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1
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).
2
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.
3
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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4
Lajeunesse, Lisa (Lisa Marie) Carleton University Dissertation Mathematics and Statistics. "Models and permutation groups." Ottawa, 1996.
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5
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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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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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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8
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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9
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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10
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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11
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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12
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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Cernes, John. "Ends of permutation groups and some centrality properties of permutational wreath products." Thesis, University of Oxford, 1996. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.339282.
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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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16
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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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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20
Ramsay, Denise. "On linearly ordered sets and permutation groups of uncountable degree." Thesis, University of Oxford, 1990. http://ora.ox.ac.uk/objects/uuid:ce9a8b26-bb4c-4c85-8231-78e89ce4109d.
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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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Coutts, Hannah Jane. "Topics in computational group theory : primitive permutation groups and matrix group normalisers." Thesis, University of St Andrews, 2011. http://hdl.handle.net/10023/2561.
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Part I of this thesis presents methods for finding the primitive permutation groups of degree d, where 2500 ≤ d < 4096, using the O'Nan-Scott Theorem and Aschbacher's theorem. Tables of the groups G are given for each O'Nan-Scott class. For the non-affine groups, additional information is given: the degree d of G, the shape of a stabiliser in G of the primitive action, the shape of the normaliser N in S[subscript(d)] of G and the rank of N. Part II presents a new algorithm NormaliserGL for computing the normaliser in GL[subscript(n)](q) of a group G ≤ GL[subscript(n)](q). The algorithm is implemented in the computational algebra system MAGMA and employs Aschbacher's theorem to break the problem into several cases. The attached CD contains the code for the algorithm as well as several test cases which demonstrate the improvement over MAGMA's existing algorithm.
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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.
23
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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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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Hendriksen, Michael Arent. "Minimal Permutation Representations of Classes of Semidirect Products of Groups." Thesis, The University of Sydney, 2015. http://hdl.handle.net/2123/14353.
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Given a finite group $G$, the smallest $n$ such that $G$ embeds into the symmetric group $S_n$ is referred to as the minimal degree. Much of the accumulated literature focuses on the interplay between minimal degrees and direct products. This thesis extends this to cover large classes of semidirect products. Chapter 1 provides a background for minimal degrees - stating and proving a number of essential theorems and outlining relevant previous work, along with some small original results. Chapter 2 calculates the minimal degrees for an infinite class of semidirect products - specifically the semidirect products of elementary abelian groups by groups of prime order not dividing the order of the base group. This is established using vector space theory, including a number of novel techniques. The utility of this research is then demonstrated by answering an existing problem in the field of minimal degrees in a new and potentially generalisable way.
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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)<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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PRANDELLI, MARIATERESA. "Algebra of sets, permutation groups and invariant factors." Doctoral thesis, Università degli Studi di Milano-Bicocca, 2019. http://hdl.handle.net/10281/241257.
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Nella tesi mi occupo del problema di trovare particolari forme diagonali per le matrici di strutture di incidenza tra t-sottoinsiemi e k-sottoinsiemi (L_t,L_k; \subseteq).
Denotato con X un insieme finito di n elementi e con L l’insieme delle parti di X, consideriamo i sottoinsiemi L_i di L, for i=0,…,n; dove L_i è l’insieme di tutti i sottoinsiemi di X di cardinalità i; i.e. gli elementi di L_i sono i-sottoinsiemi di X.
(L_t,L_k; \subseteq) è la struttura d’incidenza così definita: per x in L_t e y in L_k, x e y sono incidenti se e solo se x è contenuto in y. La sua matrice d’incidenza è denotata con W_{tk}.
R.M.Wilson trova una forma diagonale per W_{tk} con metodi puramente combinatorici. Per semplicità noi chiamiamo questo teorema ``Wilson's Theorem''.
Il cuore della tesi è il capitolo 4 dove diamo una nuova dimostrazione di Wilson's Theorem via algebra lineare.
Costruiamo una nuova struttura algebrica: sia G contenuto in Sym(n) un gruppo di permutazioni su X. L’azione di G su X induce un’azione naturale su L. Così G agisce su ogni L_i.
Questa azione partiziona ogni L_i in orbite.
Per 0≤ t≤ k≤ n, consideriamo la tactical decomposition della struttura d’incidenza (L_t,L_k; \subseteq) e definiamo due matrici X^+_{tk} and X^-_{tk} chiamate le matrici d’incidenza della tactical decomposition.
Se G={1} allora le orbite di G corrispondono ai sottoinsiemi e X^+_{tk}=W^T_{tk} è la matrice trasposta della struttura d’incidenza (L_t,L_k; \subseteq).
Nel capitolo 5 diamo alcuni nuovi risultati in merito alla matrice X^+_{tk}.
Nel capitolo 4 introduciamo un’algebra collegata all’insieme delle parti L, allo scopo di dare la nostra nuova dimostrazione di Wilson's Theorem.
Sia R uno fra Q, R or C, costruiamo lo spazio vettoriale RL delle somme formali di elementi di L a coefficienti in R
Estendiamo la relazione di inclusione da L a RL. Per fare questo definiamo le due mappe di incidenza ɛ and δ tali che le matrici ad esse associate rispetto alle basi L_t e L_k siano W^T_{tk} e W_{tk}, rispettivamente.
I risultati del capitolo 4 sono raggiunti considerando una particolare base per RL_i.
Dati 0≤ t≤ n-1 e k=t+1, costruiamo la mappa simmetrica δɛ, e decomponiamo RL_t in somma diretta di autospazi di δɛ, denotati con E_{t,i}, per i=0,…,t’, dove t’=min{t,n-t}.
Proviamo che gli autospazi E_{t,i} sono Sym(n)-invarianti e irriducibili e che
ɛ (E_{t,i})=E_{k,i}.
Osserviamo che da queste decomposizioni è immediato trovare due basi in RL_t e RL_k, rispettivamente, tali che la matrice associata a ɛ: R L_t -> R L_k è la forma diagonale di W_{tk} trovata da R.M. Wilson.
Se noi consideriamo W_{tk} matrice d’incidenza della struttura d’incidenza (L_t,L_k, \subseteq), noi possiamo anche vedere W^T_{tk} come matrice associata a ɛ ristretta allo Z-modulo ZL_t.
Questo ci suggerisce di affrontare il problema via algebra lineare. Sfortunatamente il risultato per gli Z- moduli non è immediato.
Noi troviamo un insieme di generatori S_i di autovettori per lo spazio vettoriale RL_i, con i=0, …, n, chiamati polytopes.
Per il nostro approccio un importante ruolo è giocato dallo Z-modulo ZL_i con base L_i insieme con i sottomoduli ZS_i generati dai polytopes.
E’ facile provare che valgono le seguenti restrizioni: ɛ: ZL_t -> ZL_k , ɛ:ZS_t -> ZS_k
Determiniamo i fattori invarianti della matrice W^T_{tk} trovando lo Smith group di ɛ: ZL_t -> ZL_k. Il risultato è raggiunto costruendo una base standard di polytopes e deducendo opportune basi di ZL_t e ZL_k.
Nel capitolo 5 introduciamo il sottomodulo di ZS_i che consiste degli elementi fissati da G, cioè
(ZS_i)^G, e troviamo lo Smith group di ɛ:(ZS_t)^G -> (ZS_k)^G.
Restringiamo poi la nostra attenzione al caso t+k=n e proviamo che i gruppi (ZL_k)^G/ ɛ (ZL_t)^G) e (ZS_k)^G/ ɛ (ZS_t)^G) hanno lo stesso ordine.
In realtà noi congetturiamo che per t+k=n i due gruppi sono isomorfi.<br>In this thesis we deal with the problem to find particular forms for incidence matrices of incidence structures between t-subsets vs k-subsets (L_t,L_k; \subseteq).
Denote by X a set of finite size n, say X={1,2,…,n} and by L the power set of X. We partition it into the sets L_i, for i=0,…,n; where L_i is the set of subsets of X of size i; i.e. the elements of L_i are the i-subsets of X.
(L_t,L_k; \subseteq) is the incidence structure so defined: for x in L_t and y in L_k, x and y are incident if and only if x \subseteq y. Its incidence matrix is denoted by W_{tk}.
R.M.Wilson finds a diagonal form for W_{tk} with purely combinatorics methods. For shortness we will refer to this result as ``Wilson's Theorem''.
Many other authors have dealt with the same problem.
The heart of the thesis is Chapter 4 where we give a new proof of Wilson's Theorem via linear maps.
We construct a new algebraic structure: let G \subseteq Sym(n) be a permutation group on X. The action of G on X induces a natural action on L. So G acts on any L_i.
This action partitions each L_i into orbits.
For 0≤ t≤ k≤ n, we consider the tactical decomposition of the incidence structure (L_t,L_k; \subseteq). Then we can define two matrices X^+_{tk} and X^-_{tk} called the incidence matrices of the tactical decomposition.
If G={1} then the orbits of G correspond to the subsets and X^+_{tk}=W^T_{tk} is the transpose matrix of the incidence structure (L_t,L_k; \subseteq).
In Chapter 5 we will give some new results related to the invariant factors of X^+_{tk}.
In Chapter 4 we introduce an algebra related to the boolean poset L, in order to give our new proof of Wilson's Theorem.
Let R be one of Q, R or C, we construct the vector space RL of formal sums of elements of L with coefficients in R
We want to extend the \subseteq relation from L into RL. To do this we define incidence maps: ɛ and δ such that the matrices associated to ɛ and δ, with respect to the bases L_t and L_k, are W^T_{tk} and W_{tk}, respectively.
The results of Chapter 4 are achieved considering a particular basis for RL_i.
Given 0≤ t≤ n-1 and k=t+1, we construct the symmetric maps δɛ, and we decompose RL_t into direct sum of eigenspaces of δɛ, denoted by E_{t,i}, with i=0,…,t’, where t’=min{t,n-t}.
We prove that the eigenspaces E_{t,i} are irreducible Sym(n)-invariant and that
ɛ (E_{t,i})=E_{k,i}.
We observe that from these decompositions it is immediate to find two bases in RL_t and RL_k, respectively, such that the associated matrix to ɛ: R L_t -> R L_k is the diagonal form of W_{tk} found by R.M. Wilson.
If we consider W_{tk} as incidence matrix of incidence structure (L_t,L_k, \subseteq), we can see W^T_{tk} as matrix associated to ɛ restricted to the Z-module ZL_t.
This suggested us to address the problem via linear algebra. Unluckly the result for the Z- modules is not immediate.
We give a generating set S_i of eigenvectors for the vector space RL_i, with i=0, …, n, called polytopes.
For our approach an important role is played by the Z-module ZL_i with basis L_i together with the submodule ZS_i generated by polytopes.
It is easy to prove that the following restrictions hold: ɛ: ZL_t -> ZL_k , ɛ:ZS_t -> ZS_k
We determine the invariant factors of the matrix W^T_{tk} finding the Smith group of ɛ: ZL_t -> ZL_k. The result is obtained constructing a standard basis of polytopes and deducing opportune bases of ZL_t and ZL_k.
In Chapter 5 we introduce the submodule of ZS_i which consists of elements fixed by G, that is
(ZS_i)^G, and we find the Smith group of ɛ:(ZS_t)^G -> (ZS_k)^G.
We restrict our attention to the case t+k=n and we prove that the groups (ZL_k)^G/ ɛ (ZL_t)^G) and (ZS_k)^G/ ɛ (ZS_t)^G) have the same order.
Actually we conjecture that, for t+k=n, the two groups are isomorphic.
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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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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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30
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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31
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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32
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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33
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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34
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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35
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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36
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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37
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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38
Giudici, Michael Robert. "Fixed point free elements of prime order in permutation groups." Thesis, Queen Mary, University of London, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.252086.
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39
Inglis, Nicholas Francis John. "Multiplicity-free permutation characters, distance-transitive graphs and classical groups." Thesis, University of Cambridge, 1987. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.256704.
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40
Justus, Amanda N. "Permutation Groups and Puzzle Tile Configurations of Instant Insanity II." Digital Commons @ East Tennessee State University, 2014. https://dc.etsu.edu/etd/2337.
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The manufacturer claims that there is only one solution to the puzzle Instant Insanity II. However, a recent paper shows that there are two solutions. Our goal is to find ways in which we only have one solution. We examine the permutation groups of the puzzle and use modern algebra to attempt to fix the puzzle. First, we find the permutation group for the case when there is only one empty slot at the top. We then examine the scenario when we add an extra column or an extra row to make the game a 4 × 5 puzzle or a 5 x 4 puzzle, respectively. We consider the possibilities when we delete a color to make the game a 3 × 3 puzzle and when we add a color, making the game a 5 × 5 puzzle. Finally, we determine if solution two is a permutation of solution one.
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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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Xuan, Mingzhi. "On Steinhaus Sets, Orbit Trees and Universal Properties of Various Subgroups in the Permutation Group of Natural Numbers." Thesis, University of North Texas, 2012. https://digital.library.unt.edu/ark:/67531/metadc149691/.
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In the first chapter, we define Steinhaus set as a set that meets every isometric copy of another set at exactly one point. We show that there is no Steinhaus set for any four-point subset in a plane.In the second chapter, we define the orbit tree of a permutation group of natural numbers, and further introduce compressed orbit trees. We show that any rooted finite tree can be realized as a compressed orbit tree of some permutation group. In the third chapter, we investigate certain classes of closed permutation groups of natural numbers with respect to their universal and surjectively universal groups. We characterize two-sided invariant groups, and prove that there is no universal group for countable groups, nor universal group for two-sided invariant groups in permutation groups of natural numbers.
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Pearce, Geoffrey. "Transitive decompositions of graphs." University of Western Australia. School of Mathematics and Statistics, 2008. http://theses.library.uwa.edu.au/adt-WU2008.0087.
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A transitive decomposition of a graph is a partition of the arc set such that there exists a group of automorphisms of the graph which preserves and acts transitively on the partition. This turns out to be a very broad idea, with several striking connections with other areas of mathematics. In this thesis we first develop some general theory of transitive decompositions, and in particular we illustrate some of the more interesting connections with certain combinatorial and geometric structures. We then give complete, or nearly complete, structural characterisations of certain classes of transitive decompositions preserved by a group with a rank 3 action on vertices (such a group has exactly two orbits on ordered pairs of distinct vertices). The main classes of rank 3 groups we study (namely those which are imprimitive, or primitive of grid type) are derived in some way from 2-transitive groups (that is, groups which are transitive on ordered pairs of distinct vertices), and the results we achieve make use of the classification by Sibley in 2004 of transitive decompositions preserved by a 2-transitive group.
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Emms, Josephine. "Amalgamation classes of directed graphs in model theory and infinite permutation groups." Thesis, University of East Anglia, 2012. https://ueaeprints.uea.ac.uk/39034/.
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45
Morje, Prabhav Gangadhar. "A nearly linear algorithm for Sylow subgroups of small-base permutation groups /." The Ohio State University, 1996. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487932351057768.
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46
Penrod, Keith G. "Infinite Product Group." BYU ScholarsArchive, 2007. https://scholarsarchive.byu.edu/etd/976.
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The theory of infinite multiplication has been studied in the case of the Hawaiian earring group, and has been seen to simplify the description of that group. In this paper we try to extend the theory of infinite multiplication to other groups and give a few examples of how this can be done. In particular, we discuss the theory as applied to symmetric groups and braid groups. We also give an equivalent definition to K. Eda's infinitary product as the fundamental group of a modified wedge product.
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CAMPOS, JÚNIOR Walfrido Siqueira. "Permutações." Universidade Federal Rural de Pernambuco, 2014. http://www.tede2.ufrpe.br:8080/tede2/handle/tede2/6711.
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Previous issue date: 2014-06-10<br>Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES<br>This work consists of the presentation of a simple permutation, seen as function. This function is bijective, hence admits inverse (inverse permutation). The composition of this function with the same it is also bijective (composed permutation). We will also see the permutations with repetition, circular without repetition and repetition, beyond chaotic permutations, which are those in which no element occupies its original position. The work is also part of the de nition and presentation of a group of permutations consisting of 3 properties in which the composition of functions satisfy all of them. This is our highest goal. Still show the parity of the permutation, as well as their applications in cases of determinants.<br>Este trabalho consta da apresentação de uma permutação simples, vista na forma de função. Essa função é bijetora, portanto admite inversa (permutação inversa). A composição dessa função com ela mesma, também é bijetora (permutação composta). Veremos também as permutações com repetição, circulares sem repetição e com repetição, além das permutações caóticas, que são aquelas em que nenhum elemento ocupa sua posição inicial. O trabalho consta também da definição e apresentação de um Grupo das permutações que consiste em 3 propriedades na qual a composição das funções satisfazem todas elas. Esse e o nosso maior objetivo. Mostraremos ainda a paridade da permutação, bem como suas aplicações em casos de determinantes.
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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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Dexter, Cache Porter. "Schur Rings over Infinite Groups." BYU ScholarsArchive, 2019. https://scholarsarchive.byu.edu/etd/8831.
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A Schur ring is a subring of the group algebra with a basis that is formed by a partition of the group. These subrings were initially used to study finite permutation groups, and classifications of Schur rings over various finite groups have been studied. Here we investigate Schur rings over various infinite groups, including free groups. We classify Schur rings over the infinite cyclic group.
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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.