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Elementary Properties of Groups 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 the elements in each of the cyclic subgroups of Also give the order of each element of

(2) Find the order of the element in

(3) Determine the elements in the subgroup of Determine the elements in the subgroup of What is the order of the subgroup of

(4) Assume that Find a necessary and sufficient condition for

(5) There is only one way to complete the following Cayley table so as to get a group. Find it. Why is it unique?

(6) Prove or give a counterexample: If is a fixed element of a group and is defined by for each then is one-to-one and onto.

(7) Prove or give a counterexample: If is a group and then

(8) Prove or give a counterexample: If is a group and then

(9) Prove or give a counterexample: If a group has a subgroup or order then has an element of order

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