The products or composite of two permutations and of degree denoted by , is obtained by first carrying out the operation defined by then by .
Let us suppose is the set of all permutations of degree .
be two elements of .
Hence the permutation has been written in such a way that the first row of coincide with the second row of . If the product of the permutations and is denoted multiplicatively, i.e., by , then 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 , .