I got several great suggestions for topics for ‘g’. The one that most caught my imagination was mathtuition88’s, the group action. Mathtuition88 is run by Mr Wu, a mathematics tutor in Singapore. His mathematics blog recounts his own explorations of interesting topics.
This starts from groups. A group, here, means a pair of things. The first thing is a set of elements. The second is some operation. It takes a pair of things in the set and matches it to something in the set. For example, try the integers as the set, with addition as the operation. There are many kinds of groups you can make. There can be finite groups, ones with as few as one element or as many as you like. (The one-element groups are so boring. We usually need at least two to have much to say about them.) There can be infinite groups, like the integers. There can be discrete groups, where there’s always some minimum distance between elements. There can be continuous groups, like the real numbers, where there’s no smallest distance between distinct elements.
Groups came about from looking at how numbers work. So the first examples anyone gets are based on numbers. The integers, especially, and then the integers modulo something. For example, there’s , which has two numbers, 0 and 1. Addition works by the rule that 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, and 1 + 1 = 0. There’s similar rules for , which has three numbers, 0, 1, and 2.
But after a few comfortable minutes on this, group theory moves on to more abstract things. Things with names like the “permutation group”. This starts with some set of things and we don’t even care what the things are. They can be numbers. They can be letters. They can be places. They can be anything. We don’t care. The group is all of the ways to swap elements around. All the relabellings we can do without losing or gaining an item. Or another, the “symmetry group”. This is, for some given thing — plates, blocks, and wallpaper patterns are great examples — all the ways you can rotate or move or reflect the thing without changing the way it looks.
And now we’re creeping up on what a “group action” is. Let me just talk about permutations here. These are where you swap around items. Like, start out with a list of items “1 2 3 4”. And pick out a permutation, say, swap the second with the fourth item. We write that, in shorthand, as (2 4). Maybe another permutation too. Say, swap the first item with the third. Write that out as (1 3). We can multiply these permutations together. Doing these permutations, in this order, has a particular effect: it swaps the second and fourth items, and swaps the first and third items. This is another permutation on these four items.
These permutations, these “swap this item with that” rules, are a group. The set for the group is instructions like “swap this with that”, or “swap this with that, and that with this other thing, and this other thing with the first thing”. Or even “leave this thing alone”. The operation between two things in the set is, do one and then the other. For example, (2 3) and then (3 4) has the effect of moving the second thing to the fourth spot, the (original) fourth thing to the third spot, and the original third thing to the second spot. That is, it’s the permutation (2 3 4). If you ever need something to doodle during a slow meeting, try working out all the ways you can shuffle around, say, six things. And what happens as you do all the possible combinations of these things. Hey, you’re only permuting six items. How many ways could that be?
So here’s what sounds like a fussy point. The group here is made up the ways you can permute these items. The items aren’t part of the group. They just gave us something to talk about. This is where I got so confused, as an undergraduate, working out groups and group actions.
When we move back to talking about the original items, then we get a group action. You get a group action by putting together a group with some set of things. Let me call the group ‘G’ and the set ‘X’. If I need something particular in the group I’ll call that ‘g’. If I need something particular from the set ‘X’ I’ll call that ‘x’. This is fairly standard mathematics notation. You see how subtly clever this notation is. The group action comes from taking things in G and applying them to things in X, to get things in X. Usually other things, but not always. In the lingo, we say the group action maps the pair of things G and X to the set X.
There are rules these actions have to follow. They’re what you would expect, if you’ve done any fiddling with groups. Don’t worry about them. What’s interesting is what we get from group actions.
First is group orbits. Take some ‘g’ out of the group G. Take some ‘x’ out of the set ‘X’. And build this new set. First, x. Then, whatever g does to x, which we write as ‘gx’. But ‘gx’ is still something in ‘X’, so … what does g do to that? So toss in ‘ggx’. Which is still something in ‘X’, so, toss in ‘gggx’. And ‘ggggx’. And keep going, until you stop getting new things. If ‘X’ is finite, this sequence has to be finite. It might be the whole set of X. It might be some subset of X. But if ‘X’ is finite, it’ll get back, eventually, to where you started, which is why we call this the “group orbit”. We use the same term even if X isn’t finite and we can’t guarantee that all these iterations of g on x eventually get back to the original x. This is a subgroup of X, based on the same group operation that G has.
There can be other special groups. Like, are there elements ‘g’ that map ‘x’ to ‘x’? Sure. The has to be at least one, since the group G has an identity element. There might be others. So, for any given ‘x’, what are all the elements in ‘g’ that don’t change it? The set of all the values of g for which gx is x is the “isotropy group” Gx. Or the “stabilizer subgroup”. This is a subgroup of G, based on x.
Yes, but the point?
Well, the biggest thing we get from group actions is the chance to put group theory principles to work on specific things. A group might describe the ways you can rotate or reflect a square plate without leaving an obvious change in the plate. The group action lets you make this about the plate. Much of modern physics is about learning how the geometry of a thing affects its behavior. This can be the obvious sorts of geometry, like, whether it’s rotationally symmetric. But it can be subtler things, like, whether the forces in the system are different at different times. Group actions let us put what we know from geometry and topology to work in specifics.
A particular favorite of mine is that they let us express the wallpaper groups. These are the ways we can use rotations and reflections and translations (linear displacements) to create different patterns. There are fewer different patterns than you might have guessed. (Different, here, overlooks such petty things as whether the repeated pattern is a diamond, a flower, or a hexagon. Or whether the pattern repeats every two inches versus every three inches.)
And they stay useful for abstract mathematical problems. All this talk about orbits and stabilizers lets us find something called the Orbit Stabilization Theorem. This connects the size of the group G to the size of orbits of x and of the stabilizer subgroups. This has the exciting advantage of letting us turn many proofs into counting arguments. A counting argument is just what you think: showing there’s as many of one thing as there are another. here’s a nice page about the Orbit Stabilization Theorem, and how to use it. This includes some nice, easy-to-understand problems like “how many different necklaces could you make with three red, two green, and one blue bead?” Or if that seems too mundane a problem, an equivalent one from organic chemistry: how many isomers of naphthol could there be? You see where these group actions give us useful information about specific problems.
If you should like a more detailed introduction, although one that supposes you’re more conversant with group theory than I do here, this is a good sequence: Group Actions I, which actually defines the things. Group actions II: the orbit-stabilizer theorem, which is about just what it says. Group actions III — what’s the point of them?, which has the sort of snappy title I like, but which gives points that make sense when you’re comfortable talking about quotient groups and isomorphisms and the like. And what I think is the last in the sequence, Group actions IV: intrinsic actions, which is about using group actions to prove stuff. And includes a mention of one of my favorite topics, the points the essay-writer just didn’t get the first time through. (And more; there’s a point where the essay goes wrong, and needs correction. I am not the Joseph who found the problem.)