# How can I output list of permutation products?

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].

• Related my this post.
– yode
Commented May 23, 2017 at 10:04

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.

• @jjc385 So I did. That'll teach me to answer Mathematica questions before I have my coffee. Commented May 5, 2017 at 2:37

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