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).