Abstract Algebra

Problems taken from Chapter 3 - Groups.

Problems

1. Problem 7: You must first show that $(S,\ast )$ 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 $det(A)\ne 0$.
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