Abstract Algebra Theory and Applications - Computer Science ...

218 CHAPTER 12 GROUP ACTIONS10. Find the number of ways a six-sided die can be constructed if each side ismarked differently with 1, . . . , 6 dots.11. Up to a rotation, how many ways can the faces of a cube be colored withthree different colors?12. Consider 12 straight wires of equal lengths with their ends soldered togetherto form the edges of a cube. Either silver or copper wire can be used for eachedge. How many different ways can the cube be constructed?13. Suppose that we color each of the eight corners of a cube. Using threedifferent colors, how many ways can the corners be colored up to a rotationof the cube?14. Each of the faces of a regular tetrahedron can be painted either red or white.Up to a rotation, how many different ways can the tetrahedron be painted?15. Suppose that the vertices of a regular hexagon are to be colored either red orwhite. How many ways can this be done up to a symmetry of the hexagon?16. A molecule of benzene is made up of six carbon atoms and six hydrogenatoms, linked together in a hexagonal shape as in Figure 12.4.(a) How many different compounds can be formed by replacing one or moreof the hydrogen atoms with a chlorine atom?(b) Find the number of different chemical compounds that can be formedby replacing three of the six hydrogen atoms in a benzene ring with aCH 3 radical.HH❅ H❅❅H❅❅❅HHFigure 12.4. A benzene ring17. How many equivalence classes of switching functions are there if the inputvariables x 1 , x 2 , and x 3 can be permuted by any permutation in S 3 ? What ifthe input variables x 1 , x 2 , x 3 , and x 4 can be permuted by any permutationin S 4 ?