Instructor: Laszlo Babai
Scribe: Mridul Mehta
We focus our attention on automorphism groups of regular solids. A tetrahedron is just another way of representing the complete graph on 4 vertices i.e., . The automorphism group . More generally, we have , where is the complete graph on vertices.
Next, we consider a cube. We may label the vertices of the cube so that the four in the top face are and , while the diagonally opposite ones in the bottom face are and respectively. cube . Here the nonidentity element of corresponds to the central involution (i.e., reflection about the center of the cube) which is orientation reversing, while elements of correspond to the orientation preserving automorphisms of the cube.
By considering the action on pairs of opposite vertices of the cube (the main diagonals of the cube) we obtain the map Autcube. The kernel of this map is central reflection. The image is all of , so the map is onto. To see this, we define as a rotation about the vertical axis of the cube, and as the corresponding element of induced by . Then permutes the main diagonals so that mod . Similarly, if we let be rotation about the diagonal by , then the induced permutation permutes the other three diagonals. The next exercise completes the argument.
Let be a graph with connected components and for all . Then Aut .
If and , then the wreath product naturally acts on ( ). Here , the th component of acts on , and permutes the . In this case, (where is the degree of ). This is called the ``imprimitive representation'' of .
Let be the graph of the -cube. So , and we may think of the vertices of this graph as elements of , i.e., strings of length consisting of 0s and s. We define the Hamming distance between two strings of equal length to be the number of places where they differ. Two vertices in will be adjacent if their Hamming distance is 1.
Switching coordinates of vertices independently corresponds to reflections about various planes which shows that . Similarly, permuting the positions of coordinates corresponds to rotations about different lines, which gives us .
The ``primitive representation'' of is the action of on given by .
The octahedron is the dual of the cube. Hence its automorphism group is the same as that of the cube.
AutDodecahedron . Here, as before, the element in is the central reflection and is the subgroup of the orientation preserving automorphisms.