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More on 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) Determine if the groups are isomorphic or not: and

(2) Determine if the groups are isomorphic or not: and

(3) Determine if the groups are isomorphic or not: and the symmetry group of the square.

(4) Is there a non-cyclic group of order Why or why not?

(5) Give examples of two non-isomorphic groups of order

(6) Prove that if is a cyclic group of order then

(7) Prove that if is an infinite cyclic group, then

(8) Prove that the set of all automorphisms of a group is itself a group with respect to composition.

(9) Verify that if is an Abelian group, then defined by for each is an automorphism of

(10) Prove that if is prime, then