Editing Cyclic permutation (section)

=== Properties ===
Any permutation can be expressed as the [[function composition|composition]] (product) of transpositions—formally, they are [[Generating set of a group|generators]] for the [[group (mathematics)|group]].<ref>{{harvnb|Rotman|2006|loc=p. 118, Prop. 2.35}}</ref> In fact, when the set being permuted is {{math|{{mset|1, 2, ..., ''n''}}}} for some integer {{math|''n''}}, then any permutation can be expressed as a product of '''{{visible anchor|adjacent transpositions}}''' <math>(1~2), (2~3), (3~4),</math> and so on.  This follows because an arbitrary transposition can be expressed as the product of adjacent transpositions. Concretely, one can express the transposition <math>(k~~l)</math> where <math>k < l</math> by moving {{math|''k''}} to {{math|''l''}} one step at a time, then moving {{math|''l''}} back to where {{math|''k''}} was, which interchanges these two and makes no other changes:

:<math>(k~~l) = (k~~k+1)\cdot(k+1~~k+2)\cdots(l-1~~l)\cdot(l-2~~l-1)\cdots(k~~k+1).</math>

The decomposition of a permutation into a product of transpositions is obtained for example by writing the permutation as a product of disjoint cycles, and then splitting iteratively each of the cycles of length 3 and longer into a product of a transposition and a cycle of length one less:

:<math>(a~b~c~d~\ldots~y~z) = (a~b)\cdot (b~c~d~\ldots~y~z).</math>

This means the initial request is to move <math>a</math> to <math>b,</math> <math>b</math> to <math>c,</math> <math>y</math> to <math>z,</math> and finally <math>z</math> to <math>a.</math> Instead one may roll the elements keeping <math>a</math> where it is by executing the right factor first (as usual in operator notation, and following the convention in the article [[Permutation#Product and inverse|Permutation]]). This has moved <math>z</math> to the position of <math>b,</math> so after the first permutation, the elements <math>a</math> and <math>z</math> are not yet at their final positions. The transposition <math>(a~b),</math> executed thereafter, then addresses <math>z</math> by the index of <math>b</math> to swap what initially were <math>a</math> and <math>z.</math>

In fact, the [[symmetric group]] is a [[Coxeter group]], meaning that it is generated by elements of order 2 (the adjacent transpositions), and all relations are of a certain form.

One of the main results on symmetric groups states that either all of the decompositions of a given permutation into transpositions have an even number of transpositions, or they all have an odd number of transpositions.<ref>{{harvnb|Rotman|2006|loc=p. 122}}</ref> This  permits the [[parity of a permutation]] to be a [[well-defined]] concept.