Finite permutation groups

Thesis



Two problems in the theory of finite permutation groups are considered in this thesis:

• A. transitive groups of degree p, where p = 4q+1 and p,q are prime,
• B. automorphism groups of 2-graphs and some related algebras.

Problem A should be seen in the following context: in 1963. N.Ito began a study of insoluble, transitive groups G of degree p on a set Ω, where p = 2q+1 and p,q are prime, showing among other things, that such a group G is 3-transitive. His methods involve the modular character theory of G for both the primes p and q (developed by R.Brauer). He uses this theory to prove facts about the permutation characters of G associated with Ω⁽²⁾ and Ω^{2}, the sets of ordered and unordered pairs (respectively) of distinct elements of Ω. The first part of this thesis represents an attempt to extend these methods to the case p = 4q+1. The main result obtained is

Theorem. Let G be an insoluble, transitive permutation group of degree p, where p = 4q+1 and p.q are prime with p>13. Then G is 3-transitive. Also some progress is made towards a proof that the groups in Problem A are 4-transitive.

In the second part of this thesis (Problem B) certain algebras are defined from 2-graphs as follows: let (Ω,Δ) be a 2-graph, that is, Δ is a set of 3-subsets of a finite set Ω such that every 4-subset of Ω contains an even number of elements of Δ. Write Ω= {e₁....,e_n}. Given any field F of characteristic 2, make FΩ into an algebra by defining [see text for continuation of abstract].

Authors: Liebeck, M; Liebeck, Martin W.

• Publication date: 1979
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Subjects: Permutation groups; Group theory