Oct 14
Open discussion of Rubik's Cube solutions.
questions
1. Can you solve the edges, while ignoring the corners? (If so, then you can get crosses on each face.)
2. Can you solve the corners, while ignoring the edges? (If so, then you can do the 2x2x2 cube.)
3. Can you solve the corners without disturbing the edges? (This and 1 can solve the cube.)
4. Can you solve the edges without disturbing the corners? (This and 2 can solve the cube.)
5. Can you do the top layer, ignoring everything else?
6. Can you do the middle layer without disturbing the top?
7. Can you do the bottom layer, without disturbing the other two? (That's the hard one.)
8. Pretty positions? (and what does it mean to be "pretty"?)
9. Quick paths to pretty positions?
10. Illegal pretty positions??
continue discussion of products of groups and factoring groups
First, remember what a "conjagacy class" is: two group elements x,y
are conjugate (i.e. "similar" to each other) if there is some z such that
x = z y z' <=> "x and y are conjugate"
Suppose now you have a group G, and two subgroups A and B.
We will call as subgroup "normal" if it is made up of entire conjugacy classes.
Suppose further that the only element that A and B have in common is the identity.
And suppose that we've chosen these subgroups so that order(A) * order(B) = order(G)
Then:
1. If A and B are both normal, G is the "direct product" of A and B.
2. If A is normal but B isn't, G is the "semidirect product" of A and B. In this case B is not unique; there are typically other B2, B3, ... that have the same group structure as B.
Either way,
g = a b where g is in G, a in A, and b in B,
and we write
G / A = B (the one under the division sign should be normal)
Example: D3 and it's structure. (Which subgroups are normal?)
Example: S4 and it's structure. (Which subroups are normal?)
Why do we care?
1. Part of group theory "big picture" goal is to identify and categorize all finite groups. This lets us combine smaller groups into bigger ones, and "divide" bigger ones into smaller ones.
2. In Rubik-like puzzles, the typical solution technique has subgoals, i.e. "solve top" or "solve all the edges". These subgoals are quotient groups in this same way: we think of a number of positions as all equivalent, and ignore that part.
