# Product or Composite of Two Permutations

The products or composite of two permutations and of degree denoted by is obtained by first carrying out the operation defined by and then by .

Let us suppose is the set of all permutations of degree .

Let

and

be two elements of .

Hence the permutation has been written in such a way that the first row of coincides with the second row of . If the product of the permutations and is denoted multiplicatively, i.e., by , then by definition

For replaces by and then replaces by so that replaces by. Similarly replaces by , by, …, by .

Obviously, is also a permutation of degree . Thus the product of two permutations of degree is also a permutation of degree . Therefore , .