Surjective endomorphism

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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.