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Generators and Direct Products 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) Verify the equalities for subgroups of : and

(2) Verify the equality for subgroups of :

.

(3) What is the order of

(4) Simplify the expression in

(5) Construct a Cayley table for Show that the group is cyclic.  

(6) Prove that is a subgroup of

(7) Prove that is Abelian if and only if both and are.

(8) Give an example to show that a direct product of two cyclic groups is not necessarily cyclic.

(9) Prove that if then where is the greatest common divisor of and Formulate and prove a generalization.

(10) Prove that if then where is the least common multiple of and Formulate and prove a generalization.

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