lesson5proof.html

1. Recall that one of you made a conjecture for the symmetries of mandalas, about broken patterns having symmetries which divide original number of symmetries. Today we will prove this.

2. What would this mean in terms of subgroups of a group? Let's look at an example of multiplication tables for both patterns. (Use 6fold rotational symmetry.) So we are trying to show that one number divides another and these numbers are orders of groups. Recast problem in this form.

3. Groupwork: In groups of three, strategize for five minutes about how you might show that 3 divides 15.

4. Then, we will talk about how to divide up the group of size 6 into pieces of equal size 3 using the subgroup. We do a specific case on left, the general case on right. At some point we define coset. How do we know the different pieces don't overlap? How do we know all the elements in one of the sets are distinct? So that we have counted everything exactly once. Of course, this is the gist of the proof.

5. Take five to write it down.