Two permutations and of degree are said to be equal if we have , .
If and are two permutations of degree 4, then we have. Here we see that both and replace 1 by 2, 2 by 3, 3 by 4, and 4 by 1.
If is a permutation of degree , we can write it in several ways. The interchange of columns will not change the permutation. Thus, we can write:
Therefore, if and are two permutations of the same elements of degree , then it is always possible to write in such a way that the first row of coincides with the second row of .
Total Number of Distinct Permutations of Degree
If is a finite set having distinct elements, then we shall have distinct arrangements of the elements of . Therefore there will be distinct permutations of degree if is the set consisting of all permutations of degree . If is the set containing of all permutations of degree then the set will have distinct elements.
This set is called the symmetric set of permutations of degree . Sometimes it is also denoted by .
Thus, ( is a permutation of degree ).
The set of all permutation of degree 3 will have 3!, i.e., 6 elements. Obviously: