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Quotient 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 order of the quotient groups and where

(2) Determine the order of the quotient groups and

(3) Construct the Cayley table for

(4) Prove that every quotient group of an Abelian group is Abelian.

(5) Prove that every quotient group of a cyclic group is cyclic.

(6) Assume (i) Prove that if is prime, then is cyclic. (ii) Prove or disprove the converse of (i).

(7) Prove that if and then where denotes the order of the element of

(8) Prove that is Abelian if and only if for all

(9) Prove that if is a subgroup of and the operation is well-defined on the set of all right cosets of then

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