Homework Assignment 08

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 3.5: 3, 4, 6, 14, 15, 17
  • For #3 and #4, make use of the fact that we already know that the transpositions generate $S_n$ together with what we learned about conjugation in $S_n$ (see Proposition 10 in Section 4.1). The subgroup lattice for $A_4$ in the book should make #14 and #15 go quickly.
• Section 4.2: 4
  • To clarify, you want to use the left regular representation of $Q_8$ to explicitly find two elements $a,b\in S_8$ such that $\langle a, b \rangle \cong Q_8$.
• Section 4.3: 2, 4, 5, 8, 9, 10, 24
  • For #4, try to think in terms of group actions, and then make use of Exercise 1 from Section 4.1. Also, #8 is a good one to know even if you don’t prove it.

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 3.5: 16

   You are encouraged to make use of Exercise 15 from Section 3.5 (without including a proof of it).

2. Section 4.3: 23

3. Section 4.3: 30

   Suppose $x$ and $x^{-1}$ are conjugate. Then there exists some $g\in G$ such that $gxg^{-1} = x^{-1}$. What can you say about $g^2$?