- 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.