Math 157 Final Project Presentation
Permutaion Group
Basic Background
A permutation group is a finite group G whose elements are permutations of a given finite set X and whose group operation is the composition of permutations. The number of elements of X is called the degree of G.
In Sage, a permutation is represented as either a string that defines a permutation using disjoint cycle notation, or a list of tuples, which represent disjoint cycles.
Element
The easiest way to work with permutation group elements in Sage is to write them in cycle notation. Since these are products of disjoint cycles (which commute), we do not need to concern ourselves with the actual order of the cycles.
we have to use a string of characters written with cycle notation into a symmetric group to make group elements.
There are alternate ways to create permutation group elements.
once we get Sage started, it can promote the product of elements τσ into the larger permutation group, which means we can “promote” elements into larger permutation groups
Properties of Permutation Elements
it is easier to grab an element out of a list of elements of a permutation group, and then it is already attached to a parent and there is no need for any coercion.
when multiply permutations, the movement is from left-to-right, which is our chosen convention for composing two permutation, so it is not ok to switch the order
the group of symmetries of a pentagon is not abelian, and here are some selected examples of various methods available.
A very useful method when studying the alternating group is the permutation group element method .sign( ).
It will return 1 if a permutation is even and -1 if a permutation is odd.
Matrix
Returns deg x deg permutation matrix associated to the permutation self.
Here is an example of how to use matrices in SageMath to display a permutation in array form. We can use the matrix( ) command, where the syntax is matrix
[ (list for row 1) , (list for row 2) ]
Conjugacy Class
Let G be a group. Two elements a and b of G are conjugate, if there exists an element g in G such that gag−1 = b. One says also that b is a conjugate of a and that a is a conjugate of b.
For example:
The symmetric group S3, consisting of all 6 permutations of three elements, has three conjugacy classes:
no change (abc → abc)
transposing two (abc → acb, abc → bac, abc → cba)
a cyclic permutation of all three (abc → bca, abc → cab)
Exercise
Prove Conjugation is a group automorphism, so conjugate groups will be isomorphic
Solution
If P is a 2-group which is not elementary abelian, then some non-identity element of the centre of P is a square.
Is this true?
Solution
Construct the group of symmetries of the tetrahedron (also the alternating group on 4 symbols, A4) with the command A=AlternatingGroup(4). Using tools such as orders of elements, and generators of subgroups, see if you can find all of the subgroups of A4 (each one exactly once). Do this without using the .subgroups() method to justify the correctness of your answer