A Brief Tour Through Basic Group Theory – Part 4

by Anna Mustata

Part IV – Relating Groups to Other Groups 

Attempting to make things more concrete leads us to developing the concept of group actions. We can now use this to zoom back out. We’ve been applying groups to geometric objects to find out how they can tell us more about the things we we’re already interested in, but can we use this to make them tell us more about themselves? 

Of course, I wouldn’t be asking this rhetorical question unless the answer was yes.  

Before we try to apply groups to themselves or to other groups, let’s establish a general way to talk about relations between groups. To see how we should look at these relations, we return to the key principle of keeping track of information. When we work in a very concrete setting – say, adding integers or looking at symmetries of a cube – the set associated to our group is very important. At a more abstract level, however, this information becomes irrelevant; the structure of the group is not determined by what the elements of the set are, but by how they combine with each other. If we were to write out the multiplication table (that is, a list of which element is formed by every possible combination of two other elements) for symmetries of the cube, then replace each function with a Greek letter everywhere it occurs, we would call the second group isomorphic to the first. That is to say, they are essentially the “same” group, but may represent a different concrete situation. 

A more interesting relation is the homomorphism. Like an isomorphism, a homomorphism equates elements of the underlying set of one group to elements in the underlying set of another without changing the multiplication table – that is, if a goes to a’, b to b’ and c to c’, then a*b = c means a’*b’ = c’. The difference is that multiple elements from one group can be equated to the same element in the second, and not every element in the second group must be reached. 

We could say that equating different elements in the first group to a single element in the second group is “gluing” them together, in the same way we glued functions together when performing a group action. The functions from the first group which are glued to the identity element from the second now perform the same role that a stabilizer performed for a group action. In this case, we call it a kernel. The elements “glued” to any other element in the second group are cosets of the kernel. 

We can imagine this gluing as “folding” the landscape of a group so that all elements in the kernel are brought to the “starting point” of the landscape. It makes sense to wonder how other symmetries are affected by this folding. In particular, we’ll look at the conjugacy symmetry where all similar paths radiate out from the starting point. If we consider the kernel as one of these paths, bringing it into the starting point requires by symmetry that we must bring all similar paths into the starting point as well. However, the kernel is defined as containing all elements that get brought into the center, so we must conclude that there are no other similar paths; in other words, conjugating the kernel by any other element in the original group returns the kernel. Subgroups which have this property are known as normal subgroups. 

Returning to the concept of groups acting on groups, we will see that examples of this have already snuck up on us and are fundamental to our descriptions of symmetries in groups. In our first symmetry, we “stepped off the path” of a subgroup by applying a function belonging to the larger group, but not necessarily the subgroup. This was, in fact, the larger group acting on the subgroup (or rather, on all cosets of the subgroup). The orbit was the set of all subgroups, and the stabilizer was the set of all elements in the subgroup itself. The same can be said for conjugating a subgroup. When that subgroup is normal, its orbit is itself and its stabilizer the entire group containing it. 

At this point, the concept of a homomorphism and the concept of a group action may look rather similar. That’s because a group action is a homomorphism. When we “apply” a function to a set, we are choosing a way for the elements of that set to be reordered amongst each other. The set of all possible such reorderings is, itself, a group known as the symmetric group of order n, or Sn, where n is the size of the set that is being reordered. Sn is an incredibly versatile group, since it can describe the reorderings of any set of size n. Our group action on a set of size n is then in fact a homomorphism from the group that acts on the set to the group Sn.  

We’ve discussed how a group can act on a subgroup of itself. However, observe that a group is a subgroup of itself. We define the regular action of a group on itself where applying an element of the group to another element of the group is the same as combining them by the group operation. This may seem like a pointless thing to do, but looking at the previous paragraph, we see that it gives us a homomorphism from any group to a symmetric group Sn where n is the size of the first group. If we examine this homomorphism, we see that its kernel contains only the identity element. In other words, no information is lost by gluing things together during the homomorphism; the group stays intact, so to say, and is merely placed directly over some path in Sn. Therefore, we can say that any group of size n is isomorphic to a subgroup of Sn. This is called Caley’s theorem. 

At this point, we have gone from thinking a group can be “basically anything” to seeing all groups of finite size as subgroups of very specific types of groups, with properties that can be understood by a person with some knowledge of permutations.  

There is one more thing we want to discuss in this section. So far, we’ve used homomorphisms to compare two groups we already knew about. However, we can also create new groups like this. We’ve already discussed “folding” over a kernel. Any normal subgroup can be used as a kernel; the process of folding this subgroup into the starting point is known as quotienting the larger group by the subgroup. Each coset of the subgroup becomes a single element in the new group. Anyone familiar with modular arithmetic can see that if we were to quotient the group of all integers by the subgroup of even integers described at the very beginning of this article, we would get a group with addition in mod 2. 

If Caley’s theorem allows us to narrow down what groups can be by embedding them in larger groups, quotienting allows us to simplify groups by breaking them down into pairs of smaller groups: a normal subgroup and its quotient group. A group that can’t be broken down any further is called a simple group (remember, these are the ones that have been classified in the Atlas of Finite Groups). Simple groups play a similar role in group theory as primes play when working with integers.  

Part V – Conclusion 

At this point, you’ve seen most of the essential ideas that allow you to work with groups. So, what now? 

On the concrete side, we can use these concepts to study particular groups that are of natural interest. Some such groups have been mentioned already. Symmetry groups of geometric objects tell us things about how molecules behave in chemistry and quantum mechanics. If we restrict this symmetry to rotations in two dimensions, we get complex number multiplication which allows us to find roots to equations of the form xn = 1. Groups of polynomials in multiple variables allow us to study the symmetries of these polynomials, which in turn allows us to solve polynomials in one variable of degree greater than two by looking at the symmetry of its roots. Groups of invertible matrices allows us to look at geometry – more specifically, transformations in geometry – using a simple set of algebraic rules, at a higher level than we can reach just with trigonometry and cartesian co-ordinates. The example given with colouring the faces of a cube shows that group theory can be used to solve combinatorical problems. Finally, modular arithmetic (the groups obtained by quotienting the integers by a subgroup) is very useful in cryptography. 

On the more abstract side, we now just about have the tools to start categorizing the types of groups that can exist. Based on what’s been discussed in this article, it should be possible to categorize all groups of size up to eight. Categorizing groups of larger size requires some more theorems, but still follows essentially the same principles.    

A Brief Tour Through Basic Group Theory – Part 3

by Anna Mustata

Part III – Group Actions 

So far, we have been looking for symmetries in general, vague spaces featuring things denoted by things like “g” and “*”. If you feel like we’ve lost touch from anything one might actually care to investigate, I can’t say I blame you. This is your cue to remember what I said at the beginning about the interplay between concrete and abstract, and call me back down to earth to discuss the concrete details.  

Rather than talk about symmetries of some abstract set, let’s look at something we know has symmetry: a regular polygon, say the hexagon. It’s natural to think of the hexagon as a set of points in the two-dimensional plane. We might want to try looking at its symmetry in terms of groups; however, now we hit upon a problem. We can’t use the vertices of the hexagon as objects of a group because the vertices of a shape don’t naturally interact by combining two of them with an operation. Instead, we can move in between vertices by functions such as rotations and reflections.  

So far, we’ve been seeing groups as sets of things or objects that are combined by the operation *. However, those objects can be anything, including being functions themselves. In this case, g*b simply means carrying out function b and then applying function g to the result. For such groups to have meaning, however, we must apply those functions to something. We are now working with two sets; the set S of functions in the group (for example, rotations and reflections), and the set X of objects that those functions apply to (for example, the vertices of our hexagon). Applying S to X is called a group action.  

In the hexagon example, we have the subgroup formed by rotating clockwise by 60 degrees. We can do this 6 times before returning to where we started. It has a coset formed by reflecting in the horizontal axis and then rotating. Since there are two cosets of size six, the total number of symmetries is twelve. This is formally known as the dihedral group D12.  

Much like how we find “paths” in a group by taking an element in it and applying it over and over, we find “paths” in a group action by taking one of the objects being acted on and applying the entire group of functions only to that object. We call this the orbit of the object.  

We can think of a group action as “gluing” elements of the group to the object acted on in order to compare their symmetries and find out more about either the object or the group. When we glue the identity element to an object, that object stays the same. However, you can glue multiple functions to get the “stay the same” position for a certain object. For example: a vertex stays the same if you rotate it by 0 degrees, but also if you reflect the hexagon through the axis passing through it and the vertex opposite to it. We call the set of functions that fix an object when glued to it the stabilizer of the object in X that they fix. 

A stabilizer always contains the “starting point” of the group it belongs to. Moving along the “path” of functions in the stabilizer of an object will always lead to a function that keeps that object in the “stay the same” position. Therefore, the stabilizer is a subgroup. Stepping off the path of the stabilizer means using a function that does not keep the object in its “stay the same” position, but rather moves it to a new position in its orbit. Following a path parallel to the stabilizer then keeps it in that new position. Therefore, we can “glue” each coset of the stabilizer to an object to move it to a particular position in its orbit.  

Looking back at Lagrange’s Theorem (the statement that the size of a group is the size of a subgroup times the number of cosets of that subgroup), we can say that, since each coset of the stabilizer corresponds to a position in the orbit, the size of a group is the size of the orbit of an object times the size of its stabilizer. 

Another similarity between orbits and cosets is that just like every element of a group must belong to a coset, every element of the set acted on must belong to an orbit. However, unlike cosets, not every orbit must be the same size.  

In chemistry, group actions can be used to study the structure of molecules. For a simpler example, the following question can be solved by looking at group actions and orbits:  

Suppose Alice has three colours and wants to paint a cube so that each face is a different colour. How many designs can she paint if two designs are considered the same when rotating a cube with one design makes it identical to a cube with the other? 

I won’t explain the full solution here, but the key to applying group theory to this problem is to apply the group of rotations of the cube to the set of all possible designs and count the number of orbits (since two cubes are identical if and only if they share an orbit). 

A Brief Tour Through Basic Group Theory – Part 2

by Anna Mustata

Part II – Symmetry and Subgroups 

Now we’ve established what a group is, you may be wondering why we care. That’s a perfectly fair question to ask after reading a list of rules about what you can do with some symbols? The definition of a group seems vague enough that you might think it could be almost anything, but in fact, a surprising amount of structure arises as the inevitable consequence of these very simple rules. Far from groups being “almost anything”, mathematicians have at this point classified all groups of a finite size that are “simple” (that is, the building blocks of other groups). They can be found in the Atlas of Finite Group Representations. 

So how do we find structure in what it seems should be a fairly uniform, amorphous landscape? Well, just pick an element and start walking. To take a simple example, consider the group formed by the set of all integers and the operation of addition. If you start at the element two and keep adding it to itself, then go backwards and keep subtracting it, you will eventually reach all the even numbers and only the even numbers. In other words, if you follow a path that starts from two, you can do everything the rules of groups allow you to do while only using half the space available to you. Since this path is self-contained, we can see it as a full landscape in its own right, contained in the larger group, so we call it a subgroup. Because we formed it simply by starting at two and adding or subtracting that from itself again and again, we call two the “generator” of this subgroup. If we had needed another element to reach all the corners of the landscape that we wanted to explore, the subgroup would have multiple generators – this is allowed (though it will never happen with the integers – thanks, Euclid’s algorithm!). 

This doesn’t seem to have advanced our cause much, but in fact, it gives us a strong foothold into the structure of groups, because subgroups can’t just be any set of elements from our group. Earlier I called the even numbers “half” of the integers. This does not, in fact, technically make sense – how do you define half of infinity? But it made intuitive sense for me to say this, because there is a clear correspondence between the even and odd numbers. If you imagine the integers as a sheet of paper, you could “fold” it in half so that over every even number 2n you have the odd number 2n + 1. The odd numbers do NOT form a subgroup, because the starting point – in this case 0 – is not among them. Instead, they form what we call a “coset” of the subgroup of even numbers.  

Cosets of a subgroup can be found by moving off the “path” of the subgroup (taking an element that is not in the subgroup) and then tracing out a parallel path starting from this element. Notice that we could start from any odd number; the coset formed by stepping off the path by 1 step and then moving in even steps is the same as that formed by taking three steps then moving in even steps, so there are exactly two cosets (the even subgroup and the odd coset) in the group of integers. Importantly, every element is in a coset. This gives us the kind of symmetry – that is, the recurrence of a pattern – we are looking for: whenever we find a path through the origin, we can arrange all other points in parallel paths. In particular, if the size of the group is finite, it is equal to the size of a “path” (subgroup) multiplied by the number of “parallel paths” (cosets). This is called Lagrange’s theorem.  

Already, we can see how our options for what a group can look like are limited. If the size of a group is prime, there is space for only one path. Since we have seen that starting at the identity element and doing the same thing over and over gives us a path, we can conclude that a group whose size is prime can be generated by a single element. Furthermore, we can say that any element (other than the identity) will generate the entire group. It’s also possible to prove that if a prime divides the size of a group, the group contains a subgroup of the size of that prime. 

If we are to try to imagine this type of symmetry, it would be like a wallpaper pattern where a shape or sequence is repeated side by side. However, another type of symmetry is like a flower or a snowflake, where all symmetric components radiate out from the same starting point. This is a type of symmetry worth looking at because the idea of a starting point is so important to us that it was one of the basic requirements for something to be a group. We can find such symmetries by taking advantage of our ability to retrace our steps. One thing: for me and I guess many more people the term “symmetry” appears in no more places than Geometry and literature, and so although I can intuitively understand what you are referring to in the paragraph before but it will get a bit more confusing as you get more abstract later on in the article. Do you think it would be a good idea to maybe find a place and explain a little more what does symmetry mean in algebra? Is it a kind of isomorphism between the properties of numbers? Is it the shift of position of a certain function (for example)?

For every element g of our group, we will write its inverse as g’. Then, we define the conjugate of another element h by g as g*h*g’. Intuitively, we can see how this will help us “loop” back to our starting point.  

 If, and after moving off the path of a subgroup by an element g not in it and taking a parallel path, we move back by g’, we once again get a “parallel path”, this time containing the conjugates by g of every element in the subgroup. Remember that putting a g and g’ together makes them disappear, so we can prove this symmetry by observing that (g*a*g’)*(g*b*g’) = g*a*(g’*g)*b*g’ = g*a*b*g’ – in other words, conjugating two elements and then combining them is the same as combining them and then conjugating the result, giving us a correspondence between the two paths. 

If we realize that conjugating the identity element just gives us the identity element again (g*e*g’ = g*g’ = e), we see that all these paths start from the same point, hence giving us the type of symmetry we were looking for. Since all paths lead back to the identity element, the conjugate of a subgroup is itself a subgroup. In this way, it is a stronger symmetry. However, it is “weaker” because not every element is guaranteed to belong to one of these paths. This is the payoff for repeating the same element (“e”) in multiple paths, thus not guaranteeing a “clean split” of all elements into either one path or another. 

A Brief Tour Through Basic Group Theory – Part 1 (Introduction)

by Anna Mustata

Part 0 – Disclaimer 

This article is a rather experimental attempt at explaining how I think about mathematics in an aesthetic and intuitive way, and hopefully help the reader see the purpose of studying certain things in mathematics and the reason behind what is done. Essentially, I want to portray mathematics in a way that is appealing and that shows more than the final results and technical proofs. Please keep in mind that none of the explanations I give in this article are fully-fledged proofs, just intuitive motivations for things being a certain way. Also, quotation marks denote terms which are NOT the official or technical term for what I am describing, but are a good intuitive way of thinking about it.  

Part I – Introduction 

Everything happens in a context. Sometimes, in mathematics, this context is obvious: Euclidean geometry happens on a flat, infinite plane. Hyperbolic geometry happens in the interior of a circle. 

In algebra, this is harder to see, but it can be visualized. Real numbers can be added on a number line. Integers can be added similarly if you restrict the points on this line that can be used. Complex numbers can be added in a plane with two axes, real and imaginary. But what happens when we try to get more abstract? Is there a space where we can add, say, polynomials in the variable x? At this point – as tends to happen when things get abstract – our simple geometric intuition seems to fail us. What we can’t show in pictures, we can describe mathematically through a set of rules. Such landscapes where mathematics can happen are known as groups. 

As with most areas in mathematics, diving headfirst into group theory without a guide will quickly leave you tangled in minutiae and wondering why you bothered approaching it in the first place. Therefore, let us outline a few key philosophical ideas to keep an eye out for in order to ensure that we are moving forwards in our attempt to study groups. 

The first and most straightforward is to remember to think of groups as a landscape. Even when we switch from literal, geometric landscapes to a list of rules, we can still think of the metaphorical “shape” of a group as something that determines how we can “move” through it when we do mathematics in that context.

Another thing that is true throughout mathematics is that everything is about information. Specifically – and this is likely to be best appreciated by computer scientists – it is about making sure that the amount of information needed to describe a useful idea is not greater than the amount of information that can be generated from it; otherwise, our idea is useless. Combining this with the concept of landscape leads us to look for symmetry; that is, for patterns that allow us to examine a small area of this landscape and deduce something about the shape of the rest of it. 

Finally, in order to progress, we need to recognize how we might get stuck, and how to become unstuck. Generally, there are two extremes where progress tends to peter out: too much abstraction, and too little. Too much abstraction eliminates all details we might care to know and leaves us with a meaningless, hollow outline. Not enough leaves us repeating the same process indefinitely, never catching a glimpse of the larger picture that might allow us to improve our measure of “information in – information out”. For the best effect, we must combine the two perspectives so that they fuel each other, zooming in and out much as one might when looking at a fractal: first, finding a detail, then using that detail to extrapolate a larger outline, then once more looking closely in order to fill in that outline.  
 
Keeping those things in mind, let’s dive in! 
 
Hopefully at this point you have a fairly good intuitive idea of what we mean by a group; however, this is mathematics, so let’s get a rigorous definition in place. In order to have a mathematical “landscape”, you need objects, and you need to be able to do things with them. Therefore, a group will be defined over a set S and an operation * (which can be anything that combines two objects to give a third). We don’t want to be able to walk off the edge of this mathematical landscape, so applying * to two elements in S must produce another element in S. There are three more axioms that need to be satisfied for (S, *) to be a group. 

1) * must be associative: that is, brackets don’t matter, or a*(b*c) = (a*b)*c. This is saying that any ordered sequence of elements in S uniquely define a “path” to another element in S.  

2) We also want our landscape to have a “starting point” (this would be 0 on the number line, for example). We will call this the identity element (usually written as e or Id). Moving by the identity element should not affect your path, so for any x in S, x*e = x = e*x. 

3) In order to actually make computations and do things, we need to be able to retrace our steps; so, every element x in S must have an inverse y such that x*y = e. 

A few algebraic manipulations will show that every element has exactly one inverse and that if x*y = e then y*x = e. (Note: it is not always true that x*y = y*x).