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Cosets 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 right cosets of in

(2) Determine the right cosets of in

(3) Determine the right cosets of in

(4) Determine the right cosets of in

(5) Verify that if then is a subgroup of (with as the operation in each component). Describe the right cosets of in geometrically.

(6) Prove that if is a subgroup of a group and then the following conditions are equivalent: (i) (ii) for some (iii) and (iv)

(7) Verify that if is a subgroup of an Abelian group and then the right coset of to which belongs is the same as the left coset of to which belongs.

(8) Let and Determine the right cosets and the left cosets of in Then verify that the collection of right cosets is different from the collection of left cosets.

(9) Prove that each right coset of in contains precisely one element from

(10) Prove that a subset of a group cannot be a right coset of two different subgroups of

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