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 credits 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.
1.\(S_3\) is the Only Non-Abelian Group of Order 6.
Let \(G\) be a non-Abelian group of order 6. Use a group action of \(G\) on the cosets of a carefully chosen subgroup of \(G\) to prove that \(G\cong S_3\text{.}\)
Problem Specs/Notes: This problem must be done with a group action for a Success, though there are many proofs of the result that do not use group actions.
Let \(G\) be a group of order \(pq\) where \(p\) and \(q\) are distinct primes with \(p\lt q\text{.}\) Prove that if \(p\) does not divide \(q-1\text{,}\) then \(G\) is cyclic.
Problem Specs/Notes: This problem needs careful use of the Sylow Theorems and our knowledge of properties of the set \(HK\) when \(H,K\leq G\) for a Success.
Suppose that \(R\) is a ring and that \(a^2=a\) for all \(a\in R\text{.}\) Prove that \(R\) is commutative. Such rings are called Boolean rings after the English mathematician George Boole (1815-1864) who developed the algebra of logic. Give an example of a Boolean ring with four elements and example of an infinite Boolean ring.
Prove that there is no integral domain with exactly six elements. Can your argument be adapted to show that there is no integral domain with exactly four elements? What about 15 elements? Use these observations to conjecture a general result about the number of elements in a finite integral domain.
An integral domain \(D\) is called a principal ideal domain (PID) if every ideal of \(D\) has the form \(\langle a\rangle =\{ad\mid d\in D\}\) for some \(a\) in \(D\text{.}\) Show that \(\Z\) is a principal ideal domain.
Problem Specs/Notes: This problem needs care not to accidentally assume the conclusion for a Success. It will be helpful to find use the division algorithm on this problem.
