A structural approach to the endomorphisms of certain abelian groups

[摘要] English: Given a set S, and any selfmap ƒ: S→S, the functional graph associatedwith ƒ can be described as a graph with vertex set S and directed edge setE = {(u; v) ϵ S2 : ƒ (u) = v}. A classification of all functional graphs inducedby lattice endomorphisms has recently been done by J. Szigeti ([12]). In thisdissertation, we aim to achieve a similar type of classi_cation with respect tofunctional graphs induced by endomorphisms on certain abelian groups.A method for finding all functional graphs that can be induced by endomorphismsof a group has been developed for all groups of the form Znpwith p anyprime, 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 endomorphismsand the minimal polynomial of the matrix representation of thegroup endomorphism has been found.

[发布机构] University of the Free State