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- 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*) ≅*S*_{4}to the full symmetry group of the cube:*S*(*C*) ≅*S*_{4}× <*J*> .

Identify the image and kernel of this homomorphism. - Show that the 60 indirect symmetries of the dodecahedron consist of 15 reflections and 45 rotatory reflections.
- Show that the groups
*C*_{2},*C*_{1}× <*J*> and*C*_{2}*C*_{1}are isomorphic but that they are not conjugate subgroups of*I*(**R**^{3}). - Describe figures in
**R**^{3}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. - Describe a figure in
**R**^{3}whose full symmetry group is the mixed group*D*_{n}*C*_{n}.

Find a figures whose full symmetry group is*D*_{2n}*D*_{n}.

Modify this last two example to find a figure with full symmetry group*C*_{2n}*C*_{n}. - 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. - The group of rotational symmetries of the cube/octahedron is isomorphic to
*S*_{4}. Which symmetries correspond to the alternating subgroup*A*_{4}?

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*S*_{6}. 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*S*_{6}correspond to the*opposite*symmetries of the cube?

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