Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 8715. If the diagonals of a cube are labeled as Figure 4.5, to which motion ofthe cube does the permutation (12)(34) correspond? What about the otherpermutations of the diagonals?16. Find the group of rigid motions of a tetrahedron. Show that this is the samegroup as A 4 .17. Prove that S n is nonabelian for n ≥ 3.18. Show that A n is nonabelian for n ≥ 4.19. Prove that D n is nonabelian for n ≥ 3.20. Let σ ∈ S n . Prove that σ can be written as the product of at most n − 1transpositions.21. Let σ ∈ S n . If σ is not a cycle, prove that σ can be written as the productof at most n − 2 transpositions.22. If σ can be expressed as an odd number of transpositions, show that anyother product of transpositions equaling σ must also be odd.23. If σ is a cycle of odd length, prove that σ 2 is also a cycle.24. Show that a 3-cycle is an even permutation.25. Prove that in A n with n ≥ 3, any permutation is a product of cycles oflength 3.26. Prove that any element in S n can be written as a finite product of the followingpermutations.(a) (12), (13), . . . , (1n)(b) (12), (23), . . . , (n − 1, n)(c) (12), (12 . . . n)27. Let G be a group and define a map λ g : G → G by λ g (a) = ga. Prove thatλ g is a permutation of G.28. Prove that there exist n! permutations of a set containing n elements.29. Recall that the center of a group G isZ(G) = {g ∈ G : gx = xg for all x ∈ G}.Find the center of D 8 . What about the center of D 10 ? What is the center ofD n ?30. Let τ = (a 1 , a 2 , . . . , a k ) be a cycle of length k.(a) Prove that if σ is any permutation, thenis a cycle of length k.στσ −1 = (σ(a 1 ), σ(a 2 ), . . . , σ(a k ))