Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 1693. The set of all inner automorphisms is denoted by Inn(G). Show that Inn(G)is a subgroup of Aut(G).4. Find an automorphism of a group G that is not an inner automorphism.5. Let G be a group and i g be an inner automorphism of G, and define a mapbyG → Aut(G)g ↦→ i g .Prove that this map is a homomorphism with image Inn(G) and kernel Z(G).Use this result to conclude thatG/Z(G) ∼ = Inn(G).6. Compute Aut(S 3 ) and Inn(S 3 ). Do the same thing for D 4 .7. Find all of the homomorphisms φ : Z → Z. What is Aut(Z)?8. Find all of the automorphisms of Z 8 . Prove that Aut(Z 8 ) ∼ = U(8).9. For k ∈ Z n , define a map φ k : Z n → Z n by a ↦→ ka. Prove that φ k is ahomomorphism.10. Prove that φ k is an isomorphism if and only if k is a generator of Z n .11. Show that every automorphism of Z n is of the form φ k , where k is a generatorof Z n .12. Prove that ψ : U(n) → Aut(Z n ) is an isomorphism, where ψ : k ↦→ φ k .