Two problems in the theory of finite permutation groups are considered in this thesis:<ul><li> A. transitive groups of degree p, where p = 4q+1 and p,q are prime,</li><li> B. automorphism groups of 2-graphs and some related algebras.</li></ul> 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].

If G is a group acting on a set Ω, and α, β ∈ Ω, the directed graph whose vertex set is Ω and whose edge set is the orbit (α, β)^G is called an orbital graph of G. These graphs have many uses in permutation group theory. A graph Γ is said to be primitive if its automorphism group acts primitively on its vertex set, and is said to have connectivity one if there is a vertex α such that the graph Γ\{α} is not connected. A half-line in Γ is a one-way infinite path in Γ. The ends of a locally finite graph Γ are equivalence classes on the set of half-lines: two half-lines lie in the same end if there exist infinitely many disjoint paths between them. A complete characterisation of the primitive undirected graphs with connectivity one is already known. We give a complete characterisation in the directed case. This enables us to show that if G is a primitive permutation group with a locally finite orbital graph with more than one end, then G has a connectivity-one orbital graph Γ, and that this graph is essentially unique. Through the application of this result we are able to determine both the structure of G, and its action on the end space of Γ. If α ∈ Ω, the orbits of the stabiliser G_α are called the α-suborbits of G. The size of an α-suborbit is called a subdegree. If all subdegrees of an infinite primitive group G are finite, Adeleke and Neumann claim one may enumerate them in a non-decreasing sequence (m_r). They conjecture that the growth of the sequence (m_r) is extremal when G acts distance transitively on a locally finite graph; that is, for all natural numbers m the stabiliser in G of any vertex α permutes the vertices lying at distance m from α transitively. They also conjecture that for any primitive group G possessing a finite self-paired suborbit of size m there might exist a number c which perhaps depends upon G, perhaps only on m, such that m_r ≤ c(m-2)^r-1. We show their questions are poorly posed, as there exist primitive groups possessing at least two distinct subdegrees, each occurring infinitely often. The subdegrees of such groups cannot be enumerated as claimed. We give a revised definition of subdegree enumeration and growth, and show that under these new definitions their conjecture is true for groups exhibiting exponential subdegree growth above a prescribed bound.

In this thesis a set, Ω, of cardinality N_K and a group acting on Ω, with N_K+1 orbits on the power set of Ω, is found for every infinite cardinal N_K. Let W_K denote the initial ordinal of cardinality N_K. Define N := {α₁α₂ . . . α_n∣ 0 < n < w, α_j ∈ w_K for j = 1, . . .,n, α_n a successor ordinal} R := {ϰ ∈ N ∣ length(ϰ) = 1 mod 2} and let these sets be ordered lexicographically. The order types of N and R are Κ-types (countable unions of scattered types) which have cardinality N_K and do not embed w*₁. Each interval in N or R embeds every ordinal of cardinality N_K and every countable converse ordinal. N and R then embed every K-type of cardinality N_K with no uncountable descending chains. Hence any such order type can be written as a countable union of wellordered types, each of order type smaller than w^w_k. In particular, if α is an ordinal between w^w_k and w_K+1, and A is a set of order type α then A= ⋃_n<wA_n where each A_n has order type wⁿ_k. If X is a subset of N with X and N - X dense in N, then X is orderisomorphic to R, whence any dense subset of R has the same order type as R. If Y is any subset of R then R is (finitely) piece- wise order-preserving isomorphic (PWOP) to R ⋃^. Y. Thus there is only one PWOP equivalence class of N_K-dense K-types which have cardinality N_K, and which do not embed w*₁. There are N_K+1 PWOP equivalence classes of ordinals of cardinality N</sub>K</sub>. Hence the PWOP automorphisms of R have N_K+1 orbits on θ(R). The countably piece- wise orderpreserving automorphisms of R have N₀ orbits on R if ∣k∣ is smaller than w₁ and ∣k∣ if it is not smaller.

Thesis (M. Sc.)--University of Adelaide, Dept. of Pure Mathematics, 1984. / Includes bibliographical references (leaves 87-88).

Thesis (Ph. D.)--University of Chicago, Department of Mathematics, June 2002. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--Oregon State University, 2000. / Typescript (photocopy). Includes bibliographical references (leaves 60-63). Also available on the World Wide Web.

Thesis (M.A.)--Kutztown University of Pennsylvania, 1994. / Source: Masters Abstracts International, Volume: 45-06, page: 3173. Abstract precedes thesis as [2] preliminary leaves. Typescript. Includes bibliographical references (leaves 52-53).

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2003. / Includes bibliographical references. Also available on the Internet.

Thesis (Ph. D.)--State University of New York at Binghamton, Mathematical Sciences Department, 2005. / Includes bibliographical references.

Thesis (Ph. D.)--Drexel University, 2007. / Includes abstract and vita. Includes bibliographical references (leaves [111]-116).