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Homomorphisms of Groups Homework

Directions: Write your homework legibly with a pencil. Complete the homework on time and by yourself. For each problem, write the instructions, label the solution/proof and show all steps. Do not turn in your scratch work. Staple your pages together, with the problems in the correct order, and use this page as a cover sheet with your name at the top right.

(1) Consider as a group with respect to addition. Define by for each where Prove that is a homomorphism.

(2) Define by for each Prove that is a well-defined mapping, a homomorphism, and then find the kernel of

(3) Prove that every homomorphic image of an Abelian group is Abelian.

(4) Prove that every homomorphic image of a cyclic group is cyclic.

(5) Prove that if is a homomorphism and is a subgroup of then is a subgroup of

(6)  Prove that if is a homomorphism and is a subgroup of then is a subgroup of

(7) Prove that if is a homomorphism and is a homomorphism, then is a homomorphism.

(8) Let and denote positive integers, and define by for each Prove that is a homomorphism.

(9) Prove that if and only if for all and

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