# Finding sequence of group generators that yields group element

Here is my permutation group that acts on lists of length 4, defined in terms of four generators:

permGroup = PermutationGroup[{Cycles[{{1, 3}, {2, 4}}],
Cycles[{{1, 2}}], Cycles[{{3, 4}}], Cycles[{{2, 4}}]}]


Given a list of four entires out of order, I can get the permutation required using FindPermutation or the position of the element in the group using GroupElementPosition:

outOfOrderExample = {3,1,2,4};

GroupElementPosition[permGroup, FindPermutation[outOfOrderExample,{1,2,3,4}]]
(*13*)


But, how do I get the precise sequence of group generators that when multiplied together puts the list outOfOrderExample back in order? Obviously, the answer is not unique. I just need any of the shortest answer.

An acceptable answer would be a list of numbers identifying the generators in permGroup, such as {2, 3, 4, 3} for this example. You can check the answer is correct by doing something like

Permute[outOfOrderExample,
PermutationProduct@@First[permGroup][[{2, 3, 4, 3}]]]


Use GroupElementToWord. There are some examples in the documentation, especially in the Neat Examples section which mentions Rubik's cube.

Below I express the permutation that takes {3,1,2,4} to {1,2,3,4} in terms of a 'word' or sequence of the group generators (and their inverses).

perms = {
Cycles[{{1, 3}, {2, 4}}],
Cycles[{{1, 2}}],
Cycles[{{3, 4}}],
Cycles[{{2, 4}}]
};

permGroup = PermutationGroup[perms];
generators = GroupGenerators[permGroup];

outOfOrderExample = {3, 1, 2, 4};
reversePerm = FindPermutation[outOfOrderExample, Sort@outOfOrderExample];
GroupElementQ[permGroup, reversePerm]
ge2word = GroupElementToWord[permGroup, reversePerm];
(** result: {-4, 3,-1} - e.g 3 means apply perm 3,
while -4 means apply inverse of perm 4 **)

reversePermSequence =
If[Positive@#, Identity, InversePermutation][
generators[[Abs[#]]]] & /@ ge2word;

Fold[Permute, outOfOrderExample, reversePermSequence]
(*result {1,2,3,4}*)

• Thank you! I did not know what the function was called. To be clear, -4 even though inverse of perm 4, is the same as 4 because it is self inverse. Right? Oct 6 '20 at 21:45
• @QuantumDot Yes the language of group theory and most modern algebra is needlessly byzantine. And yes, it just so happens that InversePermutation[perms[[4]]] is the same as perms[[4]]. Oct 6 '20 at 21:48
• I made some minor edits which uses the group generators instead of the original perms and fixes the order, since I noticed a bug when trying some different groups. I realize the generators are the same, but this is just so anybody passing by with a different set of perms finds this useful. Oct 6 '20 at 22:04