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Permutations 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) Assume and compute and

(2) Assume and compute and

(3) Write each of the following as a single cycle or a product of disjoint cycles?


    

(4) Write each of the following as a single cycle or a product of disjoint cycles?

(5) How many elements of map to How many elements of map to

(6) Using cycle notation prove is non-Abelian when

(7) Show that every element of is a two-cycle or can be written as a product of two-cycles.

(8) Prove that if contains at least three elements, then in non-Abelian.

(9) If then composition, as an operation on is not commutative. However, is Abelian. Explain.

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