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Finite permutation groupsLiebeck, Martin W. January 1979 (has links)
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<sup>(2)</sup> and Ω<sup>{2}</sup>, 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= {e<sub>1</sub>....,e<sub>n</sub>}. Given any field F of characteristic 2, make FΩ into an algebra by defining [see text for continuation of abstract].
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Subdegree growth rates of infinite primitive permutation groupsSmith, Simon Mark January 2005 (has links)
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)<sup>G</sup> 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<sub>α</sub> 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<sub>r</sub>). They conjecture that the growth of the sequence (m<sub>r</sub>) 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<sub>r</sub> ≤ c(m-2)<sup>r-1</sup>. 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.
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On linearly ordered sets and permutation groups of uncountable degreeRamsay, Denise January 1990 (has links)
In this thesis a set, Ω, of cardinality N<sub>K</sub> and a group acting on Ω, with N<sub>K+1</sub> orbits on the power set of Ω, is found for every infinite cardinal N<sub>K</sub>. Let W<sub>K</sub> denote the initial ordinal of cardinality N<sub>K</sub>. Define N := {α<sub>1</sub>α<sub>2</sub> . . . α<sub>n</sub>∣ 0 < n < w, α<sub>j</sub> ∈ w<sub>K</sub> for j = 1, . . .,n, α<sub>n</sub> 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<sub>K</sub> and do not embed w*<sub>1</sub>. Each interval in N or R embeds every ordinal of cardinality N<sub>K</sub> and every countable converse ordinal. N and R then embed every K-type of cardinality N<sub>K</sub> 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<sup>w</sup><sub>k</sub>. In particular, if α is an ordinal between w<sup>w</sup><sub>k</sub> and w<sub>K+1</sub>, and A is a set of order type α then A= ⋃<sub>n<w</sub>A<sub>n</sub> where each A<sub>n</sub> has order type w<sup>n</sup><sub>k</sub>. 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 ⋃<sup>.</sup> Y. Thus there is only one PWOP equivalence class of N<sub>K</sub>-dense K-types which have cardinality N<sub>K</sub>, and which do not embed w*<sub>1</sub>. There are N<sub>K+1</sub> PWOP equivalence classes of ordinals of cardinality N</sub>K</sub>. Hence the PWOP automorphisms of R have N<sub>K+1</sub> orbits on θ(R). The countably piece- wise orderpreserving automorphisms of R have N<sub>0</sub> orbits on R if ∣k∣ is smaller than w<sub>1</sub> and ∣k∣ if it is not smaller.
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An elementary characterization of the simple groups PSL (3, 3) and M 11 in terms of the centralizer of an involution /Doyle, John. January 1984 (has links) (PDF)
Thesis (M. Sc.)--University of Adelaide, Dept. of Pure Mathematics, 1984. / Includes bibliographical references (leaves 87-88).
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The large scale geometry of nilpotent-by-cycle groups /Ahlin, Ashley Reiter. January 2002 (has links)
Thesis (Ph. D.)--University of Chicago, Department of Mathematics, June 2002. / Includes bibliographical references. Also available on the Internet.
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Logical implications between different flavors of asphericity /Biskup, Igor Marko. January 2000 (has links)
Thesis (Ph. D.)--Oregon State University, 2000. / Typescript (photocopy). Includes bibliographical references (leaves 60-63). Also available on the World Wide Web.
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Kernel-trace approach to congruences on regular and inverse semigroupsSondecker, Victoria L. January 1994 (has links)
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).
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Quasi-isometric rigidity of higher rank S-arithmetic lattices /Wortman, Kevin. January 2003 (has links)
Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 2003. / Includes bibliographical references. Also available on the Internet.
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Solvability in groups of piecewise-linear homeomorphisms of the unit intervalBleak, Collin. January 2005 (has links)
Thesis (Ph. D.)--State University of New York at Binghamton, Mathematical Sciences Department, 2005. / Includes bibliographical references.
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Non-commutative harmonic analysis on certain semi-direct product groups /Aafif, Amal. Boyer, Robert Paul. Krandick, Werner J. January 2007 (has links)
Thesis (Ph. D.)--Drexel University, 2007. / Includes abstract and vita. Includes bibliographical references (leaves [111]-116).
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