Exercises 5.6 Sage Exercises
1.
Create the full symmetric group G = SymmetricGroup(10)
.
2.
Create elements of G
with the following (varying) syntax. Pay attention to commas, quotes, brackets, parentheses. The first two use a string (characters) as input, mimicking the way we write permuations (but with commas). The second two use a list of tuples.
a = G("(5,7,2,9,3,1,8)")
b = G("(1,3)(4,5)")
c = G([(1,2),(3,4)])
d = G([(1,3),(2,5,8),(4,6,7,9,10)])
Compute
Compute the orders of each of these four individual elements (
a
throughd
) using a single permutation group element method.Use the permutation group element method
.sign()
to determine if are even or odd permutations.-
Create two cyclic subgroups of
with the commands:H = G.subgroup([a])
K = G.subgroup([d])
List, and study, the elements of each subgroup. Without using Sage, list the order of each subgroup of
Then use Sage to construct a subgroup of with order 10. More complicated subgroups can be formed by using two or more generators. Construct a subgroup
of with the commandL = G.subgroup([b,c])
. Compute the order of and list all of the elements of
3.
Construct the group of symmetries of the tetrahedron (also the alternating group on 4 symbols, A=AlternatingGroup(4)
. Using tools such as orders of elements, and generators of subgroups, see if you can find all of the subgroups of .subgroups()
method to justify the correctness of your answer (though it might be a convenient way to check your work).
Provide a nice summary as your answerβnot just piles of output. So use Sage as a tool, as needed, but basically your answer will be a concise paragraph and/or table. This is the one part of this assignment without clear, precise directions, so spend some time on this portion to get it right. Hint: no subgroup of
4.
The subsection The Motion Group of a Cube describes the
From the list of elements of the group, can you locate the ten rotations about axes? (Hint: the identity is easy, the other nine never send any symbol to itself.)
Can you identify the six symmetries that are a transposition of diagonals? (Hint:
[g for g in cube if g.order() == 2]
is a good preliminary filter.)Verify that any two of the quarter-turns (
above
,front
,right
) are sufficient to generate the whole group. How do you know each pair generates the entire group?Can you express one of the diagonal transpositions as a product of quarter-turns? This can be a notoriously difficult problem, especially for software. It is known as the βword problem.β
Number the six faces of the cube with the numbers
through (any way you like). Now consider the same three symmetries we used before (quarter-turns about face-to-face axes), but now view them as permutations of the six faces. In this way, we construct each symmetry as an element of Verify that the subgroup generated by these symmetries is the whole symmetry group of the cube. Again, rather than using three generators, try using just two.
5.
Save your work, and then see if you can crash your Sage session by building the subgroup of b
and d
of orders N
as part of your submitted worksheet.
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N = G.subgroup([b,d])
N.list()
What is the order of