A structural approach to the endomorphisms of certain abelian groups

Abstract

English: 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.

Afrikaans: Gegewe 'n versameling S, en enige selfafbeelding ƒ: S →S, kan die funksionele grafiek geassosieer met ƒ beskryf word as die grafiek met nodus versameling S en gerigte randversameling E = {(u; v) ϵ S2 : ƒ (u) = v}. 'n Klassifikasie van alle funksionele grafi eke wat deur tralie endomor fismes geinduseer word, was onlangs deur J. Szigeti gedoen ([12]). In hierdie verhandeling beoog ons om 'n soortgelyke tipe klassi fikasie te bekom met betrekking tot die funksionele gra fieke eke wat deur endomor fismes van sekere abelse groepe geinduseer word. 'n Metode vir die bepaling van alle funksionele grafieke wat geinduseer word deur endomorsmes van 'n groep is ontwikkel vir alle groepe van die vorm Znp met p enige priem, n ϵ N, en Zn vir enige n ϵ N, sowel as alle sikliese groepe. 'n Belangrike verband tussen die funksionele gra fieke wat ooreenstem met groep endomor smes en die minimale polinoom van die matriks voorstelling van die endomor fisme gevind.