Instructions: You may type up or handwrite your work, but it must be neat, professional, and organized and it must be saved as a PDF file and uploaded to the appropriate Gradescope assignment. Use a scanner or scanning app to convert handwritten work on paper to PDF. I encourage you to type your work using the provided template.
All tasks below must have a complete solution that represents a good-faith attempt at being right to receive engagement credits. If your submission is complete and turned in on time, you will receive full engagement credit for the assignment. All other submissions will receive zero engagement credit. Read the guidelines at Grading Specifications carefully.
To abide by the class academic honesty policy, your work should represent your own understanding in your own words. If you work with other students, you must clearly indicate who you worked with in your submission. The same is true for using tools like generative AI although I strongly discourage you from using such tools since you need to build your own understanding here to do well on exams.
For these problems a justification is not required for credit, but it may be useful for your own understanding to include one. True/False problems should be marked True if the statement is always true, and False otherwise. Multiple choice problems may have more than one correct answer if that is indicated in the problem statement; be sure to select all that apply. Fill-in problems require a short answer such as a number, word, or phrase.
Let \(H=\{z \in \C^* \mid |z|=1\}\text{.}\) Use the First Isomorphism Theorem to show that \(\C^*/H\) is isomorphic to \(\R^+\text{,}\) the group of positive real numbers under multiplication.
Problem Specs/Notes: This problem needs a good choice of homomorphism, including clear justification of why your map is a homomorphism, why its kernel is \(H\text{,}\) and why the image is \(\R^+\) for a Success. Solutions that only show the isomorphism class of \(\C^*/H\) without explicitly using the First Isomorphism Theorem will not receive a Success.
How many homomorphisms are there from \(\Z_{380}\) to \(\Z_{252}\text{?}\) Use Lagrange’s Theorem and the First Isomorphism Theorem to fully justify your answer.
Problem Specs/Notes: This problem needs a clear justification of the number of homomorphisms invoking the relevant theorems for a Success. It may be useful to factor the numbers into their prime factorizations.
