Question

We define SO(3) to be the group of 3 × 3 orthogonal matrices whose determinant is 1. This is the group of rotations in three-space, and you can visualize each element as a rotation about some axis by some angle.

(1) Check that SO(3) satisfies the three axioms of a group. You may take for granted that matrix multiplication is associative, as well as any standard properties of transposes and determinants.

(2) Prove that any reflection about a two-plane (or rather the matrix representation of such a linear transformation) is not included in SO(3). Hint: what is the determinant of such a matrix? Note that after a change of basis you can take the plane to be the xy-plane.

(3) Show that SO(3) is nonabelian.

(4) Consider the cube in R 3 whose set of eight vertices is {(i, j, k) : i, j, k ∈ {1, −1}}. Let H ⊂ SO(3) be the subgroup consisting of those rotations which map this cube to itself setwise.3 What is the order of H? You should provide some justification for your answer.

(5) Observe that each element of H determines a permutation of the set of 8 vertices. Give an example of such a permutation which does not arise from an element of H. Note: if you wish to identify such a permutation with an element of S8, you will first need to choose an ordering of the vertices.

(6) Similarly, each element of H determines a permutation of the set of 6 faces. Give an example of such a permutation which does not arise from an element of H.

Answer 1

The solution is given below. For question a, notice that determinant of a matrix and its transpose is equal. For othogonal matrices the transpose is the inverse. This is enough to show that SO3 is a group.

For question b, we change the basis and find the matrix with respect to the new basis. Finding the determinant of the new matrix is much easier and it equals the original determinant since the two matrices are similar.

For question c, there is an explicit counterexample.

Question d is by a counting argument described in the solution.