Read: Start Section 3.3.
Turn in: 3.32, 3.34, 3.37, 3.41, 3.42
- For 3.32, you do not need to do the part comparing the subgroup lattice for $S_3$ with the one for $D_3$.
- The discussion of “matchings” in the book is a little bit vague, but it’s helpful in building intuition for the idea of “isomorphism”, which is coming soon. Let me know if you have questions.
- I think 3.37 is tough, but it is possible to find a matching if you choose the correct generating sets. Note that we have seen the Cayley diagram for $D_4$ with respect to the generating set $\{r,s\}$ and the Cayley diagram for $\operatorname{Spin}_{3\times 3}$ with respect to the generating set $\{s_{11}, s_{22}, s_{12}\}$ (see Figure 2.3 on page 34), but you’ll need to find a different generating set for one of the groups in order to produce a matching. Try finding a generating set for $D_4$ that consists of two reflections and then draw the corresponding Cayley diagram.
Extra practice: 3.35, 3.38–3.40
- Regarding 3.38–3.40, if you think there is a matching between the two groups, you should justify this by choosing a generating set for each of the groups, and showing that the Cayley diagrams can be matched up. If you think there is not a matching, then you should explain why not. In this case, the two paragraphs right before Problem 3.35 are really helpful, and you can make use of them even though they have not been proven.
