Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 15141. Let G be a group and g ∈ G. Define a map i g : G → G by i g (x) = gxg −1 .Prove that i g defines an automorphism of G. Such an automorphism is calledan inner automorphism. The set of all inner automorphisms is denotedby Inn(G).42. Prove that Inn(G) is a subgroup of Aut(G).43. What are the inner automorphisms of the quaternion group Q 8 ? Is Inn(G) =Aut(G) in this case?44. Let G be a group and g ∈ G. Define maps λ g : G → G and ρ g : G → G byλ g (x) = gx and ρ g (x) = xg −1 . Show that i g = ρ g ◦ λ g is an automorphismof G. The map ρ g : G → G is called the right regular representationof G.45. Let G be the internal direct product of subgroups H and K. Show that themap φ : G → H × K defined by φ(g) = (h, k) for g = hk, where h ∈ H andk ∈ K, is one-to-one and onto.46. Let G and H be isomorphic groups. If G has a subgroup of order n, provethat H must also have a subgroup of order n.47. If G ∼ = G and H ∼ = H, show that G × H ∼ = G × H.48. Prove that G × H is isomorphic to H × G.49. Let n 1 , . . . , n k be positive integers. Show thatk∏i=1Z niif and only if gcd(n i , n j ) = 1 for i ≠ j.∼ = Zn1···n k50. Prove that A × B is abelian if and only if A and B are abelian.51. If G is the internal direct product of H 1 , H 2 , . . . , H n , prove that G is isomorphicto ∏ i H i.52. Let H 1 and H 2 be subgroups of G 1 and G 2 , respectively. Prove that H 1 ×H 2is a subgroup of G 1 × G 2 .53. Let m, n ∈ Z. Prove that 〈m, n〉 ∼ = 〈d〉 if and only if d = gcd(m, n).54. Let m, n ∈ Z. Prove that 〈m〉 ∩ 〈n〉 ∼ = 〈l〉 if and only if d = lcm(m, n).