Lab8-Cycle_index.ipynb

Kernel: SageMath 10.0

Lab 8 - Cycle index

Math 737

Exercise 1: Use the code below to create the cyclic group of order 6. By comparing to the example from class, what does each command compute?

C6=CyclicPermutationGroup(6)

C6.conjugacy_classes()

C6.cycle_index()

C6.profile_series()

Exercise 2: Use similar commands on the dihedral group to compute its conjugacy classes, cycle index, and profile series. How many 6 bead bracelets with 2 colors are there if bracelets are considered to be the same under rotation or reflection? How many of these have 3 beads of each color?

D6 = DihedralGroup(6); D6

Dihedral group of order 12 as a permutation group

Exercise 3: Consider 2-colorings of the faces of the cube, as in the AoC textbook Section 6.2. Type in the permutation group generators given in the table in the middle of p. 196 to generate a permutation group (there is an example below of how to do this). Use the commands from the prior exercise to compute the number of orbits under this group action with all possible combinations of the number of faces of each color. Check that your answers match the computations in the book.

p = Permutation([(1,2),(3,4)]); p

[2, 1, 4, 3]

mygroup = PermutationGroup([p]); mygroup

Permutation Group with generators [(1,2)(3,4)]

Exercise 4: Find the number of elements in the group you made in Exercise 3. Guess what permutation group this group is isomorphic to. Check your guess by using the command .isomorphism_to()

C6.isomorphism_to(C6)

Permutation group endomorphism of Cyclic group of order 6 as a permutation group
  Defn: [(1,2,3,4,5,6)] -> [(1,2,3,4,5,6)]