Chapter 8: Software and Calculations

The function  Orbit[G,x]  will compute the orbit of x under the permutation group or
linear group G. Here x is a positive natural number or a vector. For
example,

A5  =  Group[P[{1, 2, 3}] ,  P[{3, 4, 5}]]

So the orbit of 2 under (Subscript[A, 5]) can be computed by

Orbit[A5,  2]

Similarly, if you set

ChoosePrime[5]

and take

F20 = Group[L[{1, 1}, {0, 1}] ,  L[{2, 0}, {0, 3}]]

then the orbit of the vector (2,3) is

Orbit[F20, {2, 3}]

(see exercise 4.4). Stabilizer[G,x] will compute the stabilizer of x in the group G . So for example

Stabilizer[A5, 3]

The conjugacy class of an element can be calculated with the function  ConjugacyClass.Let's use it to compute the conjugacy classes in (Subscript[A, 5]).  To begin, we know that the conjugacy classes in (Subscript[S, 5])correspond to the cycle types of permutations of degree 5. The even cycle types are

{1}, {3}, {5}, {2,2} .

The function  CycleTypes computes the number of permutations in each cycle type:

CycleTypes[A5]

Now two elements in (Subscript[A, 5])may be conjugate by an element in (Subscript[S, 5]) but not by an element in (Subscript[A, 5]). So a conjugacy class of (Subscript[S, 5])may break up into more than one conjugacy class in (Subscript[A, 5]). We begin with a 3-cycle:

ConjugacyClass[A5,  P[{1, 2, 3}]]

These are all 20 3-cycles. Next we look at the conjugacy class of a 5-cycle:

ConjugacyClass[A5,  P[{1, 2, 3,  4,  5}]]

This is only half of the 5-cycles! One that is missing is  (1 2 3 5 4) . So let's compute its conjugacy class:

ConjugacyClass[A5,  P[{1, 2, 3, 5,  4}]]

These are the remaining 5-cycles. Lastly we look at the conjugacy class of a product of two transpositions:

ConjugacyClass[A5,  P[{1, 2}, {3, 4}]]

These are all 15 permutations of type {2, 2}. So these four sets together with {(1)} are the conjugacy classes of (Subscript[A, 5]).

The centre of a group can be computed with the function  Centre . For example,

D4  =  Group[P[{1,  2,  3,  4}],  P[{1,  3}]]

and

Centre[D4]

For a matrix α in \(TraditionalForm\`\(\(GL\)\((\)\(2, 𝔽\_p\)\)\)) the corresponding linear fractional transformation  (Subscript[s, α]) is computed by the function FLTPermutation . Let's repeat the calculation of 
	
					(Subscript[f, 3]):  GL(2,  (Subscript[𝔽, 3])) ⟶ (Subscript[S, 4]) 

using this function:

ChoosePrime[3]

a  =  L[{2, 1}, {0, 1}]

b  =  L[{2, 2}, {0, 1}]

c  =  L[{0, 1}, {1, 0}]

FLTPermutation[a]

FLTPermutation[b]

FLTPermutation[c]