Section 1.5. Homomorphisms 45 (i) Prove that if a ∈ G has infinite order, then so does f(a), and if a has finite order n, then so does f(a). Conclude that if G has an element of some order n and H does not, then G ∼ = H. (ii) Prove that if G ∼ = H, then, for every divisor d of |G|, both G and H have the same number of elements of order d. (iii) If a ∈ G, then its conjugacy class is {gag−1 : g ∈ G}. If G and H are isomorphic groups, prove that they have the same number of conjugacy classes. Indeed, if G has exactly c conjugacy classes of size s, then so does H. 1.50. Prove that A4 and D12 are nonisomorphic groups of order 12. 1.51. (i) Find a subgroup H of S4 with H = V and H ∼ = V. (ii) Prove that the subgroup H in part (i) is not a normal subgroup. 1.52. Let G = {x1,.. . , xn} be a monoid, and let A = [aij ] be a multiplication table of G that is, aij = aiaj . Prove that G is a group if and only if A is a Latin square, that is, each row and column of A is a permutation of G. ∗ 1.53. Let G = {f : R → R : f(x) = ax + b, where a = 0}. Prove that G is a group under composition that is isomorphic to the subgroup of GL(2, R) consisting of all matrices of the form [ a b 0 1 ]. ∗ 1.54. (i) If f : G → H is a homomorphism and x ∈ G has order k, prove that f(x) ∈ H has order m, where m | k. (ii) If f : G → H is a homomorphism and (|G|, |H|) = 1, prove that f(x) = 1 for all x ∈ G. 1.55. (i) Prove that cos θ − sin θ sin θ cos θ k = cos kθ − sin kθ sin kθ cos kθ . Hint. Use induction on k ≥ 1. (ii) Prove that the special orthogonal group SO(2, R), consisting of all 2 × 2 orthog- onal matrices of determinant 1, is isomorphic to the circle group S1. (Denote the transpose of a matrix A by A if A = A−1, then A is orthogonal.) Hint. Consider ϕ : cos α − sin α sin α cos α → (cos α, sin α). 1.56. Let G be the additive group of all polynomials in x with coeﬃcients in Z, and let H be the multiplicative group of all positive rationals. Prove that G ∼ = H. Hint. List the prime numbers p0 = 2,p1 = 3,p2 = 5,... , and define ϕ(e0 + e1x + e2x2 + · · · + enxn) = p00 e · · · pnn e . ∗ 1.57. (i) Show that if H is a subgroup with bH = Hb = {hb : h ∈ H} for every b ∈ G, then H must be a normal subgroup. (ii) Use part (i) to give a second proof of Proposition 1.68(ii): if H ⊆ G has index 2, then H G. 1.58. (i) Prove that if α ∈ Sn, then α and α−1 are conjugate. (ii) Give an example of a group G containing an element x for which x and x−1 are not conjugate.

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