How can I output list of permutation products?

Question

I have A = Permutations[{1, 2, 3, 4}]. And c = Cycles[{{1, 2, 3, 4}}]. And I need to output for all $a \in A: a \cdot с \cdot a^{-1}$. How can I do this?

P.S. $a^{-1}$ is InversePermutation[a].

Answer 1

Well, first I'm going to use the fact that the set of permutations you have is just a SymmetricGroup:

g = SymmetricGroup[4]
PermutationCycles /@ Permutations[{1, 2, 3, 4}] === GroupElements[g]
(* True *)

Then it's a simple matter of Mapping over GroupElements:

Map[PermutationProduct[#, c, InversePermutation[#]] &,
  GroupElements@g]
(* {Cycles[{{1, 2, 3, 4}}], ... Cycles[{{1, 4, 3, 2}}]} *)

You can also use GroupBy to break these up as equivalence classes, which is often what one wants to do in this situation:

GroupBy[GroupElements@g,
  PermutationProduct[#, c, InversePermutation[#]] &]

This will return an Association where the keys are the unique products $ a\cdot c \cdot a^{-1} $ generated, and the values are the group elements $ a $ that yield that product.

Answer 2

edited to add missing definition

Perhaps the following?:

A = PermutationCycles /@ Permutations[{1,2,3,4}];
Thread @ PermutationProduct[
    A,
    Cycles[{{1,2,3,4}}],
    Thread @ InversePermutation @ A
]

{Cycles[{{1, 2, 3, 4}}], Cycles[{{1, 2, 4, 3}}], Cycles[{{1, 3, 2, 4}}], Cycles[{{1, 4, 2, 3}}], Cycles[{{1, 3, 4, 2}}], Cycles[{{1, 4, 3, 2}}], Cycles[{{1, 3, 4, 2}}], Cycles[{{1, 4, 3, 2}}], Cycles[{{1, 2, 4, 3}}], Cycles[{{1, 2, 3, 4}}], Cycles[{{1, 4, 2, 3}}], Cycles[{{1, 3, 2, 4}}], Cycles[{{1, 4, 2, 3}}], Cycles[{{1, 3, 2, 4}}], Cycles[{{1, 4, 3, 2}}], Cycles[{{1, 3, 4, 2}}], Cycles[{{1, 2, 3, 4}}], Cycles[{{1, 2, 4, 3}}], Cycles[{{1, 2, 3, 4}}], Cycles[{{1, 2, 4, 3}}], Cycles[{{1, 3, 2, 4}}], Cycles[{{1, 4, 2, 3}}], Cycles[{{1, 3, 4, 2}}], Cycles[{{1, 4, 3, 2}}]}