Homework Assignment 10

Problems to try

Please try to solve these problems, but you do not turn them in. You can also use these results in future problems. I’m happy to talk about these problems (or any others) if you have questions.

• Section 4.4: 1, 3, 6, 12
  • Two things regarding #1: (1) The book is using $\varphi_g$ for what we are denoting by $\gamma_g$. (2) When proving $\sigma\varphi_g\sigma^{-1} = \varphi_{\sigma(g)}$, you are proving that they are equal as functions, so you need to show that $\forall x \in G$, $\sigma\varphi_g\sigma^{-1} (x) = \varphi_{\sigma(g)}(x)$.
• Section 4.5: 13, 18, 33, 34

Additional Problem. Prove that there are no simple groups of order 144.

This is a fun one that brings many things together. Here’s a possible path. Look at the possibilities for $n_3$. Remember that if $n_3$ is small, then $N_G(P)$, for $P\in \operatorname{Syl}_3(G)$, has small index, so in this case you should investigate the core of $N_G(P)$. Try to force $n_3$ to be 1 or 16. Of course if $n_3 = 1$, then you are done. If $n_3 = 16$, try counting elements to show that there must exist distinct $P,Q\in \operatorname{Syl}_3(G)$ such that $P\cap Q$ is nontrivial. Now investigate $N_G(P\cap Q)$. Try to show it has small index in $G$, and then consider the core of $N_G(P\cap Q)$.

---

Problems to submit

These are the problems to turn in for a grade. Please write in complete sentences, use correct punctuation, and organize your work. Once your finish these problems, please follow the directions in the Canvas assignment to submit them. If you have any questions (about the math or writing or submission process or anything), please let me know!

1. Section 4.5: 16

   Try counting elements.

2. Section 4.5: 24

3. Section 4.5: 32