Exercises 6

1. The embedding of a regular tetrahedron in a cube shown on the right, gives a map from the full symmetry group of the tetrahedron: S(T) ≅ S₄ to the full symmetry group of the cube: S(C) ≅ S₄ × < J > .
   Identify the image and kernel of this homomorphism.

   Solution to question 1

2. Show that the 60 indirect symmetries of the dodecahedron consist of 15 reflections and 45 rotatory reflections.

   Solution to question 2

3. Show that the groups C₂ , C₁ × < J > and C₂ C₁ are isomorphic but that they are not conjugate subgroups of I(R³).

   Solution to question 3

4. Describe figures in R³ whose direct symmetry groups are each of the examples in the classification of rotational symmetry groups and which have no opposite symmetries.
   [Hint: think about "painting designs" on the faces of some of the figures which do have opposite symmetries.]

   Describe figures whose full symmetry groups are G × < J > with G each of the possible direct symmetry groups and where J is central inversion.

   Solution to question 4

5. Describe a figure in R³ whose full symmetry group is the mixed group D_nC_n.
   Find a figures whose full symmetry group is D_2nD_n.
   Modify this last two example to find a figure with full symmetry group C_2nC_n.

   Solution to question 5

6. Figures are formed by placing two identical oblong tiles on top of each other with their centres in the same vertical line.
   If the angles between their long axes are as shown:
   (i) 0 (ii) π/2 (iii) π/3,
   find the orders of their direct and full symmetry groups and identify these groups.
   If the upper tile is coloured white and the lower one is coloured black, determine the direct and full symmetry groups in the three cases.

   Solution to question 6

7. The group of rotational symmetries of the cube/octahedron is isomorphic to S₄. Which symmetries correspond to the alternating subgroup A₄?
   Every symmetry of the cube determines a permutation of the 6 faces of the cube and hence gives a group homomorphism into the permutation group S₆. Write down the permutations corresponding to rotation by π/2 about centres of faces, π about centres of edges and 2π/3 about diagonals.
   Which elements of S₆ correspond to the opposite symmetries of the cube?

   Solution to question 7