Central homomorphism

Definition
Suppose $$G_1$$ and $$G_2$$ are groups. A homomorphism of groups $$\varphi:G_1 \to G_2$$ is termed a central homomorphism if it satisfies the condition that $$\varphi(Z(G_1)) \le Z(G_2)$$, where $$Z(G_1)$$ and $$Z(G_2)$$ denote respectively the centers of $$G_1$$ and $$G_2$$ respectively.

Facts

 * Every surjective homomorphism is central.
 * An injective homomorphism is central if and only if the image is a subgroup whose center is contained in the center of the whole group.
 * We can define the category of groups with central homomorphisms.
 * Every central homomorphism defines a homoclinism of groups. This results in a functor from the category of groups with central homomorphisms to the category of groups with homoclinisms.