# [GAP Forum] Cyclic permutation of five elements

Stefan Kohl kohl at mathematik.uni-stuttgart.de
Tue Mar 30 17:38:03 BST 2004

```Dear Stephen C. Lipp and Forum,

> Problem 1: Given five elements find a cyclic set of transpositions which
> will yield all permutations of these elements with respect to a given
> permutation, namely (2,3,4,5).
>
> Unfortunately, I am not speaking of permutations in the traditional
> sense, but I am speaking of the indexing of the set.  Hence, if I apply
> the following transpositions
>
> [<3,4>, <4,5>, <2,3>, <1,2>, <4,5>, <2,3>, <3,4>, <2,3>, <5,1>, <4,5>,
>  <1,2>, <3,4>, <2,3>, <3,4>, <1,2>, <5,1>, <1,2>, <4,5>, <5,1>, <4,5>,
>  <2,3>, <3,4>, <1,2>, <3,4>, <4,5>, <2,3>, <4,5>, <3,4>, <1,2>, <4,5>]
>
> to the set [1,2,3,4,5] I get the cycle
>
> [[1,2,4,3,5], [1,2,4,5,3], [1,4,2,5,3], [4,1,2,5,3], [4,1,2,3,5],
> [4,2,1,3,5],
>  [4,2,3,1,5], [4,3,2,1,5], [5,3,2,1,4], [5,3,2,4,1], [3,5,2,4,1],
> [3,5,4,2,1],
>  [3,4,5,2,1], [3,4,2,5,1], [4,3,2,5,1], [1,3,2,5,4], [3,1,2,5,4],
> [3,1,2,4,5],
>  [5,1,2,4,3], [5,1,2,3,4], [5,2,1,3,4], [5,2,3,1,4], [2,5,3,1,4],
> [2,5,1,3,4],
>  [2,5,1,4,3], [2,1,5,4,3], [2,1,5,3,4], [2,1,3,5,4], [1,2,3,5,4],
> [1,2,3,4,5]]
>
> The set of transpositions was obtained using an algorithm on a
> spreadsheet with a random number generator.
>
> This cycle contains all 5!/4 permutations of 5 elements where the last 4
> elements are allowed to rotate.  The terminology I am using would
> indicate there exists an isomorphism between the index transpositions
> and group permutations.  Does such an isomorphism exist and, if so, what
> is it?

Let me try to guess what you want to say:

Apparently you describe a walk through the Cayley graph of the
symmetric group S_5 with the set of transpositions as generating set
which passes each element of the conjugacy class of the 4-cycles
exactly once.

The isomorphism you are looking for is probably just the
action isomorphism for the action of S_5 on the set {1,2,3,4,5}.

Please don't hesitate to ask in case I have misunderstood you or
if you have further questions.

Hope this helps,

Stefan Kohl

```