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Isomorphism 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) Prove that is isomorphic to the multiplicative group of all rational numbers of the form for

(2) Prove that is isomorphic to the multiplicative group of all rational numbers of the form for

(3) Assume that is a group with respect to multiplication and that is an isomorphism with



Replace each of the following by the appropriate letter,

(4) Use the mapping to show that First verify that is well-defined.

(5) Prove that if and are groups and if and are isomorphisms then is an isomorphism.

(6) Prove that defines an isomorphism, of the group (addition) of all real numbers onto the group of all positive real numbers (multiplication). What is the inverse of the mapping Is the inverse an isomorphism?

(7) Prove that if and are relatively prime, then

(8) Assume that is an isomorphism and is a subgroup of and that Prove that is a subgroup of What does this mean?

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