Cyclic Permutations

A permutation of the type
\[\left( {\begin{array}{*{20}{c}} {{a_1}}&{{a_2}}&{{a_3}}& \cdots &{{a_{n – 1}}}&{{a_n}} \\ {{a_2}}&{{a_3}}&{{a_4}}& \cdots &{{a_n}}&{{a_1}} \end{array}} \right)\]

is called a cyclic permutation or a cycle. It is usually denoted by the symbol $$\left( {{a_1},{a_2}, \ldots ,{a_n}} \right)$$.

Thus if $$f$$ is a permutation of degree $$n$$ of a set $$S$$ having $$n$$ distinct elements, and if it is possible to arrange some of the elements (say $$m$$ in number) of the set $$S$$ in a row such that the $$f – $$ image of each element in this row is the element following it and the $$f – $$ image of the last element in the row is the first element and the remaining $$\left( {n – m} \right)$$ elements of the set $$S$$ remain invariant under $$f$$, then $$f$$ is called a cycle permutation or a cycle of length $$m$$.

The number of objects permuted by the cycles is called the length of the cycle. Thus by the cycle of length of e we mean a permutation in which the image of each element remains unchanged under a permutation $$f$$. Consequently the cycle permutation of length one is the identity permutation.

One Row Symbol: One row symbol is used to denote a cycle permutation. In the notation the elements of $$S$$ are arranged in such a way that the image of each element in this row is the element which follows it and that of the last element is the first element. These elements of $$X$$ which remain invariant need not be written in the row.


Let $$f = \left( {\begin{array}{*{20}{c}} 1&2&3&4&5&6 \\ 2&4&1&3&5&6 \end{array}} \right)$$ be a cyclic permutation.

Since the elements 1, 2, 3, 4 are such that $$f\left( 1 \right) = 2$$, $$f\left( 2 \right) = 4$$, $$f\left( 4 \right) = 3$$ and $$f\left( 3 \right) = 1$$  and two remaining elements 5 and 6 remain invariant under $$f$$, $$f$$ is a cycle of length 4 or a 4-cycle and can be expressed as $$f\left( {1\,\,2\,\,4\,\,3} \right)$$.

Transposition: A cycle of length two is called a transposition. Thus the cycle $$\left( {1,\,3} \right)$$ is a transposition. It is 2-cyclic such that the image of 1 is 3 and the image of 3 is 1 and the remaining missing elements are invariant.

Disjoint Cycles: Two cycles are said to be disjoint if when they are expressed in one row notations, they have no element in common.