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Dissertations / Theses on the topic 'Permutation groups. Finite groups'
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1
Walton, Jacqueline. "Representing the quotient groups of a finite permutation group." Thesis, University of Warwick, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.340088.
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Xu, Jing. "On closures of finite permutation groups /." Connect to this title, 2005. http://theses.library.uwa.edu.au/adt-WU2006.0023.
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Xu, Jing. "On closures of finite permutation groups." University of Western Australia. School of Mathematics and Statistics, 2006. http://theses.library.uwa.edu.au/adt-WU2006.0023.
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[Formulae and special characters in this field can only be approximated. See PDF version for accurate reproduction] In this thesis we investigate the properties of k-closures of certain finite permutation groups. Given a permutation group G on a finite set Ω, for k ≥ 1, the k-closure G(k) of G is the largest subgroup of Sym(Ω) with the same orbits as G on the set Ωk of k-tuples from Ω. The first problem in this thesis is to study the 3-closures of affine permutation groups. In 1992, Praeger and Saxl showed if G is a finite primitive group and k ≥ 2 then either G(k) and G have the same socle or (G(k),G) is known. In the case where the socle of G is an elementary abelian group, so that G is a primitive group of affine transformations of a finite vector space, the fact that G(k) has the same socle as G gives little information about the relative sizes of the two groups G and G(k). In this thesis we use Aschbacher’s Theorem for subgroups of finite general linear groups to show that, if G ≤ AGL(d, p) is an affine permutation group which is not 3-transitive, then for any point α ∈ Ω, Gα and (G(3) ∩ AGL(d, p))α lie in the same Aschbacher class. Our results rely on a detailed analysis of the 2-closures of subgroups of general linear groups acting on non-zero vectors and are independent of the finite simple group classification. In addition, modifying the work of Praeger and Saxl in [47], we are able to give an explicit list of affine primitive permutation groups G for which G(3) is not affine. The second research problem is to give a partial positive answer to the so-called Polycirculant Conjecture, which states that every transitive 2-closed permutation group contains a semiregular element, that is, a permutation whose cycles all have the same length. This would imply that every vertex-transitive graph has a semiregular automorphism. In this thesis we make substantial progress on the Polycirculant Conjecture by proving that every vertex-transitive, locally-quasiprimitive graph has a semiregular automorphism. The main ingredient of the proof is the determination of all biquasiprimitive permutation groups with no semiregular elements. Publications arising from this thesis are [17, 54].
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Benjamin, Ian Francis. "Quasi-permutation representations of finite groups." Thesis, University of Liverpool, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.250561.
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Vauhkonen, Antti Kalervo. "Finite primitive permutation groups of rank 4." Thesis, Imperial College London, 1993. http://hdl.handle.net/10044/1/58543.
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In this thesis we classify finite primitive permutation groups of rank 4. According to the 0' Nan-Scott theorem, a finite primitive permutation group is an affine group, an almost simple group, or has either simple diagonal action, product action or twisted wreath action. In Chapter 1 we completely determine the primitive rank 4 permutation groups with one of the last three types of actions up to permutation equivalence. In Chapter 2 we use Aschbacher's subgroup structure theorem for the finite classical groups to reduce the classification of affine primitive rank 4 permutation groups G of degree p*^ (p prime) to the case where a point stabilizer G in G satisfies soc(G/Z(G ))=L for some ^ 0 0 0 non-abelian simple group L. In Chapter 3 we classify all such groups G with L a simple group of Lie type over a finite field of characteristic p. Finally, in Chapter 4 we determine all the faithful primitive rank 4 permutation representations of the finite linear groups up to permutation equivalence.
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Torres, Bisquertt María de la Luz. "Symmetric generation of finite groups." CSUSB ScholarWorks, 2005. https://scholarworks.lib.csusb.edu/etd-project/2625.
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Advantages of the double coset enumeration technique include its use to represent group elements in a convenient shorter form than their usual permutation representations and to find nice permutation representations for groups. In this thesis we construct, by hand, several groups, including U₃(3) : 2, L₂(13), PGL₂(11), and PGL₂(7), represent their elements in the short form (symmetric representation) and produce their permutation representations.
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Bamblett, Jane Carswell. "Algorithms for computing in finite groups." Thesis, University of Oxford, 1994. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.240616.
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Kasouha, Abeir Mikhail. "Symmetric representations of elements of finite groups." CSUSB ScholarWorks, 2004. https://scholarworks.lib.csusb.edu/etd-project/2605.
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This thesis demonstrates an alternative, concise but informative, method for representing group elements, which will prove particularly useful for the sporadic groups. It explains the theory behind symmetric presentations, and describes the algorithm for working with elements represented in this manner.
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Menezes, Nina E. "Random generation and chief length of finite groups." Thesis, University of St Andrews, 2013. http://hdl.handle.net/10023/3578.
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Part I of this thesis studies P[subscript(G)](d), the probability of generating a nonabelian simple group G with d randomly chosen elements, and extends this idea to consider the conditional probability P[subscript(G,Soc(G))](d), the probability of generating an almost simple group G by d randomly chosen elements, given that they project onto a generating set of G/Soc(G). In particular we show that for a 2-generated almost simple group, P[subscript(G,Soc(G))](2) 53≥90, with equality if and only if G = A₆ or S₆. Furthermore P[subscript(G,Soc(G))](2) 9≥10 except for 30 almost simple groups G, and we specify this list and provide exact values for P[subscript(G,Soc(G))](2) in these cases. We conclude Part I by showing that for all almost simple groups P[subscript(G,Soc(G))](3)≥139/150. In Part II we consider a related notion. Given a probability ε, we wish to determine d[superscript(ε)] (G), the number of random elements needed to generate a finite group G with failure probabilty at most ε. A generalisation of a result of Lubotzky bounds d[superscript(ε)](G) in terms of l(G), the chief length of G, and d(G), the minimal number of generators needed to generate G. We obtain bounds on the chief length of permutation groups in terms of the degree n, and bounds on the chief length of completely reducible matrix groups in terms of the dimension and field size. Combining these with existing bounds on d(G), we obtain bounds on d[superscript(ε)] (G) for permutation groups and completely reducible matrix groups.
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Lemieux, Stephane R. "Minimal degree of faithful permutation representations of finite groups." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0015/MQ48492.pdf.
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Lemieux, Stephane R. (Stephane Robert) Carleton University Dissertation Mathematics and Statistics. "Minimal degree of faithful permutation representations of finite groups." Ottawa, 1999.
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Rashwan, Osama Agami. "On the composition factors of some permutation modules." Thesis, University of East Anglia, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.323354.
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Romero, Angie Tatiana Suárez. "Computação em grupos de permutação finitos com GAP." Universidade Federal de Goiás, 2018. http://repositorio.bc.ufg.br/tede/handle/tede/8220.
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Conselho Nacional de Pesquisa e Desenvolvimento Científico e Tecnológico - CNPq
Cayley’s theorem allows us to represent a finite group as a permutations group of a finite set of points. In general, an action of a finite group G in a finite set, is described as an application of the group G in the symmetric group Sym(Ω). In this work we will describe some algorithms for permutation groups and implement them in the GAP system. We begin by describing a way of representing groups in computers, we calculate orbits, stabilizers in the basic form and by means of Schreier’s vectors. Later we make algorithms to work with primitive and transitive groups, thus arriving at the concept of BSGS, base and strong generator set, for permutation groups with the algorithm SCHREIERSIMS. In the end we work with group homomorphisms, we find the elements of a group through backtrack searches.
O Teorema de Cayley nos permite representar um grupo finito como grupo de permutações de um conjunto finito de pontos. De forma geral, uma ação de um grupo finito G em um conjunto finito Ω, é descrita como uma aplicação do grupo G no grupo simétrico Sym(Ω). Neste trabalho vamos descrever alguns algoritmos para grupos de permutação e implementa-los no sistema GAP. Começamos descrevendo uma maneira de representar grupos em computadores, calculamos órbitas, estabilizadores na forma básica e por meio de vetores de Schreier. Posteriormente fazemos algoritmos para trabalhar com grupos transitivos e primitivos, chegando assim ao conceito de, base e conjunto gerador forte (BSGS) para grupos de permutação finitos com o algoritmo SCHREIER-SIMS. No final trabalhamos com homomorfismos de grupos e encontramos os elementos de um grupo mediante pesquisas backtrack.
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Kuzucuoglu, M. "Barely transitive permutation groups." Thesis, University of Manchester, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.233097.
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Lajeunesse, Lisa (Lisa Marie) Carleton University Dissertation Mathematics and Statistics. "Models and permutation groups." Ottawa, 1996.
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Guy, Jean-Pierre. "Groupes isomorphes au groupe de multiplication d'un quasigroupe." Toulouse 3, 1993. http://www.theses.fr/1993TOU30015.
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Probleme: soit g un groupe abstrait, est-il possible de construire un quasigroupe dont le groupe de multiplication est isomorphe a g? une reponse negative sera apportee pour les groupes hamiltoniens, de heineken-mohamed, des quaternions generalises et dicycliques d'ordre 4n. Une reponse positive sera apportee pour les groupes symetriques, alternes, diedraux, les groupes de mathieu de degre 11, 12 et 23, les groupes lineaires generaux et projectifs lineaires, certains p-groupes (semi-diedraux,. . . ), les groupes de coxeter de type bn. Le probleme pose pouvant se ramener a l'etude des groupes de multiplication de boucles, l'auteur construira des boucles commutatives, a l'aide de leurs translations a gauche, dont le groupe de multiplication est isomorphe soit au groupe (4,4|2,,2n+1) de degre 4n+2, soit au groupe (2,4,4;n+1) de degre 4n+4. Nous montrerons que certains d'entre eux sont transitifs minimaux, i. E. Sans sous groupe propre transitif. D'autre part, une boucle commutative dont le groupe de multiplication est isomorphe au p-sous groupe de sylow du groupe symetrique d'ordre p#2 sera construite par l'intermediaire de sa table de multiplication. Enfin, il sera montre que si les groupes abeliens, de fischer decentres, alternes sont representables en groupe de multiplication d'une boucle, une telle representation n'existe pas pour les groupes diedraux, les groupes de frobenius, les groupes de permutations dont le stabilisateur d'un element est de cardinal 1, 2 ou 3
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Spiga, Pablo. "P elements in permutation groups." Thesis, Queen Mary, University of London, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.413152.
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McNab, C. A. "Some problems in permutation groups." Thesis, University of Oxford, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.382633.
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Astles, David Christopher. "Permutation groups acting on subsets." Thesis, University of East Anglia, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.280040.
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Yang, Keyan. "On Orbit Equivalent Permutation Groups." The Ohio State University, 2008. http://rave.ohiolink.edu/etdc/view?acc_num=osu1222455916.
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Fawcett, Joanna Bethia. "Bases of primitive permutation groups." Thesis, University of Cambridge, 2013. https://www.repository.cam.ac.uk/handle/1810/252304.
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Marion, Claude Miguel Emmanuel. "Triangle groups and finite simple groups." Thesis, Imperial College London, 2009. http://hdl.handle.net/10044/1/4371.
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This thesis contains a study of the spaces of homomorphisms from hyperbolic triangle groups to finite groups of Lie type which leads to a number of deterministic, asymptotic,and probabilistic results on the (p1, p2, p3)-generation problem for finite groups of Lie type. Let G₀ = L(pn) be a finite simple group of Lie type over the finite field Fpn and let T = Tp1,p2,p3 be the hyperbolic triangle group (x,y : xp1 = yp2 = (xy)p3 = 1) where p1, p2, p3 are prime numbers satisfying the hyperbolic condition 1/p1 + 1/p2 + 1/p3 < 1. In general, the size of Hom(T,G₀) is a polynomial in q, where q = pn, whose degree gives the dimension of Hom(T,G), where G is the corresponding algebraic group, seen as a variety. Computing the precise size of Hom(T,G₀) or giving an asymptotic estimate leads to a number of applications. One can for example investigate whether or not there is an epimorphism in Hom(T,G₀). This is equivalent to determining whether or not G₀ is a (p1, p2, p3)-group. Asymptotically, one might be interested in determining the probability that a random homomorphism in Hom(T,G₀) is an epimorphism as |G₀|→∞ . Given a prime number p, one can also ask wether there are finitely, or infinitely many positive integers n such that L(pn) is a (p1, p2, p3)-group. We solve these problems for the following families of finite simple groups of Lie type of small rank: the classical groups PSL2(q), PSL3(q), PSU3(q) and the exceptional groups 2B2(q), 2G2(q), G2(q), 3D4(q). The methods involve the character theory and the subgroup structure of these groups. Following the concept of linear rigidity of a triple of elements in GLn(Fp), used in inverse Galois theory, we introduce the concept for a hyperbolic triple of primes to be rigid in a simple algebraic group G. The triple (p1, p2, p3) is rigid in G if the sum of the dimensions of the subvarieties of elements of order p1, p2, p3 in G is equal to 2 dim G. This is the minimum required for G(pn) to have a generating triple of elements of these orders. We formulate a conjecture that if (p1, p2, p3) is a rigid triple then given a prime p there are only finitely many positive integers n such that L(pn) is a (p1, p2, p3)-group. We prove this conjecture for the classical groups PSL2(q), PSL3(q), and PSU3(q) and show that it is consistent with the substantial results in the literature about Hurwitz groups (i.e. when (p1, p2, p3) = (2, 3, 7)). We also classify the rigid hyperbolic triples of primes in algebraic groups, and in doing so we obtain some new families of non-Hurwitz groups.
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Maroti, Attila. "Permutation groups and representation theoretic invariants." Thesis, University of Birmingham, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.403013.
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Tracey, Gareth M. "Minimal generation of transitive permutation groups." Thesis, University of Warwick, 2017. http://wrap.warwick.ac.uk/97251/.
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This thesis discusses upper bounds on the minimal number of elements d(G) required to generate a finite group G. We derive explicit upper bounds for the function d on transitive and minimally transitive permutation groups, in terms of their degree n. In the transitive case, bounds obtained first by Kovács and Newman, then by Bryant, Kovács and Robinson, and finally by Lucchini, Menegazzo and Morigi, show that d(G) = O(n/ √log n), for a transitive permutation group G of degree n. In this thesis, we find best possible estimates for the constant involved.
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Sheikh, Atiqa. "Orbital diameters of primitive permutation groups." Thesis, Imperial College London, 2017. http://hdl.handle.net/10044/1/58869.
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Let G be a transitive permutation group acting on a finite set X. Recall that G is primitive if there are no non-trivial equivalence relations on X which are preserved by G. An orbital graph of G is a graph with vertex set X and edges {x,y}, where (x,y) belongs to a fixed orbit of the natural action of G on the set X x X. A well-known result by D.G. Higman asserts that G is primitive if and only if all the orbital graphs are connected. For a primitive group G, we define the orbital diameter of G to be the maximum of the diameters of all orbital graphs of G. Let C be an infinite class of finite primitive permutation groups. This gives rise to an infinite family of orbital graphs. It may be that the diameters of these orbital graphs tend to infinity. More interestingly, it may be that the diameters of all the orbital graphs are bounded above by some fixed constant; if this is the case, then we say that C is bounded. Previous results by M. W. Liebeck, D. Macpherson and K. Tent focus attention on classes of almost simple primitive permutation groups which are bounded. In the thesis we analyse the orbital diameters of three families of groups, as follows. Firstly, we analyse the alternating and symmetric groups. For the primitive actions of these groups, we give necessary numerical conditions for the orbital diameter to be bounded above by some constant c and we make the result precise for c=5. For each primitive action, we also describe either all or an infinite family of orbital graphs of diameter 2. Then we analyse the almost simple groups with socle isomorphic to the projective special linear group PSL(2,q). For the primitive actions of these groups we give necessary conditions for the orbital diameter to be bounded above by 2 and we also give information about orbital graphs of diameter 2. Lastly, we analyse a large family of simple groups known as the simple groups of Lie type which consist of the simple classical groups and exceptional groups. In particular, we analyse a class of primitive actions known as parabolic actions, giving a precise description of the actions for which the orbital diameter is at most 2. For the simple classical groups, we also describe infinite families of orbital graphs of diameter 2.
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Fiddes, Ceridwyn. "The cyclizer function on permutation groups." Thesis, University of Bath, 2003. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.425697.
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Schaefer, Artur. "Synchronizing permutation groups and graph endomorphisms." Thesis, University of St Andrews, 2016. http://hdl.handle.net/10023/9912.
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The current thesis is focused on synchronizing permutation groups and on graph endo- morphisms. Applying the implicit classification of rank 3 groups, we provide a bound on synchronizing ranks of rank 3 groups, at first. Then, we determine the singular graph endomorphisms of the Hamming graph and related graphs, count Latin hypercuboids of class r, establish their relation to mixed MDS codes, investigate G-decompositions of (non)-synchronizing semigroups, and analyse the kernel graph construction used in the theorem of Cameron and Kazanidis which identifies non-synchronizing transformations with graph endomorphisms [20]. The contribution lies in the following points: 1. A bound on synchronizing ranks of groups of permutation rank 3 is given, and a complete list of small non-synchronizing groups of permutation rank 3 is provided (see Chapter 3). 2. The singular endomorphisms of the Hamming graph and some related graphs are characterised (see Chapter 5). 3. A theorem on the extension of partial Latin hypercuboids is given, Latin hyper- cuboids for small values are counted, and their correspondence to mixed MDS codes is unveiled (see Chapter 6). 4. The research on normalizing groups from [3] is extended to semigroups of the form < G, T >, and decomposition properties of non-synchronizing semigroups are described which are then applied to semigroups induced by combinatorial tiling problems (see Chapter 7). 5. At last, it is shown that all rank 3 graphs admitting singular endomorphisms are hulls and it is conjectured that a hull on n vertices has minimal generating set of at most n generators (see Chapter 8).
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Hyatt, Matthew. "Quasisymmetric Functions and Permutation Statistics for Coxeter Groups and Wreath Product Groups." Scholarly Repository, 2011. http://scholarlyrepository.miami.edu/oa_dissertations/609.
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Eulerian quasisymmetric functions were introduced by Shareshian and Wachs in order to obtain a q-analog of Euler's exponential generating function formula for the Eulerian polynomials. They are defined via the symmetric group, and applying the stable and nonstable principal specializations yields formulas for joint distributions of permutation statistics. We consider the wreath product of the cyclic group with the symmetric group, also known as the group of colored permutations. We use this group to introduce colored Eulerian quasisymmetric functions, which are a generalization of Eulerian quasisymmetric functions. We derive a formula for the generating function of these colored Eulerian quasisymmetric functions, which reduces to a formula of Shareshian and Wachs for the Eulerian quasisymmetric functions. We show that applying the stable and nonstable principal specializations yields formulas for joint distributions of colored permutation statistics. The family of colored permutation groups includes the family of symmetric groups and the family of hyperoctahedral groups, also called the type A Coxeter groups and type B Coxeter groups, respectively. By specializing our formulas to these cases, they reduce to the Shareshian-Wachs q-analog of Euler's formula, formulas of Foata and Han, and a new generalization of a formula of Chow and Gessel.
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Penrod, Keith. "Infinite product groups /." Diss., CLICK HERE for online access, 2007. http://contentdm.lib.byu.edu/ETD/image/etd1977.pdf.
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Turner, Simon. "The cyclizer series of infinite permutation groups." Thesis, University of Bath, 2013. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.577751.
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The cyclizer of an infinite permutation group G is the group generated by the cycles involved in elements of G, along with G itself. There is an ascending subgroup series beginning with G, where each term in the series is the cyclizer of the previous term. We call this series the cyclizer series for G. If this series terminates then we say the cyclizer length of G is the length of the respective cyclizer series. We study several innite permutation groups, and either determine their cyclizer series, or determine that the cyclizer series terminates and give the cyclizer length. In each of the innite permutation groups studied, the cyclizer length is at most 3. We also study the structure of a group that arises as the cyclizer of the innite cyclic group acting regularly on itself. Our study discovers an interesting innite simple group, and a family of associated innite characteristically simple groups.
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Cox, Charles. "Infinite permutation groups containing all finitary permutations." Thesis, University of Southampton, 2016. https://eprints.soton.ac.uk/401538/.
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Groups naturally occu as the symmetries of an object. This is why they appear in so many different areas of mathematics. For example we find class grops in number theory, fundamental groups in topology, and amenable groups in analysis. In this thesis we will use techniques and approaches from various fields in order to study groups. This is a 'three paper' thesis, meaning that the main body of the document is made up of three papers. The first two of these look at permutation groups which contain all permutations with finite support, the first focussing on decision problems and the second on the R? property (which involves counting the number of twisting conjugacy classes in a group). The third works with wreath products C}Z where C is cyclic, and looks to dermine the probability of choosing two elements in a group which commute (known as the degree of commutativity, a topic which has been studied for finite groups intensely but at the time of writing this thesis has only two papers involving infinite groups, one of which is in this thesis).
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Mazhar, Siddiqua. "Composition of permutation representations of triangle groups." Thesis, University of Newcastle upon Tyne, 2017. http://hdl.handle.net/10443/3857.
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A triangle group is denoted by (p, q, r) and has finite presentation (p, q, r) = hx, y|xp = yq = (xy)r = 1i. In the 1960’s Higman conjectured that almost every triangle group has among its homomorphic images all but finitely many of the alternating groups. This was proved by Everitt in [6]. In this thesis, we combine permutation representations using the methods used in the proof of Higman’s conjecture. We do some experiments by using GAP code and then we examine the situations where the composition of a number of coset diagrams for a triangle group is imprimitive. Chapter 1 provides the introduction of the thesis. Chapter 2 contains some basic results from group theory and definitions. In Chapter 3 we describe our construction that builds compositions of coset diagrams. In Chapter 4 we describe three situations that make the composition of coset diagrams imprimitive and prove some results about the structure of the permutation groups we construct. We conduct experiments based on the theorems we proved and analyse the experiments. In Chapter 5 we prove that if a triangle group G has an alternating group as a finite quotient of degree deg > 6 containing at least one handle, then G has a quotient Cdeg−1 p o Adeg. We also prove that if, for an integer m 6= deg − 1 such that m > 4 and the alternating group Am can be generated by two product of disjoint p-cycles, and a triangle group G has a quotient Adeg containing two disjoint handles, then G also has a quotient Am o Adeg.
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Schimanski, Nichole Louise. "Orthomorphisms of Boolean Groups." PDXScholar, 2016. http://pdxscholar.library.pdx.edu/open_access_etds/3100.
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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.
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Ruengrot, Pornrat. "Perfect isometry groups for blocks of finite groups." Thesis, University of Manchester, 2011. https://www.research.manchester.ac.uk/portal/en/theses/perfect-isometry-groups-for-blocks-of-finite-groups(092f1a9a-1583-4e8e-b285-a77c49e48765).html.
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Our aim is to investigate perfect isometry groups, which are invariants for blocks of finite groups. There are two subgoals. First is to study some properties of perfect isometry groups in general. We found that every perfect isometry has essentially a unique sign. This allowed us to show that, in many cases, a perfect isometry group contains a direct factor generated by -id. The second subgoal is to calculate perfect isometry groups for various blocks. Notable results include the perfect isometry groups for blocks with defect 1, abelian p-groups, extra special p-groups, and the principal 2-block of the Suzuki group Sz(q). In the case of blocks with defect 1, we also showed that every perfect isometry can be induced by a derived equivalence. With the help of a computer, we also calculated perfect isometry groups for some blocks of sporadic simple groups.Apart from perfect isometries, we also investigated self-isotypies in the special case where C_G(x) is a p-group whenever x is a p-element. We applied our result to calculate isotypies in cyclic p-groups and the principal 2-blocks of the Suzuki group Sz(q).
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Zou, Tingxiang. "Structures pseudo-finies et dimensions de comptage." Thesis, Lyon, 2019. http://www.theses.fr/2019LYSE1083/document.
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Cette thèse porte sur la théorie des modèles des structures pseudo-finies en mettant l’accent sur les groupes et les corps. Le but est d'approfondir notre compréhension des interactions entre les dimensions de comptage pseudo-finies et les propriétés algébriques de leurs structures sous-jacentes, ainsi que de la classification de certaines classes de structures en fonction de leurs dimensions. Notre approche se fait par l'étude d'exemples. Nous avons examiné trois classes de structures. La première est la classe des H-structures, qui sont des expansions génériques. Nous avons donné une construction explicite de H-structures pseudo-finies comme ultraproduits de structures finies. Le deuxième exemple est la classe des corps aux différences finis. Nous avons étudié les propriétés de la dimension pseudo-finie grossière de cette classe. Nous avons montré qu'elle est définissable et prend des valeurs entières, et nous avons trouvé un lien partiel entre cette dimension et le degré de transcendance transformelle. Le troisième exemple est la classe des groupes de permutations primitifs pseudo-finis. Nous avons généralisé le théorème classique de classification de Hrushovski pour les groupes stables de permutations d'un ensemble fortement minimal au cas où une dimension abstraite existe, cas qui inclut à la fois les rangs classiques de la théorie des modèles et les dimensions de comptage pseudo-finies. Dans cette thèse, nous avons aussi généralisé le théorème de Schlichting aux sous-groupes approximatifs, en utilisant une notion de commensurabilitéThis thesis is about the model theory of pseudofinite structures with the focus on groups and fields. The aim is to deepen our understanding of how pseudofinite counting dimensions can interact with the algebraic properties of underlying structures and how we could classify certain classes of structures according to their counting dimensions. Our approach is by studying examples. We treat three classes of structures: The first one is the class of H-structures, which are generic expansions of existing structures. We give an explicit construction of pseudofinite H-structures as ultraproducts of finite structures. The second one is the class of finite difference fields. We study properties of coarse pseudofinite dimension in this class, show that it is definable and integer-valued and build a partial connection between this dimension and transformal transcendence degree. The third example is the class of pseudofinite primitive permutation groups. We generalise Hrushovski's classical classification theorem for stable permutation groups acting on a strongly minimal set to the case where there exists an abstract notion of dimension, which includes both the classical model theoretic ranks and pseudofinite counting dimensions. In this thesis, we also generalise Schlichting's theorem for groups to the case of approximate subgroups with a notion of commensurability
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Sanders, Paul Anthony. "Some 2-groups and their automorphism groups." Thesis, University of Oxford, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.329987.
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Smith, Simon Mark. "Subdegree growth rates of infinite primitive permutation groups." Thesis, University of Oxford, 2005. http://ora.ox.ac.uk/objects/uuid:1baa0e15-363a-4163-b21b-59fcd62d210b.
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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 Gα 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 (mr). They conjecture that the growth of the sequence (mr) 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 mr ≤ 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.
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JÃnior, Raimundo de AraÃjo Bastos. "Commutators in finite groups." Universidade Federal do CearÃ, 2010. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=5496.
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Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgicoOs problemas que abordaremos estÃo diretamente associados à existÃncia de elementos no subgrupo derivado que nÃo sÃo comutadores. Nosso objetivo serà apresentar os resultados de Tim Bonner, que sÃo estimativas para a razÃo entre o comprimento do derivado e a ordem do grupo (limitaÃÃo superior e determinaÃÃo do "comportamento assintÃtico"), culminando com uma prova da conjectura de Bardakov.
The problems which we address in this work are directly related to the existence of elements in the derived subgroup that are not commutators. Our purpose is to present the results of Tim Bonner [1]. In his paper, one finds estimates for the ratio between the commutator length and the order of group (more precisely, upper limits and the establishment of its asymptotic behavior), leading to the proof of Bardakov's Conjecture.
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Stavis, Andreas. "Representations of finite groups." Thesis, Karlstads universitet, 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:kau:diva-69642.
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Representation theory is concerned with the ways of writing elements of abstract algebraic structures as linear transformations of vector spaces. Typical structures amenable to representation theory are groups, associative algebras and Lie algebras. In this thesis we study linear representations of finite groups. The study focuses on character theory and how character theory can be used to extract information from a group. Prior to that, concepts needed to treat character theory, and some of their ramifications, are investigated. The study is based on existing literature, with excessive use of examples to illuminate important aspects. An example treating a class of p-groups is also discussed.
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Majid, Shahn, and Andreas Cap@esi ac at. "Riemannian Geometry of Quantum Groups and Finite Groups with." ESI preprints, 2000. ftp://ftp.esi.ac.at/pub/Preprints/esi902.ps.
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Howden, David J. A. "Computing automorphism groups and isomorphism testing in finite groups." Thesis, University of Warwick, 2012. http://wrap.warwick.ac.uk/50060/.
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We outline a new method for computing automorphism groups and performing isomorphism testing for soluble groups. We derive procedures for computing polycyclic presentations for soluble automorphism groups, allowing for much more efficient calculations. Finally, we demonstrate how these methods can be extended to tackle some non-soluble groups. Performance statistics are included for an implementation of these algorithms in the MAGMA [BCP97] language.
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Helleloid, Geir T. "Automorphism groups of finite p-groups : structure and applications /." May be available electronically:, 2007. http://proquest.umi.com/login?COPT=REJTPTU1MTUmSU5UPTAmVkVSPTI=&clientId=12498.
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Iniguez-Goizueta, Ainhoa. "Word fibres in finite p-groups and pro-p groups." Thesis, University of Oxford, 2016. https://ora.ox.ac.uk/objects/uuid:3a9cfc11-d171-4876-82b3-7dff012c3a70.
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Roney-Dougal, Colva Mary. "Permutation groups with a unique nondiagonal self-paired orbital." Thesis, Queen Mary, University of London, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.246981.
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Sharp, Graham R. "Recognition algorithms for actions of permutation groups on pairs." Thesis, University of Oxford, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.244602.
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Treacher, Helen. "The reconstruction index of semi-2-regular permutation groups." Thesis, University of East Anglia, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.429591.
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Blackford, J. Thomas. "Permutation groups of extended cyclic codes over Galois Rings /." The Ohio State University, 1999. http://rave.ohiolink.edu/etdc/view?acc_num=osu1488186329502909.
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Badar, Muhammad. "Dynamical Systems Over Finite Groups." Thesis, Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-17948.
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In this thesis, the dynamical system is used as a function on afinite group, to show how states change. We investigate the'numberof cycles' and 'length of cycle' under finite groups. Using grouptheory, fixed point, periodic points and some examples, formulas tofind 'number of cycles' and 'length of cycle' are derived. Theexamples used are on finite cyclic group Z_6 with respectto binary operation '+'. Generalization using finite groups ismade. At the end, I compared the dynamical system over finite cyclic groups with the finite non-cyclic groups and then prove the general formulas to find 'number of cycles' and 'length of cycle' for both cyclic and non-cyclic groups.
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Bidwell, Jonni, and n/a. "Computing automorphisms of finite groups." University of Otago. Department of Mathematics & Statistics, 2007. http://adt.otago.ac.nz./public/adt-NZDU20070320.162909.
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In this thesis we explore the problem of computing automorphisms of finite groups, eventually focusing on some group product constructions. Roughly speaking, the automorphism group of a group gives the nature of its internal symmetry. In general, determination of the automorphism group requires significant computational effort and it is advantageous to find situations in which this may be reduced.
The two main results give descriptions of the automorphism groups of finite direct products and split metacyclic p-groups. Given a direct product G = H x K where H and K have no common direct factor, we give the order and structure of Aut G in terms of Aut H, Aut K and the central homomorphism groups Hom (H, Z(K)) and Hom (K, Z(H)). A similar result is given for the the split metacyclic p-group, in the case where p is odd. Implementations of both of these results are given as functions for the computational algebra system GAP, which we use extensively throughout.
An account of the literature and relevant standard results on automorphisms is given. In particular we mention one of the more esoteric constructions, the automorphism tower. This is defined as the series obtained by repeatedly taking the automorphism group of some starting group G₀. There is interest as to whether or not this series terminates, in the sense that some group is reached that is isomorphic to its group of automorphisms. Besides a famous result of Wielandt in 1939, there has not been much further insight gained here. We make use of the technology to construct several examples, demonstrating their complex and varied behaviour.
For the main results we introduce a 2 x 2 matrix description for the relevant automorphism groups, where the entries come from the homorphism groups mentioned previously. In the case of the direct product, this is later generalised to an n x n matrix (when we consider groups with any number of direct factors) and the common direct factor restriction is relaxed to the component groups not having a common abelian direct factor. In the case of the split metacyclic p-group, our matrices have entries that are not all homomorphisms, but are similar. We include the code for our GAP impementation of these results, which we show significantly expedites computation of the automorphism groups.
We show that this matrix language can be used to describe automorphisms of any semidirect product and certain central products too, although these general cases are much more complicated. Specifically, multiplication is no longer defined in such a natural way as is seen in the previous cases and the matrix entries are mappings much less well-behaved than homomorphisms. We conclude with some suggestion of types of semidirect products for which our approach may yield a convenient description of the automorphisms.
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Jackson, Jack Lee. "Splitting in finite metacyclic groups." Diss., The University of Arizona, 1999. http://hdl.handle.net/10150/289018.
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It is well known that all finite metacyclic groups have a presentation of the form G = ‹a,x,aᵐ = 1,xˢaᵗ = 1,aˣ = aʳ›. The primary question that occupies this dissertation is determining under what conditions a group with such a presentation splits over the given normal subgroup ‹a›. Necessary and sufficient conditions are given for splitting, and techniques for finding complements are given in the cases where G splits over ‹a›. Several representative examples are examined in detail, and the splitting theorem is applied to give alternate proofs of theorems of Dedekind and Blackburn.