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. 

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