# Surjective endomorphism

From Groupprops

## Definition

An endomorphism of a group is termed a **surjective endomorphism** if it is surjective as a set map; equivalently, its image is the whole group.

Surjective endomorphisms of a group correspond to isomorphisms between the group and its quotient groups.

## Facts

- A group is termed Hopfian if and only if every surjective endomorphism of the group is an automorphism. An equivalent formulation is that the quotient by any nontrivial normal subgroup is
*not*isomorphic to the whole group. Finite groups, slender groups, and groups satisfying the ascending chain condition on normal subgroups are examples of Hopfian groups. - If a subgroup property satisfies the image condition, i.e., it is preserved on taking homomorphic images, then it is preserved (positively) under surjective endomorphisms. For instance, the image of a normal subgroup under a surjective endomorphism is again a normal subgroup. The image of a subnormal subgroup under a surjective endomorphism is again a subnormal subgroup.
- If a subgroup property satisfies the inverse image condition, i.e., the inverse image of a subgroup satisfying the property, under a homomorphism, also satisfies the property, then it is preserved under inverse images for surjective endomorphisms. In particular, if a property satisfies both the image condition and the inverse image condition, then a subgroup has the property if and only if its image under a given surjective endomorphism does.
- A subgroup that is invariant under surjective endomorphisms is termed a strictly characteristic subgroup (this is also sometimes called a distinguished subgroup). The property of being strictly characteristic is somewhere in between the property of being a fully characteristic subgroup (also called a fully invariant subgroup) and a characteristic subgroup.