for x. Solve a Ixa = c for x. 36. Let a and b belong to a group G. Find an x in G such that xabx-1 = ba. 37. Let G be a finite group. Show that the number of elements x of G such that x3 = e is odd. Show that the number of elements x of G such that x2 # e is even. 38. Give an example of a group with elements a, b, c, d, and x such that axb = exd but ab # cd. (Hence "middle cancellation" is not valid in groups.) 39. Suppose that G is a group with the property that for every choice of elements in G, axb = exd implies ab = cd. Prove that G is Abelian. ("Middle cancellation" implies commutativety.) 40. Find an element X in D, such that Ro VXH = D'. 41. Suppose F, and F, are distinct reflections in a dihedral group D,. Prove that F F2 # Ro. 42. Suppose F, and F, are distinct reflections in a dihedral group D,, such that F F, = F,F . Prove that F F2 = R 180- 43. Let R be any fixed rotation and F any fixed reflection in a dihedral group. Prove that RkFRk = F. 44. Let R be any fixed rotation and F any fixed reflection in a dihedral group. Prove that FR*F = R . Why does this imply that D,, is non-Abelian? 45. In the dihedral group D,,, let R = R360/ and let F be any reflection. Write each of the following products in the form Ri or R'F, where Osi<.>