# 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.

Example:
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.