Thurs Feb 3
* isometries in S4, A4, C4, D4, S5, A5, C5, D5
* properties of individual elements:
- cycle length
- commutator of something? (define "communtator")
* discuss solid cube group (starting this was the homework assignment)
1 Identity (cycle length = 1)
6 F, R, U, F', R', U' (cycle length = 4; 90 degree face-centered rotation)
3 F2, R2, U2 (cycle length = 2; 180 degree face-centered rotation)
8 FR etc (cycle length = 3; 120 degree corner-centered rotation)
6 RRU etc (cycle length = 2; 180 degree edge-centered rotation)
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sum = 24
* treat R,F,U,R',F',U' as generators; draw the graph for this 24 element group
* Are R,F,U generators? R,F? R? Any single element?
* odd/even
* notation on rubik's cube; commutators; cycle length (assign)
* time allowing :
matrix multiplication rules
permutation matrices
* for Thurs Feb 10
1.Find the commutator subgroup of the solid cube group.
2.Do R, F generate the cube group? R,F,U? R??
3.What is the cycle length of the Rubik's commutator FRF'R' ? FR ?
- next time:
* if c = a b a' then we say "c and b are conjugate".
* conjugacy classes
* conjugacy classes of solid cube