Please answer the following questions. All statements must be explicitly justified.

Introduction: Groups are an Important algebraic structure. Abelian groups have additional structure, and studying roots of unity have that structure. Requirements: Your submission must be your originai work. No more than a combined totai of 30% of the submission and no more than a 10% match to any one individuai source can be directiy quoted or ciosely paraphrased from sources, even if cited correctly. Use the Turnitin Originaiity Report availabie in Taskstream as a guide for this measure of originaiity. You must use the rubric to direct the creation of your submission because it provides detailed criteria that wiii be used to evaiuate your work. Each requirement beiow may be evaiuated by more than one rubric aspect. The rubric aspect tities may contain h yperiinks to reievant portions of the course. Given: A set S with the operation * is an Apelian group If the following five properties are shown to be true: . closure property: For all rand tin S, r*t is also in 5 - commutative property: For all r and t in S, r*t=t*r . identity property: There exists an element e in 5 so that for every 5 In S, s'e=s . inverse property: For every s in 5, there exists an element x in 5 so that s*x=e . associative property: For every q, r, and t in S, q*(r*t)=(q'r)*t A. Prove that the set 6 (the fifth roots of unity) is an Abelian group under the operation * (complex multiplication) by using the definition given above to prove the following are true: 1. closure property 2 commutative property 3. Identity property 4. Inverse property 5 associative property Note: A Cayiey tabie wiii not be suFficient to explain the associative property because this property invoives three arbitrary eiernents, not two. To prove the associative property, express the three arbitrary eiements of G using variabies