We investigate non-geometric sharp permutation groups of type {0,k} whose point stabilizers are certain Frobenius groups. We show that if a point stabilizer has a cyclic Frobenius kernel whose order is a power of a prime and Frobenius complement cyclic of prime order, then the point stabilizer is isomorphic to the symmetric group on 3 letters, and there is up to permutation isomorphism, one such permutation group. Further, we determine a significant structural description of non-geometric sharp permutation groups of type {0,k} whose point stabilizers are Frobenius groups with elementary abelian Frobenius kernel K and Frobenius complement L with |L| = |K|-1. As a result of this structural description, it is shown that the smallest non-solvable Frobenius group cannot be a point stabilizer in a non-geometric sharp permutation group of type {0,k}.

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The problem of edge disjoint path routing arises from applications in distributed memory parallel computing. We examine this problem in both the directed hypercube and two-dimensional grid topologies. Complexity results are obtained for these problems where the routing must consist entirely of shortest length paths. Additionally, approximation algorithms are presented for the case when the routing request is of a special form known as a permutation. Permutations simply require that no vertex in the graph may be used more than once as either a source or target for a routing request. Szymanski conjectured that permutations are always routable in the directed hypercube, and this remains an open problem.

Ce mémoire est centré sur l'étude des coefficients de corrélation partiels en régression linéaire multiple, à travers les tests de permutation. Ces tests sont nécessaires lorsque les suppositions du modèle linéaire ne sont pas verifiées, et l'application des tests classiques est erronnée. On présente les bases théoriques de trois méthodes de la littérature, Manly (1991), Freedman et Lane (1983) et Kennedy (1995), et on fait une étude de simulation afin de les comparer. On ajoute aux comparaisons le test paramétrique, ainsi qu'une méthode qu'on propose. On regarde l'erreur de type I et la puissance de ces tests. Un dernier volet du mémoire est la présentation des tests de Mantel (1967) et Smoose et al. (1986) qui sont des généralisations de ces méthodes de permutation pour la régression multiple à des matrices de distances. A titre d'exemple, ces différentes techniques de permutation sont appliquées sur des matrices de distances génétiques en relation avec des distances environnementales et des distances géographiques. ______________________________________________________________________________ MOTS-CLÉS DE L’AUTEUR : Échangeabilité, Test de permutation, Test de Mantel, Test de Mantel partiel, Régression multiple, Corrélation partielle, Résidus.

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].

An orthomorphism, π, of a group, (G, +), is a permutation of G with the property that the map x → -x + π(x) is also a permutation. In this paper, we consider orthomorphisms of the additive group of binary n-tuples, Zn2. We use known orthomorphism preserving functions to prove a uniformity in the cycle types of orthomorphisms that extend certain partial orthomorphisms, and prove that extensions of particular sizes of partial orthomorphisms exist. Further, in studying the action of conjugating orthomorphisms by automorphisms, we find several symmetries within the orbits and stabilizers of this action, and other orthomorphism-preserving functions. In addition, we prove a lower bound on the number of orthomorphisms of Zn2 using the equivalence of orthomorphisms to transversals in Latin squares. Lastly, we present a Monte Carlo method for generating orthomorphisms and discuss the results of the implementation.

Parametric methods are effective and appropriate when data sets are obtained by well-defined random sampling procedures, the population distribution for responses is well-defined, the null sampling distributions of suitable test statistics do not depend on any unknown entity and well-defined likelihood models are provided for by nuisance parameters. Permutation testing methods, on the other hand, are appropriate and unavoidable when distribution models for responses are not well specified, nonparametric or depend on too many nuisance parameters; when ancillary statistics in well-specified distributional models have a strong influence on inferential results or are confounded with other nuisance entities; when the sample sizes are less than the number of parameters and when data sets are obtained by ill-specified selection-bias procedures. In addition, permutation tests are useful not only when parametric tests are not possible, but also when more importance needs to be given to the observed data set, than to the population model, as is typical for example in biostatistics. The different types of permutation methods for analysis of variance, multiple linear regression and principal component analysis are explored. More specifically, one-way, twoway and three-way ANOVA permutation strategies will be discussed. Approximate and exact permutation tests for the significance of one or more regression coefficients in a multiple linear regression model will be explained next, and lastly, the use of permutation tests used as a means to validate and confirm the results obtained from the exploratory PCA will be described. / Dissertation (MSc)--University of Pretoria, 2012. / Statistics / unrestricted