1 First of all, we need to know what homomorphism is. In group theory, if a mapping f from G to H satisfies that for any g 1, g2 belongs to G and has F (G1* G2) = F (G1) * F (G2), then we call F a homomorphism from G to H. If any H belongs to H, there is F (E).
2. If F is both injective and injective, then we call F a homomorphism from G to H. Next, we need to prove that there is a homomorphism between the group G and each of its quotient groups. Assuming that H is a subgroup of G, we can define a mapping F: G->; Construct a homomorphism from g to h.
3. Specifically, we can define f(g)=gh, where G is the element in G and H is the unit element in H. Obviously, this mapping is homomorphism from G to H, so we need to prove that this homomorphism is surjective. Suppose an element H' belongs to H, so that H' is not equal to the unit element in H.
4. Then, we can find an element G' belongs to G, so gh'=h'. Since H' is not equal to the unit element in H and gh' is not equal to the unit element in H, we can get that f(gh')=f(g')*f(h')=f(g')*h' is not equal to the unit element in H, which shows that F is not surjective.
Group-related knowledge
1, group is a basic structure in algebra, which consists of a set and a binary operation. Group theory appears in many branches of mathematics, such as abstract algebra and geometry. The concept and research methods of groups have an important influence on other branches of abstract algebra.
2. Groups have the following basic properties to satisfy: the law of association, the existence and uniqueness of unit elements, and the inverse of each element. If the number of elements in group G is finite, it is called a finite group, otherwise it is called an infinite group. In addition, topological groups, also known as continuous groups, are groups with topological space structure, which require the binary operation and inverse function of groups to be continuous.
It is worth mentioning that the concept of group also plays an important role in the study of physics and chemistry, because many different physical structures, such as crystal structure and hydrogen atom structure, can be studied by group theory.