Unformatted text preview: 5. Let G be the symmetry group of a cube (with no re²ections allowed), i.e. the group of space rotations which preserve a cube. Show that G ≃ S 4 . Hint: a cube has four diagonals which connect opposite vertices and go through the center of the cube. Any element of G induces a permutation of the set of diagonals. It is enough to show that every permutation of the set of diagonals is realized by some element of G . (Why? What is the order of G ?) (If it helps, you can use without proof the fact that an isometry of space which ±xes the origin and is represented by a matrix of determinant +1 is a rotation about some axis). 6. How challenging did you ±nd this assignment? How long did it take?...
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- Spring '08