A structural approach to the endomorphisms of certain abelian groups

Given a set S, and any selfmap ƒ: S→S, the functional graph associated with ƒ can be described as a graph with vertex set S and directed edge set E = {(u; v) ϵ S2 : ƒ (u) = v}. A classification of all functional graphs induced by lattice endomorphisms has recently been done by J. Szigeti ([12]). In this dissertation, we aim to achieve a similar type of classi_cation with respect to functional graphs induced by endomorphisms on certain abelian groups. A method for finding all functional graphs that can be induced by endomorphisms of a group has been developed for all groups of the form Znp with p any prime, n ϵ N, and Zn for any n ϵ N, as well as all cyclic groups. A deep connection between the functional graphs corresponding to group endomorphisms and the minimal polynomial of the matrix representation of the group endomorphism has been found.