Sep 15
Questions about anything?
Aside :
definitions page is a reference for math vocabulary.
discuss homework
1. Topspin [R^2, F] commutator.
2. Triangle symmetries group D3 mulutiplication table.
3. start thinking about solid cube
- how many elements in the group?
- how many different kinds (describe them) and how many of each kind
- how many generators needed for the group?
permutations
top spin
We looked at to look at commutator of RR and F, which is
[RR, F] = do(RR) do(F) undo(RR) undo(F) = RR F LL F
To make the notation simple, I'm going to start it
in such a way that only number 1 and up move around.
And I'll use the ~ character to mark the edges of the wheel.
start state : ... 1 2 ~ 3 4 5 6 ~ ...
after [RR,F] : ... 4 3 ~ 6 5 1 2 ~ ...
Let's follow where the six pieces go.
1 becomes 4
4 becomes 5
5 becomes 1
This is called an "orbit" or "cycle.
This one has length (or "order") three, and is written
(1, 4, 5).
Looking at the others pieces,
2 becomes 3
3 becomes 6
6 turns into 2
or just
(2, 3, 6).
In math, moving around things like this
is called a "permutation". They play a huge
role in group theory.
This week, read about them here :
http://en.wikipedia.org/wiki/Permutation
There are several notations for writing down a permutation.
The one we just did is either
(a) 1 2 3 4 5 6
4 3 6 5 1 2
or just (b) 4 3 6 5 1 2
or (c) (1,4,5)(2,3,6)
which is what we just wrote down.
So the new vocabulary is:
- permutation = rearrangement of the integers [1,2,3,4,...]
- orbit (or cycle) = sequence of positions that loop around
- order = length of an orbit
Permutations form groups. And it turns out that all
finite groups can are made up of pieces of permutation groups.
So they're kind of the foundation of discrete group theory.
Play around with this notation, and do some a few examples by hand.
- What is the notation for the identify element as a permutation? Answer: ()
- What does the topspin R (right rotation) look like?
- What is the topspin F (flip) look like?
symmetric group
How many elements are in S3? S4? Discuss the factorial (n!) notation.
Is either (or both) of these another name for a group we've already seen?
GAP
Doing all these permutations and symbol manipulations by hand gets a bit old.
We need a calculator.
GAP (groups, algorithms, programming) application
a calculator for group theory.
It's installed on cs.marlboro.edu; if you would like I can give you
an account there, and show you how to connect from your computer.
Lines end with ;
Variable assignment is done with :=
$ gap
# arithmetic
gap> alpha = 3;
gap> beta = 4;
gap> alpha + beta;
7
# topspin permutation
gap> L := (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20);
# turn to left: 1 turns into 2, etc
gap> ListPerm(L); # see how the list [1,2,3,...] is transformed.
[2,3,4,...,20,1]
gap> L^2 # L*L = L^2 = do L twice
gap> L := R^-1; # inverse
gap> F := (1,4)(2,3); # flip
# order of group operations in gap is RIGHT TO LEFT!
gap> ListPerm(F * L); # left, then flip. (RIGHT TO LEFT!)
I'll do some in-class playing around with GAP in class to demonstrate it. Unless everyone gets access and feels OK, I will not be using it for assignments ... but for those who like to work that way, it's there. Things to try included Order() to get length of cycle.
GAP can define groups from generators, and do all sorts of tricky stuff, which we may see later.
I will be using GAP in class occasionally to do the group theory, rather than writing out things by hand.
what next
That was a lot of technical mumbo-jumbo. This week's homework will let you play around with it.
Depending on time and how much your eyes are glazing over, we can start talking about how all this applies to the Rubik's Cube ... or we'll start in on that next time.
![[paper clip]](/courses/source/wiki_images/paper_clip_tilt.png)
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