Vertex-transitive digraphs with extra automorphisms that preserve the natural arc-colouring

Diamond open accessIn a Cayley digraph on a group G, if a distinct colour is assigned to each arc-orbit under the left-regular action of G, it is not hard to show that the elements of the left-regular action of G are the only digraph automorphisms that preserve this colouring. In this paper, we show that the equivalent statement is not true in the most straightforward generalisation to G-vertex-transitive digraphs, even if we restrict the situation to avoid some obvious potential problems. Specifically, we display an infinite family of 2-closed groups G, and a G-arc-transitive digraph on each (without any digons) for which there exists an automorphism of the digraph that is not an element of G (it is an automorphism of G). Since the digraph is G-arc-transitive, the arcs would all be assigned the same colour under the colouring by arc-orbits, so this digraph automorphism is colour-preserving.Ye

Investigating Abstract Algebra Students' Representational Fluency and Example-Based Intuitions

The quotient group concept is a difficult for many students getting started in abstract algebra (Dubinsky et al., 1994; Melhuish, Lew, Hicks, and Kandasamy, 2020). The first study in this thesis explores an undergraduate, a first-year graduate, and second-year graduate students' representational fluency as they work on a "collapsing structure", quotient, task across multiple registers: Cayley tables, group presentations, Cayley digraphs to Schreier coset digraphs, and formal-symbolic mappings. The second study characterizes the (partial) make-up of two graduate learners' example-based intuitions related to orbit-stabilizer relationships induced by group actions. The (partial) make-up of a learner's intuition as a quantifiable object was defined in this thesis as a point viewed in R17, 12 variable values collected with a new prototype instrument, The Non-Creative versus Creative Forms of Intuition Survey (NCCFIS), 2 values for confidence in truth value, and 3 additional variables: error to non-error type, unique versus common, and network thinking. The revised Fuzzy C-Means Clustering Algorithm (FCM) by Bezdek et al. (1981) was used to classify the (partial) make-up of learners' reported intuitions into fuzzy sets based on attribute similarity

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