Abstract Algebra

Problems taken from Chapter 3 - Groups.

Problems

1. Problem 7: You must first show that \((S,*)\) is a group and then prove that it is abelian.
2. Problem 11: Hint - Recall from linear algebra that \(A\) is invertible if and only if \(\det(A)\neq 0\).
3. Problem 26: Hint - Write out the multiplication table for \(U(n)\) for a few values of \(n\), for example, \(n=3\), \(n=4\), and maybe even \(n=5\). You should see a pattern. Then prove that the pattern that you see holds for any \(n\). I don't need to see your multiplication tables but only the general proof. Proof by example is not a proof.
4. Problem 27
5. Problem 33