ABSTRACT ALGEBRA Competency 210.4.2: Groups - The graduate analyzes the characteristics of and proves theorems involving groups. Task 5: Homomorphisms Introduction: A group homomorphism is a map between groups that preserves the operation. Identifying homomorphisms is a fundamental skill that is necessary for competence in abstract algebra. Requirements: Your submission must be your original work. No more than a combined total of 30% of the submission and no more than a 10% match to any one individual source can be directly quoted or closely paraphrased from sources, even if cited correctly. Use the Turnitin Originality Report available in Taskstream as a guide for this measure of originality. You must use the rubric to direct the creation of your submission because it provides detailed criteria that will be used to evaluate your work. Each requirement below may be evaluated by more than one rubric aspect. The rubric aspect titles may contain hyperlinks to relevant portions of the course. Given: G is the group of 31st roots of unity under complex multiplication. Z is the group of integers mod 31 under modular addition. 31 2 Let elements in G be represented using exponential notation as 31 where is an integer ranging from 0 to 30. Consider the function defined by ( 2 31 ) = []31 from G -> Z31 . A. Prove that the given function is operation preserving. B. Acknowledge sources, using APA-formatted in-text citations and references, for content that is quoted, paraphrased, or summarized. C. Demonstrate professional communication in the content and presentation of your submission