Composition of permutation representations of triangle groups

Abstract

A triangle group is denoted by (p, q, r) and has finite presentation (p, q, r) = hx, y|xp = yq = (xy)r = 1i. In the 1960’s Higman conjectured that almost every triangle group has among its homomorphic images all but finitely many of the alternating groups. This was proved by Everitt in [6]. In this thesis, we combine permutation representations using the methods used in the proof of Higman’s conjecture. We do some experiments by using GAP code and then we examine the situations where the composition of a number of coset diagrams for a triangle group is imprimitive. Chapter 1 provides the introduction of the thesis. Chapter 2 contains some basic results from group theory and definitions. In Chapter 3 we describe our construction that builds compositions of coset diagrams. In Chapter 4 we describe three situations that make the composition of coset diagrams imprimitive and prove some results about the structure of the permutation groups we construct. We conduct experiments based on the theorems we proved and analyse the experiments. In Chapter 5 we prove that if a triangle group G has an alternating group as a finite quotient of degree deg > 6 containing at least one handle, then G has a quotient Cdeg−1 p o Adeg. We also prove that if, for an integer m 6= deg − 1 such that m > 4 and the alternating group Am can be generated by two product of disjoint p-cycles, and a triangle group G has a quotient Adeg containing two disjoint handles, then G also has a quotient Am o Adeg.