Finite permutation groups - Liebeck, M; Liebeck, Martin W. (1979)
﻿ 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....,en}. Given any field F of characteristic 2, make FΩ into an algebra by defining [see text for continuation of abstract].