Read: Finish Section 3.2 and start Section 3.3.
Turn in: 3.32, 3.34, 3.37
- 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 a little 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 (say $s$ and $sr$) and then draw the corresponding Cayley diagram. (Remember that we have Jovanny’s Group Table for \(D_4\).)
Extra practice: 3.35