Graphs of algebraic objects

Thesis

A rack on [𝑛] can be thought of as a set of permutations (𝑓_𝑥)_{𝑥 ∈ [𝑛]} such that 𝑓_(𝑥)𝑓y = 𝑓_y^-1𝑓_x𝑓_y for all 𝑥 and y; additionally, a rack such that (𝑥)𝑓_𝑥 = 𝑥 for all 𝑥 is a quandle, and a quandle such that 𝑓_x² = ɩ for all 𝑥 is a kei. These objects can be represented by loopless, edge-coloured directed multigraphs on [𝑛]; we put an edge of colour y between 𝑥 and z if and only if (𝑥)𝑓_y = z. This thesis studies racks and kei via these graphical representations.

In 2013, Blackburn showed that for any ε > 0 and 𝑛 sufficiently large, the number of isomorphism classes of kei on [𝑛] is at least 2^{(1/4 - ε)𝑛2}, while the number of isomorphism classes of racks on [𝑛] is at most 2^(c+ε)𝑛2 for a constant c. The main result of this thesis is that the lower bound is asymptotically correct; for any ε > 0 the number of isomorphism classes of racks on [𝑛] is at most 2^{(1/4 + ε)𝑛2}, for 𝑛 sufficiently large. We also show that in almost all of these racks almost all components of the corresponding graph have size 2. Additionally, we show several more detailed results regarding kei, including a full classification of all kei in which each component has size equal to an odd prime.

This thesis also contains several partial results on another topic. The commuting graph of a finite group is defined to be the graph on the non-trivial cosets of G/Z(G) where x̄ȳ is an edge if and only if 𝑥 and y commute. In 2012, Hegarty and Zhelezov proposed a family of random groups and published some results on the corresponding family of random graphs; this thesis contains some generalisations and extensions of these results.

Authors: Ashford, M

• DOI: 10.5287/ora-6qbkpmbnj
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Enumeration; Racks; Commuting Graph
• Subjects: Mathematics; Combinatorial analysis