Digraph Groups and Related Groups

Tez Türü: Doktora

Tezin Yürütüldüğü Kurum: University of Essex, Matematik Bilimleri Bölümü, Matemarık, İngiltere

Tez Danışmanı: Gerald Williams

Tezin Onay Tarihi: 2022

Tezin Dili: İngilizce

Açık Arşiv Koleksiyonu: AVESİS Açık Erişim Koleksiyonu

Desteklendiği Program: Diğer

Özet:

This thesis investigates finite digraph groups and related groups like the generalization

of Johnson and Mennicke groups. Cuno and Williams introduced the

term "digraph group" for the first time in [9], 2020. The groups are defined by

non-empty presentations and each relator is in the form R(x, y), where x and y

are distinct generators and R(., .) is defined by some fixed cyclically reduced word

R(a, b) that involves both a and b. There is a directed graph associated with each

of these presentations, where the vertices correspond to the generators and the arcs

correspond to the relators. In Chapter 2, we investigate Cayley digraph groups

to determine whether they are finite cyclic and provide formulae to calculate the

order. In Chapters 3 and 4, the girth of the underlying undirected graph is at

least 4. We show that the resulting groups are non-trivial and cannot be finite of

rank 3 or higher under the condition |V | = |A| − 1 in Chapter 3. We investigate

when the corresponding digraph groups are finite cyclic for |V | 6 |A| in Chapter 4

and we are able to show that the corresponding group of strongly connected and

semi-connected digraphs under certain standard conditions which are known to be

necessary for the digraph group to be finite ((i)−(iv) defined in Preamble 4.1). We

generalise Johnson and Mennicke groups, which are non-cyclic finite groups defined

in terms of a digraph that is a directed triangle to digraphs that are n−vertex

tournaments in Chapter 5. In Chapter 6 we use GAP to perform a computational

investigation into digraph groups with particular relators and we obtain results

whether the corresponding digraph groups are cyclic, abelian, perfect or not. We

also provide their size, derived series, derived length and facts about isomorphism

between them. The relators used correspond to the those used in the Mennicke and

Johnson groups, and some new fixed relators. We obtain digraph presentations of

various 2-groups, 3-groups and perfect groups.