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Title: Finite permutation groups
Author: Liebeck, Martin W.
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 1979
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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].
Supervisor: Neumann, P. M. Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Group theory ; Permutation groups