Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 851 22 1434332 1431 24Figure 4.6. Transpositions in the motion group of a cubeProof. From Proposition 4.11, we already know that the motion group ofthe cube has 24 elements, the same number of elements as there are in S 4 .There are exactly four diagonals in the cube. If we label these diagonals 1,2, 3, and 4, we must show that the motion group of the cube will give usany permutation of the diagonals (Figure 4.5). If we can obtain all of thesepermutations, then S 4 and the group of rigid motions of the cube must bethe same. To obtain a transposition we can rotate the cube 180 ◦ about theaxis joining the midpoints of opposite edges (Figure 4.6). There are six suchaxes, giving all transpositions in S 4 . Since every element in S 4 is the productof a finite number of transpositions, the motion group of a cube must be S 4 .□Exercises1. Write the following permutations in cycle notation.(a)( 1 2 3 4 52 4 1 5 3)(b)( 1 2 3 4 54 2 5 1 3)(c)( 1 2 3 4 53 5 1 4 2)(d)( 1 2 3 4 51 4 3 2 5)2. Compute each of the following.