DIGRAPH GROUPS AND RELATED GROUPS

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| \leq |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.