Suppose is a finite set having distinct elements. Then a one-one mapping of onto itself is called a permutation of degree . The number of elements in the finite set is known as the degree of permutation.
Symbol for a Permutation
Let be a finite set having distinct elements. If in one-one mapping, then is a permutation of degree .
Let , , , …, where i.e. is one arrangement of the elements .
It is customary to write a permutation in a two line symbol. In this notation we write:
i.e. each element in the second row is the image of the element of the first row lying directly above it.
If is a finite set of order four then,
etc., are all permutations of degree four. Here in the permutation the elements 1, 2, 3, 4 are replaced respectively by the elements 2, 4, 1, 3. Thus . Similarly, and .
Thus for a permutation on , we just put the elements of in one row in any order we like and below each element of this row we put down its image under, or to obtain another row of elements of .