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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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