I would like to permute the elements of the set $A$ using symmetric group. How may I be able to do that? For example, in Sage,
G = SymmetricGroup(3) for i in G: print [j*i*j^(-1) for j in G] gives all the conjugacy classes for $S_3$ group. How may I able to do same with SymmetricGroup[3] is Mathematica.
Furthermore, I would like find the elements of $S_3$ that leaves the set $A=\{1, 2\}$ invariant or leaves each element fixed. How may I be able to do that?
I find the OP questions somewhat unclear. In any case, the answers can be found in the tutorials "Permutations" and "Permutation Groups".
I would like to permute the elements of the set A using symmetric group.
In[44]:= Permute[{1,2,3},SymmetricGroup[3]] Out[44]= {{1, 2, 3}, {1, 3, 2}, {2, 1, 3}, {2, 3, 1}, {3, 1, 2}, {3, 2, 1}} The Mathematica equivalent of the Sage code:
G = SymmetricGroup(3) for i in G: print [j*i*j^(-1) for j in G] should be written with other operators/functions, otherwise j*i*j^(-1) will simplify to i. Here I use NonCommutativeMultiply and Inverse:
In[42]:= Permute[NonCommutativeMultiply[j, i, Inverse[j]], SymmetricGroup[3]] Out[42]= {j ** i ** Inverse[j], j ** Inverse[j] ** i, i ** j ** Inverse[j], i ** Inverse[j] ** j, Inverse[j] ** j ** i, Inverse[j] ** i ** j} Furthermore, I would like find the elements of $S_3$ that leaves the set $A=\{1,2\}$ invariant [...]
This is unclear to me because:
In[47]:= GroupStabilizer[SymmetricGroup[3], {1, 2}] Out[47]= PermutationGroup[{}] hence
In[36]:= Permute[{1,2,3}, GroupStabilizer[SymmetricGroup[3], {1, 2}]] Out[36]= {{}} May be the following examples are helpful illustrations.
Fixing the 3d element using $S_3$:
In[50]:= Permute[Range[3], GroupStabilizer[SymmetricGroup[3], {3}]] Out[50]= {{1, 2, 3}, {2, 1, 3}} Fixing the elements {1,2} using $S_4$:
In[49]:= Permute[Range[4], GroupStabilizer[SymmetricGroup[4], {1, 2}]] Out[49]= {{1, 2, 3, 4}, {1, 2, 4, 3}} 