How are surjective group homomorphisms important

Question

I see that surjective homomorphisms are kind of intermediate between group homomorphisms and group isomorphisms. To some extent I understand the importance of latter two. I would like to ask about the significance of surjective group homomorphisms.

Answer 1

If $f:G\rightarrow H$ is a surjective group homomorphism, then by the first isomorphism theorem $\frac{G}{\ker(f)}\cong H$. Another way to formulate this is to say that $H$ is a quotient group of $G$. That's the main importance of surjective group morphisms.

This is in some sense dual to having a injective group morphisms. If $f:G\rightarrow H$ is injective then $G\cong \text{im}(f)$ is a subgroup of $H$. There is a nice categorical way of thinking about these hings. But if you don't know categories yet, you need not learn this language right now.

Answer 2

Surjective group homomorphisms represent the possible quotients of a group and so reflect the structure of the group.