10-1 Through 10-20 Feedback
• We are getting a little lax in signing up on the To Do pages after class. Please make sure you sign up for what you plan to contribute each day.
• HW8 problem 3, $3+\langle 3\rangle =0+\langle 3 \rangle$
• HW8 problem 10, in this case, $|H\cap K=gcd(|H|,|K|)$, but isn’t always the case: Consider $D_8$. Using the notation from the book, $H=\{\mu_1, \rho_0\}$ and $K=\{ mu_1, \rho_0\}$ are both subgroups of order 2, but $H\cap K=\{\rho_0\}$ has order 1.
• HW8 problem 10, you left out the case $G$ contains no elements of order 4 or 8. Then all nonidentity elements have order 2.
• we still need a formal definition of addition modulo n
• Lemma Property of Cosets: I corrected the proof of part (3) to not assume the group is abelian. In the proof of part (6), it’s unclear whether $\phi$ is well-defined. It would be better to prove this part as we did in class, by showing that $|aH|=|H|$.
• HW8 Problem 10: Your generalization isn’t correct. Consider the subgroups $H=\{\rho_0, \mu_1\}$ and $K=\{\rho_0, \mu_2\}$. Then $K\cap H = \{\rho_0\}$ has size 1, but $\gcd(|H|, |K|) =2$.
• HW8 Problem 3: The coset $3+\langle 3\rangle$ is unnecessary, as it is equal to $0+\langle 3\rangle$.
• HW5 Problem 1(a) The set given is not equal to $SL(n, \mathbb{R})$, as $SL(n, \mathbb{R}) = \{A \in GL(n, \mathbb{R}) \mid \det A =1 \}$
• HW7 Problem 11 is missing the problem statement.
• Theorem 11.12 still needs to be added
• HW9 Problem 2: The $\textrm{lcm}(15,25)=75$, not 150. This should be corrected. Also, you can link to Theorem 11.5 in this problem.
• We didn’t discuss decomposable and indecomposable in class, so I will not expect you to know these for any quizzes or exams.
• Nice use of the underbrace in the proof of Theorem 11.5!