In abstract algebra, the multiplication operation of permutation is usually called "synthesis" or "connection". The definition of this operation is based on the sequential relationship between two permutation elements.

First of all, we need to be clear about what substitution is. In mathematics, permutation is a kind of bijection from set to itself, that is, each element is mapped to one and only one element, and the image of each element is mapped only once. For example, if we have a set of three elements {1, 2,3}, then an arrangement is to map 1 to 2, 2 to 3, and 3 to 1.

Then, we define the multiplication operation of two permutations. Suppose we have two permutations p and q, and their elements are the same. We can regard P as the first operation and Q as the second operation. So the product of p and q is to apply p to the set first and then q to the result. Specifically, if P maps element A to B and Q maps element C to D, then the product of P and Q maps element A to D and element B to C. ..

This multiplication operation satisfies the commutative law and associative law. This is because whether we execute P or Q first, the result is the same. In addition, if we perform p and q twice in succession, the result is that the results of performing p and q once respectively are multiplied again.

However, this multiplication operation does not satisfy the distribution law. That is to say, we can't guarantee (P*Q)*R=P*(Q*R) for any permutation of p, q and r, because multiplication changes the order of elements, while the distribution law requires the order of elements to remain unchanged.

Generally speaking, in abstract algebra, the multiplication of two permutations is a compound operation based on the order of elements. Although it does not satisfy all the operation rules (such as distribution law), it plays an important role in studying the properties and applications of permutation.