dc.contributor.advisor | Meyer, J. H. | |
dc.contributor.author | De Klerk, Ben-Eben | |
dc.date.accessioned | 2017-08-07T07:38:17Z | |
dc.date.available | 2017-08-07T07:38:17Z | |
dc.date.issued | 2016-09 | |
dc.identifier.uri | http://hdl.handle.net/11660/6521 | |
dc.description.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. | en_ZA |
dc.description.abstract | 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. | af |
dc.language.iso | en | en_ZA |
dc.publisher | University of the Free State | en_ZA |
dc.subject | Abelian Group | en_ZA |
dc.subject | Automorphism | en_ZA |
dc.subject | Endomorphism | en_ZA |
dc.subject | Conjugacy classes | en_ZA |
dc.subject | Functional graph | en_ZA |
dc.subject | Tree | en_ZA |
dc.subject | Finite field | en_ZA |
dc.subject | Cyclotomic polynomial | en_ZA |
dc.subject | Minimal polynomial | en_ZA |
dc.subject | Thesis (Ph.D. (Mathematics and Applied Mathematics))--University of the Free State, 2016 | en_ZA |
dc.title | A structural approach to the endomorphisms of certain abelian groups | en_ZA |
dc.type | Thesis | en_ZA |
dc.rights.holder | University of the Free State | en_ZA |