Let m, m' be two multiplication sets, that is to say, m and m' are two algebraic systems with closed combination method (generally written as multiplication), σ is the mapping from m to m', and the image of the product of any two elements is the product of the images of these two elements, that is, for any two elements A and B in M, σ (A B) = σ (A) is satisfied. That is to say, when a→σ(a), b→σ(b) and a b→σ (a b), then this mapping σ is called the homomorphism from m to m'. This concept actually turns bijection in isomorphic concept into general mapping. If σ is a mapping from m to m', it is said that σ is a homomorphism from m to m'; If σ is a mapping from m to m', it is said that σ is a homomorphism from m to m', also called m and m' homomorphism.

Homomorphism and isomorphism of groups are important means to study the relationship between groups. Isomorphic mapping is a mapping that maintains operations between groups. Two groups with isomorphic mappings can be regarded as the same group because they have the same group structure. The most basic and important task in algebra is to find out the classification of various algebraic systems in the sense of isomorphism.

Homomorphic mapping only needs operation preservation, which is obviously more flexible than isomorphic mapping, and can study the relationship between two groups with different structures. Especially important are several homomorphism theorems, just as the basic theorem of morphism tells us that under the condition of total homomorphism, two groups contain a group isomorphism (g 1/ section ≌G2)! When dealing with some isomorphic problems, we often use this theorem in reverse, that is, we first construct a homomorphism. Because the operation-preserving mapping can study some relations between two algebraic systems, we can study them in some simple ways for more complex algebraic systems. In addition, automorphism and automorphism of groups are also important means to study groups.