Descending Endomorphisms of Groups

We define a descending endomorphism of a group as an endomorphism that in-duces a corresponding endomorphism in any homomorphic image of the group, such that the composition of the descending endomorphism with the homomorphism equals the composition of the homomorphism with the induced endomorphism. After proving that descending endomor-phisms of a certain class of Abelian groups, including all finitely generated Abelian groups, are universal power endomorphisms, we characterise the descending endomorphisms of direct products of groups, and thus obtain a procedure to determine all the descending endomorphisms of a direct product using the descending endomorphisms of the direct factors. As a natural outcome of this theory, we also obtain a characterisation of the direct products whose normal subgroups are direct products of normal subgroups of the direct factors.