There is something odd about the experience of learning group theory. At first, one is told that the great virtue of groups is their abstractness: many mathematical structures, from number systems, to sets of permutations, to symmetries, to automorphisms of other algebraic structures, to invariants of geometric objects (these last two are examples you won’t meet for a while) have important properties in common, and these are encapsulated in a small set of axioms that lead to a rich theory with applications throughout mathematics. So far so good — understanding about abstraction is wonderful and mind-expanding and the definition of a group is one of the best examples.
But then one studies group actions (and later group representations). They appear to be doing the reverse of abstraction: we take an abstract group and find a way of thinking of it as a group of symmetries. And that is supposed to help us understand the group better — so much so that group actions are an indispensable part of group theory.
So is abstraction good or bad? Well, both the views above are correct. Abstraction does indeed play a very important clarifying role, by showing us that many apparently different phenomena are basically the same, and isolating the aspects of those phenomena that really matter. However, if a group is defined for us in an abstract way (I’ll say more precisely what I mean by this later), then showing that it is isomorphic to a group of symmetries can make it much easier to answer questions about that group.
In this post, and one or two further ones, I want to discuss what a group action actually is, the orbit-stabilizer theorem and how to remember its proof, and how to use group actions to prove facts about groups.
What is a group action?
There are two ways of defining group actions. I don’t know which one you were given in lectures, but most people give the first and then mention the second more as an aside.
Definition 1. Let be a group and let be a set. An action of on is a function with the following properties: for every , and for every and every .
Definition 2. Let be a group and let be a set. Let be the group of all permutations of . An action of on is a homomorphism .
I much prefer the second of these, because I find it far more intuitive: an action of a group is a way of thinking of the elements of as symmetries of some sort. It’s tempting to say that an action of is a way of regarding as a symmetry group, but that’s not quite correct, because we allow different elements of to give us the same symmetry. For example, here’s an action of the permutation group on the set . There are two permutations of , namely the identity and : map all even permutations to the identity and all odd permutations to .
What do we have to check in order to be sure that that is an action? If and are elements of and and are the associated permutations of , then we need to equal . That is an easy consequence of facts about what you get when you multiply even and odd permutations together, which correspond closely to facts about what happens when you multiply the identity and together. (For example, odd times odd is even, and identity.)
If is a homomorphism from to , that means that for each , is a permutation of . (Here I am doing something easy but important: making sure I am clear in my mind about what kinds of objects things are. It’s a very good habit to get into.) That means that I will find myself writing slightly odd expressions like : since is a permutation of , which is in turn a kind of function from to , I must be able to apply to elements of .
If one is careful, it can be nice to imagine that the elements of themselves are “doing the transformation” to . That is, instead of turning into a bijection, which in turn does things to elements of , we allow ourselves (once we have carefully defined what the action is) to write expressions like , and simply understand that this is shorthand for . Then the main property that an action has to have is that is the same as for every and every . (The first of these means you multiply and and then apply the transformation that corresponds to , whereas the second means that you apply the transformation that corresponds to and then the transformation that corresponds to .)
There is one example that is a good one to have in your head, as it gives you a very good idea of what an action is. Let be the alternating group and let be a regular tetrahedron, and label its vertices 1, 2, 3 and 4. For each permutation in and each position of , we can find a rotation that permutes the vertices of according to . For example, to achieve the permutation we do a half turn about the line that joins the midpoints of the edges linking 1 to 2 and 3 to 4, and to achieve the permutation we do rotation through 120 degrees through the line that joins vertex 4 to the middle of the opposite face. (Note that these axes depend on the position of rather than being fixed. See the discussion in the post on permutations.)
The action just described is a faithful action, which means that different elements of the group correspond to different transformations. (More formally, the homomorphism from to is an injection.) However, we can also use this set-up to define a second action of . Let be the set that consists of the three lines that join midpoints of opposite edges. (That is, is a set with three elements, each of which is a line.) Then any rotation of the tetrahedron will also permute these three lines, so acts on . This action is not faithful: for example, a half turn about one of the lines fixes all three lines (the other two rotate through 180 degrees but they still map to themselves). In a later post, we shall see that this gives us a very clear explanation of an important fact about the group .
I want to end this post by elaborating on what I meant by “defining a group in an abstract way”. You should by now have met the dihedral groups. The dihedral group of order can be defined in two rather different ways. The first way is concrete: it is the group of symmetries of an -gon. By that I don’t really mean that its elements have to be symmetries of -gons, but rather that any group that is isomorphic to the group of symmetries of the -gon counts as an instantiation of the abstract group .
Another way of defining groups is by using generators and relations. This is called giving a presentation of the group. In the case of the group the usual presentation uses two generators, and , say, and the relations and . It’s not hard to use these relations to reduce every product you can make out of , , and to an element of the form or with . For example, if (so we are talking about the symmetry group of the pentagon), and I take the product , then I can do a series of obvious simplifications as follows. First, since , I can change every power of to either or . If I do that, I get . In a similar way, I can change all powers of so that they are between 0 and 4. That gets me to . Thirdly, the relation implies that . That is, I can move an from the right of a to the left of that if I change the to a . From that it follows that I can do the same with powers of . For example,
Going back to the expression , I can use the above fact to change it to
Applying the little fact again, we get
So we have a simple algorithm for putting all “words” (that is, expressions made out of , , and ) into a standard form. With a bit more effort, one can show that no two expressions in standard form are equal. For example, if we would deduce that (by multiplying both sides on the right by ), which is false.
Actually, why is it false? How can we be sure that there isn’t some strange way of using the relations and to show that ? One quick answer is that we can find a concrete group — the symmetry group of a pentagon — and two elements of that group — one reflection and one rotation — that satisfy the given relations. If those relations implied that then they would enable us to deduce that a reflection of the pentagon was equal to a rotation of the pentagon, which is just plain false.
That same argument shows that has at least 10 elements, and since it has at most 10 elements (since there are 10 distinct standard forms) it has exactly 10 elements. By using the standard-form algorithm we can build up the multiplication table. For example, , and so on.
So now we have two ways of thinking about . Either it is the group with generators and and relations and , or it is the group of symmetries of an -gon.
The main point of what I want to say in this section is that there is a danger that you will become too keen on abstraction. Certain facts about are obvious if you think of it as a symmetry group and quite a lot less obvious if you argue directly from a presentation. For example, contains (a copy of) as a subgroup. One proof of that fact consists in arguing as follows. Let and be the generators of and let and be the generators of . Then the function that takes to and to is an isomorphism from to its image, which is a subgroup of . Checking this is slightly fiddly, though not too hard.
How much more transparent, however, is the following argument. is the group of symmetries of a regular hexagon. If you join alternate vertices of the hexagon you get an equilateral triangle. All the symmetries of that triangle are given in an obvious way by symmetries of the hexagon, so the symmetry group of the triangle is a subgroup of the symmetry group of the hexagon.
I shall have more to say about group actions, but for now I’ll content myself with this message (which I’ve said a few times, but let me say it once more).
Abstraction is great, but don’t get too carried away with it. In particular, if you know that a group is isomorphic to a group of symmetries, that gives you direct access to a lot of information about it. Don’t throw that information away (unless for some reason you like complicated fiddly proofs that don’t tell you why a result is true).