# [GAP Forum] Cyclic permutation of five elements

Stephen C. Lipp stephen.lipp at edi-nola.com
Tue Mar 30 14:49:25 BST 2004

```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?

Please forgive me if my use of the terminology is poor.  It has been a
very long time since I made use of the terminology of group theory.

```