3. Consider the map o : (Z10, +) -gt; (Z10, +) such that o(x) = 2x. (a) Show that o is a homomorphism. (b) Find kero and Imo. (c) Give a reason why

3. Consider the map φ : (Z10, +) → (Z10, +) such that φ(x) = 2x.

(a) Show that φ is a homomorphism.

(b) Find kerφ and Imφ.

(c) Give a reason why ker φ Z10.

(d) Find a integer k for which the map θ is an isomorphism, where 

θ : (Z10, +) → (Z10, +) 

x → kx

(e) Suppose there is a homomorphism f : (Z10, +) → (Z11, +).

(i) What are the possible values for the order of ker f?

(ii) What are the possible values for the order of Im f?

(iii) What can you deduce about f? (Give reasons for your answers.) 

3. Consider the map o : (Z10, +) -> (Z10, +) such that o(x) = 2x.(a) Show that o is a homomorphism.(b) Find kero and Imo.(c) Give a reason why ker d 4 Z10.(d) Find a integer k for which the map 0 is an isomorphism, where0 : (210, +) -> (210, + )x > kx(e) Suppose there is a homomorphism f : (Z10, +) -> (Z11, +).(i) What are the possible values for the order of ker f?(ii) What are the possible values for the order of Im f?(iii) What can you deduce about f?(Give reasons for your answers. )