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.