Sep 22
homework
Discuss the homework:
1. S4 and the octahedral group (symmetries of a solid cube)
- are the groups the the same?
- what does it mean for two groups to be the same anyway?
Aside:
- a) C4 : Numbers and addition mod 4 : (0, 1, 2, 3) with + and wrap around
- b) rotational symmetries of an rectangle with different width/height.
- Both have 4 elements. The same group? Why or why not?
2. Topspin. We've done most of what I want to do with it ... any questions?
3. 2 x 2 x 2 Rubik's cube as a group
- Can it be written as a permutation? (yes)
- Discuss the cyclic notation from the homework.
Aside: can you do that for the solid cube?
practice
Let's review the connections between the puzzles, groups, and our vocabulary.
- what is an "element of the group" ?
- what is a "position of the puzzle" ?
- how many are there of each?
- what is a "generator" ?
- what is a "commutator" ? (answer 1: commutator of A and B is "A B undo(A) undo(B)"
- what is the "order" of a group element?
- what is a cycle? why do they matter?
- can you give several examples for topspin ?
- why are the commutators interesting? (answer 1 : trying to "do nothing" ... and failing.")
odd and even
Turns out that all permutations are even odd or even, which is defined in terms of the number of pair swaps.
- Point 1: any permutation can be done in terms of pair swaps.
- Point 2: the parity (odd or even) of the pair swaps is fixed for a given permutation.
Do some examples and discuss. (Harder: prove this.)
What implications does this have for Rubik's Cube? (What is being permuted, and what is the parity of the generators?)
For TopSpin with 20 pieces? 19 pieces? Is flip odd or even? Is a left rotation odd or even?
similarity
What does it mean for two group elements to be similar? Discuss.
Technical definition: A and B are similar if there is an X such that
A = X B undo(X)
What's that all about, and what's the intuition? Turns out this gets used lots in solving puzzles. Once you have an operation that does something interesting (perhaps a repeated commutator), you often do something like it by
(a) putting the pieces you want to move into the places that get changed
(b) doing the operator move sequence
(c) undoing (a).
Talk about the rotations of a solid cube in those terms.
counting groups with small number of elements
Class exercise:
Groups with 1 element? 2? 3? ...
This goes back to notions of "sameness" between groups.
Basic idea: two groups are the same if you can match up the elements so that the group operation table looks the same.
What pattern is there between the small groups and the size of the group?
"nesting" of group within each other
Turns out this happens in several different ways.
And you can use this to build new groups from old groups.
Example: how many groups are there with 6 elements?
And what are the differences between them?
Do some handwaving to introduce the ideas of
- subgroups
- factor groups ... and how those tie into solving puzzles.
what's next
Let's explore the 2 x 2 x 2 cube (i.e. the corners of the rubik's cube).
Work through it as a group theory thing
(warning: can be defined a few different ways)
- generators?
- commutators?
- size of the group?
- interesting operators?
