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Cayley's Theorem 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) Write the permutation associated with each element of by the isomorphism in the proof of Cayley's theorem.

(2) Write the permutation associated with each element of the symmetry group of the square by the isomorphism in the proof of Cayley's theorem.

(3) Write the permutation associated with each element of by the isomorphism in the proof of Cayley's theorem.

(5) Prove Cayley's theorem.

(6) Assume that is a group, and that and are defined by and for each For each define by Define by for each

    (a) Prove that each is an isomorphism of onto
    
    (b) Prove that if and only if for all

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