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 non 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 cycle. Thus by the cycle of length on e we mean a permutation in which the image of each element remains unchanged under a permutation f. Consequently 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 a 2-cyclic such that the image of 1 is 3 and image of 3 is 1 and the remaining missing elements are invariant.

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