Homework 06

Read: Section 2.5, and start of Section 2.6.

• There is a lot to read in Section 2.6, so it’s good to get a start now. I’m only asking you to work up to 2.67, but if you can find some extra time, I recommend reading more now (so you have less to read next time).

Turn in: 2.61(choose 2 of the 3 parts), 2.63, 2.64, 2.65, 2.67

• For 2.65, you should be able to create a proof without making a table. Try something like this:

  “Suppose that the element $b$ appears twice in the row corresponding to $g$. This means that there are two columns, that correspond to two group elements $h_1$ and $h_2$, such that…” Complete the sentence and keep going to get a contradiction.

• For 2.67, remember that earlier we found that the order of $\operatorname{Spin}_{3\times3}$ (i.e. the number of net actions) is equal to the number of scrambled Spinpossible boards since each net action can be associated to the board that results when the action is applied to the solved board. The same principle applies to $\operatorname{Spin}_{1\times2}$.

Extra practice: 2.66