Grupe in Cayleyjevi digrafi

Petek, Ana (author), Šparl, Primož (mentor)

Vsebina diplomskega dela po eni strani sodi na področje teorije grup, po drugi strani pa na področje teorije grafov. V diplomskem delu obravnavamo Cayleyjeve digrafe različnih končnih grup in se ukvarjamo s prepoznavanjem lastnosti grup iz njihovih upodobitev s pomočjo Cayleyjevih (di)grafov. Ko govorimo o lastnostih grup, mislimo predvsem na lastnosti kot so redi elementov, generatorji grupe, podgrupe, podgrupe edinke, odseki in podobno. Cayleyjevi digrafi so za takšno obravnavo zelo primerni, saj nam neposredno ali posredno pokažejo vse lastnosti grupe in nam dajejo jasno sliko o njeni kompleksnosti. Cayleyjevi (di)grafi so ime dobili po Arthurju Cayleyju, ki jih je prvič omenil leta 1878. Omenjene grafe naravno dobimo iz grup, saj so njihova vozlišča elementi grupe. Zaradi simetrije, ki jo dopuščajo, ti grafi iz vsakega vozlišča "izgledajo"povsem enako. Pri iskanju lastnosti grup preučujemo strukture njihovih Cayleyjevih (di)grafov in njegovih lastnosti. Grupe so namreč abstrakten pojem, zato nam je v veliko pomoč, da lahko lastnosti grup obravnavamo na njihovih Cayleyjevih (di)grafih in s tem obravnavo teh abstraktnih objektov vizualiziramo. V diplomskem delu uvodoma pojasnimo osnovne pojme teorije grup, ki jih bo bralec potreboval pri razumevanju diplomskega dela. Za tem ponovimo osnovne pojme teorije grafov in vpeljemo pojem digrafa. Potem bralca seznanimo s pojmom Cayleyjevega (di)grafa, kateremu v diplomskem delu posvetimo največ pozornosti. Pri tem predvsem pokažemo, na kakšen način se lastnosti grup odražajo v njihovih Cayleyjevih (di)grafih in kako lahko navedene lastnosti razberemo.

• Language: Slovenian
• Keywords: grupa
• Work type: Bachelor thesis/paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: PEF - Faculty of Education
• Year: 2016
• PID: 20.500.12556/RUL-85731
• COBISS.SI-ID: 11175753
• Publication date in RUL: 20.09.2017

PETEK, Ana, 2016, Grupe in Cayleyjevi digrafi [online]. Bachelor’s thesis. [Accessed 14 April 2025]. Retrieved from: http://pefprints.pef.uni-lj.si/3763/

Secondary language

• Language: English
• Title: Groups and Cayley digraphs

Abstract

On one hand the content of this thesis falls within the scope of Group theory, and on the other hand in the field of Graph theory. The thesis deals with Cayley digraphs of different finite groups and with identifying properties of groups from their representation as Cayley (di)graphs. By the properties of groups we particularly refer to the properties such as the orders of elements, generators of group, the subgroups, the normal subgroups, the cosets, and so on. Cayley digraphs are very appropriate for such consideration as they directly or indirectly reveal various properties of the groups, and give a clear insight into the complexity of the group. Cayley (di)graphs are named after Arthur Cayley, who first mentioned them in 1878. These graphs are naturally obtained from groups with their vertices being the elements of the group in question. Due to the symmetry these graphs \look\ exactly the same from each vertex. When investigating the properties of the groups we examine the structure of their Cayley (di)graphs and their characteristics. Groups are in fact abstract objects. Being able to investigate their properties via their Cayley (di)graphs is thus of great help since it enables us to visualise these abstract objects. At the beginning of the thesis we define the basic concepts of Group theory that are needed to understand the thesis. After that we explain the basic concepts of Graph theory and introduce a concept of the digraph. Afterwards we introduce the concept of Cayley (di)graphs, which play a central role in the thesis. In particular, we indicate how the properties of the group are re ected in their Cayley (di)graphs and how these characteristics can be determined.

• Keywords: group