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

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

(2) Prove that any reflection about a two-plane (or rather thematrix representation of such a linear transformation) is notincluded 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 thexy-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 subgroupconsisting of those rotations which map this cube to itselfsetwise.3 What is the order of H? You should provide somejustification for your answer.

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

(6) Similarly, each element of H determines a permutation of theset of 6 faces. Give an example of such a permutation which doesnot arise from an element of H.