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 .

Let

and

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 , .

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