we rearrange (2) to read:

r = a + b – 5x

Then if we replace r in (4) with this definition, we get:

|a + b|5 + c = a + b – 5x + c

So

| |a + b|5 + c |5 = |a + b – 5x + c|5    …(5)

Here is the sleight of hand…

Now – by definition – the term 5x in the above has no impact on the equality. For example, | 38 |5 = 3, because 38 = (5 × 7) + 3. So if we add (or subtract) any multiple of 5 to 38, the result is precisely the same. E.g. |38 + 15|5 = |53|5 = 3 again [as 53 = (5 × 10) + 3].

So if we go back to (5) above, we can replace 5x by any other multiple of 5 without changing the equation. In particular we can replace it with 5y to get.

| |a + b|5 + c |5 = |a + b – 5y + c|5    …(6)

Then we can rearrange (3) to note that s = b + c – 5y. If we substitute this in (6) we get:

| |a + b|5 + c |5 = |a + s|5

Going back to the definition of s in (3) above, this is the same as saying:

| |a + b|5 + c |5 = | a + |b + c|5 |5

Which is the same as (1) and what we were required to prove (I wonder what that phrase would be in Latin?).

So we have proved associativity is true and our construct is indeed a Group, generally denoted by ℤ5.

The more attentive of readers will have noted that the same proof holds for any value n instead of 5, so more generally ℤn under addition Moduluo 5 forms a Group.

 
[9]
 
Chordates include all vertebrates and several related (and more ancient) species such as lancelets, a sort of proto-fish, examples of which dating to 525 million years ago have been discovered.
 
[10]
 
The reason for calling this Clock Arithmetic is probably fairly obvious. A traditional clock returns to 1 o’clock after passing midday or midnight. Similarly a ℤn returns to 0 after passing n – 1. 10 o’clock plus 4 hours equals 2 o’clock, operating exactly like ℤ12.

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