A permutation of the type
is called a cyclic permutation or a cycle. It is usually denoted by the symbol .
Thus if is a permutation of degree of a set having distinct elements, and if it is possible to arrange some of the elements (say in number) of the set in a row such that the image of each element in this row is the element following it and the image of the last element in the row is the first element and the remaining elements of the set remain invariant under , then is called a cycle permutation or a cycle of length .
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 . 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 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 which remain invariant need not be written in the row.
Let be a cyclic permutation.
Since the elements 1, 2, 3, 4 are such that , , and and two remaining elements 5 and 6 remain invariant under , is a cycle of length 4 or a 4-cycle and can be expressed as .
Transposition: A cycle of length two is called a transposition. Thus the cycle 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.