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1
Mkiva, Soga Loyiso Tiyo. "The non-cancellation groups of certain groups which are split extensions of a finite abelian group by a finite rank free abelian group." Thesis, University of the Western Cape, 2008. http://etd.uwc.ac.za/index.php?module=etd&action=viewtitle&id=gen8Srv25Nme4_8520_1262644840.
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<p> <br></p>
<p align="left">The groups we consider in this study belong to the class <font face="F30">X</font><font face="F25" size="1"><font face="F25" size="1">0 </font></font><font face="F15">of all finitely generated groups with finite commutator subgroups.</font></p>
2
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.
3
George, Timothy Edward. "Symmetric representation of elements of finite groups." CSUSB ScholarWorks, 2006. https://scholarworks.lib.csusb.edu/etd-project/3105.
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The purpose of the thesis is to give an alternative and more efficient method for working with finite groups by constructing finite groups as homomorphic images of progenitors. The method introduced can be applied to all finite groups that possess symmetric generating sets of involutions. Such groups include all finite non-abelian simple groups, which can then be constructed by the technique of manual double coset enumeration.
4
Wegner, Alexander. "The construction of finite soluble factor groups of finitely presented groups and its application." Thesis, University of St Andrews, 1992. http://hdl.handle.net/10023/12600.
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Computational group theory deals with the design, analysis and computer implementation of algorithms for solving computational problems involving groups, and with the applications of the programs produced to interesting questions in group theory, in other branches of mathematics, and in other areas of science. This thesis describes an implementation of a proposal for a Soluble Quotient Algorithm, i.e. a description of the algorithms used and a report on the findings of an empirical study of the behaviour of the programs, and gives an account of an application of the programs. The programs were used for the construction of soluble groups with interesting properties, e.g. for the construction of soluble groups of large derived length which seem to be candidates for groups having efficient presentations. New finite soluble groups of derived length six with trivial Schur multiplier and efficient presentations are described. The methods for finding efficient presentations proved to be only practicable for groups of moderate order. Therefore, for a given derived length soluble groups of small order are of interest. The minimal soluble groups of derived length less than or equal to six are classified.
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Bujard, Cédric. "Finite subgroups of the extended Morava stabilizer groups." Thesis, Strasbourg, 2012. http://www.theses.fr/2012STRAD010/document.
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L'objet de la thèse est la classification à conjugaison près des sous-groupes finis du groupe de stabilisateur (classique) de Morava S_n et du groupe de stabilisateur étendu G_n(u) associé à une loi de groupe formel F de hauteur n définie sur le corps F_p à p éléments. Une classification complète dans S_n est établie pour tout entier positif n et premier p. De plus, on montre que la classification dans le groupe étendu dépend aussi de F et son unité associée u dans l'anneau des entiers p-adiques. On établit un cadre théorique pour la classification dans G_n(u), on donne des conditions nécessaires et suffisantes sur n, p et u pour l'existence dans G_n(u) d'extensions de sous-groupes finis maximaux de S_n par le groupe de Galois de F_{p^n} sur F_p, et lorsque de telles extensions existent on dénombre leurs classes de conjugaisons. On illustre nos méthodes en fournissant une classification complète et explicite dans le cas n=2<br>The problem addressed is the classification up to conjugation of the finite subgroups of the (classical) Morava stabilizer group S_n and the extended Morava stabilizer group G_n(u) associated to a formal group law F of height n over the field F_p of p elements. A complete classification in S_n is provided for any positive integer n and prime p. Furthermore, we show that the classification in the extended group also depends on F and its associated unit u in the ring of p-adic integers. We provide a theoretical framework for the classification in G_n(u), we give necessary and sufficient conditions on n, p and u for the existence in G_n(u) of extensions of maximal finite subgroups of S_n by the Galois group of F_{p^n} over F_p, and whenever such extension exist we enumerate their conjugacy classes. We illustrate our methods by providing a complete and explicit classification in the case n=2
6
McDougall-Bagnall, Jonathan M. "Generation problems for finite groups." Thesis, University of St Andrews, 2011. http://hdl.handle.net/10023/2529.
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It can be deduced from the Burnside Basis Theorem that if G is a finite p-group with d(G)=r then given any generating set A for G there exists a subset of A of size r that generates G. We have denoted this property B. A group is said to have the basis property if all subgroups have property B. This thesis is a study into the nature of these two properties. Note all groups are finite unless stated otherwise. We begin this thesis by providing examples of groups with and without property B and several results on the structure of groups with property B, showing that under certain conditions property B is inherited by quotients. This culminates with a result which shows that groups with property B that can be expressed as direct products are exactly those arising from the Burnside Basis Theorem. We also seek to create a class of groups which have property B. We provide a method for constructing groups with property B and trivial Frattini subgroup using finite fields. We then classify all groups G where the quotient of G by the Frattini subgroup is isomorphic to this construction. We finally note that groups arising from this construction do not in general have the basis property. Finally we look at groups with the basis property. We prove that groups with the basis property are soluble and consist only of elements of prime-power order. We then exploit the classification of all such groups by Higman to provide a complete classification of groups with the basis property.
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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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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 TecnolÃgico<br>Os 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.<br>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.
9
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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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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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.
12
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).
13
Martin, Stuart. "Quivers and the modular representation theory of finite groups." Thesis, University of Oxford, 1988. http://ora.ox.ac.uk/objects/uuid:59d4dc72-60e5-4424-9e3c-650eb2b1d050.
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The purpose of this thesis is to discuss the rôle of certain types of quiver which appear in the modular representation theory of finite groups. It is our concern to study two different types of quiver. First of all we construct the ordinary quiver of certain blocks of defect 2 of the symmetric group, and then apply our results to the alternating group and to the theory of partitions. Secondly, we consider connected components of the stable Auslander-Reiten quiver of certain groups G with normal subgroup N. The main interest lies in comparing the tree class of components of N-modules, with the tree class of components of these modules induced up to G.
14
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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15
Wetherell, Chris. "Subnormal structure of finite soluble groups." View thesis entry in Australian Digital Theses Program, 2001. http://thesis.anu.edu.au/public/adt-ANU20020607.121248/index.html.
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Brown, Scott. "Finite reducible matrix algebras." University of Western Australia. School of Mathematics and Statistics, 2006. http://theses.library.uwa.edu.au/adt-WU2006.0079.
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[Truncated abstract] A matrix is said to be cyclic if its characteristic polynomial is equal to its minimal polynomial. Cyclic matrices play an important role in some algorithms for matrix group computation, such as the Cyclic Meataxe of Neumann and Praeger. In 1999, Wall and Fulman independently proved that the proportion of cyclic matrices in general linear groups over a finite field of fixed order q has limit [formula] as the dimension approaches infinity. First we study cyclic matrices in maximal reducible matrix groups, that is, the stabilisers in general linear groups of proper nontrivial subspaces. We modify Wall’s generating function approach to determine the limiting proportion of cyclic matrices in maximal reducible matrix groups, as the dimension of the underlying vector space increases while that of the invariant subspace remains fixed. This proportion is found to be [formula] note the change of the exponent of q in the leading term of the expansion. Moreover, we exhibit in each maximal reducible matrix group a family of noncyclic matrices whose proportion is [formula]. Maximal completely reducible matrix groups are the stabilisers in a general linear group of a nontrivial decomposition U1⊕U2 of the underlying vector space. We take a similar approach to determine the limiting proportion of cyclic matrices in maximal completely reducible matrix groups, as the dimension of the underlying vector space increases while the dimension of U1 remains fixed. This limiting proportion is [formula]. ... We prove that this proportion is[formula] provided the dimension of the fixed subspace is at least two and the size q of the field is at least three. This is also the limiting proportion as the dimension increases for separable matrices in maximal completely reducible matrix groups. We focus on algorithmic applications towards the end of the thesis. We develop modifications of the Cyclic Irreducibility Test - a Las Vegas algorithm designed to find the invariant subspace for a given maximal reducible matrix algebra, and a Monte Carlo algorithm which is given an arbitrary matrix algebra as input and returns an invariant subspace if one exists, a statement saying the algebra is irreducible, or a statement saying that the algebra is neither irreducible nor maximal reducible. The last response has an upper bound on the probability of incorrectness.
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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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Clark, Jonathan Owen. "Cohomology of some finite groups." Thesis, University of Oxford, 1993. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.240535.
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Flavell, Paul. "Some topics on finite groups." Thesis, University of Oxford, 1990. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.257944.
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Craven, David Andrew. "Algebraic modules for finite groups." Thesis, University of Oxford, 2007. http://ora.ox.ac.uk/objects/uuid:7f641b33-d301-4445-8269-a5a33f4b7e5e.
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The main focus of this thesis is algebraic modules---modules that satisfy a polynomial equation with integer co-efficients in the Green ring---in various finite groups, as well as their general theory. In particular, we ask the question `when are all the simple modules for a finite group G algebraic?' We call this the (p-)SMA property. The first chapter introduces the topic and deals with preliminary results, together with the trivial first results. The second chapter provides the general theory of algebraic modules, with particular attention to the relationship between algebraic modules and the composition factors of a group, and between algebraic modules and the Heller operator and Auslander--Reiten quiver. The third chapter concerns itself with indecomposable modules for dihedral and elementary abelian groups. The study of such groups is both interesting in its own right, and can be applied to studying simple modules for simple groups, such as the sporadic groups in the final chapter. The fourth chapter analyzes the groups PSL(2,q); here we determine, in characteristic 2, which simple modules for PSL(2,q) are algebraic, for any odd q. The fifth chapter generalizes this analysis to many groups of Lie type, although most results here are in defining characteristic only. Notable exceptions include the small Ree groups, which have the 2-SMA property for all q. The sixth and final chapter focuses on the sporadic groups: for most groups we provide results on some simple modules, and some of the groups are completely analyzed in all characteristics. This is normally carried out by restricting to the Sylow p-subgroup. This thesis develops the current state of knowledge concerning algebraic modules for finite groups, and particularly for which simple groups, and for which primes, all simple modules are algebraic.
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Soicher, L. H. "Presentations of some finite groups." Thesis, University of Cambridge, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.332999.
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King, Carlisle. "Generation of finite simple groups." Thesis, Imperial College London, 2018. http://hdl.handle.net/10044/1/63863.
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Let G be a finite simple group. Well-known results of Miller, Steinberg and Aschbacher-Guralnick prove that G can be generated by a pair of elements - we say that G is 2-generated. In this thesis, we consider some variations of this result. A natural refinement of the 2-generation result is to ask, for a pair of integers (a,b), whether a finite simple group G is generated by an element of order a and an element of order b. The smallest pair of interest is (2,3) (pairs of elements of order 2 generate dihedral groups). Work of Liebeck-Shalev and Lubeck-Malle shows that, apart from a few known infinite families, all but finitely many finite simple groups are (2,3)-generated. The first major result of this thesis proves that every finite simple group can be generated by an element of order 2 and an element of prime order. An equivalent statement of the 2-generation theorem is that every finite simple group is an image of F_2, the free group on 2 generators. More generally, given a finitely presented group H, one can ask which finite simple groups are images of H. A result of Liebeck and Shalev shows that given a pair (A,B) of nontrivial finite groups such that at least one of A or B is not a 2-group, every finite simple classical group of sufficiently large rank is an image of A star B, the free product of A and B. The second major result of this thesis generalizes this result by proving the same conclusion holds for pairs (A,B) of any nontrivial finite groups, not both of order 2.
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Mohammed, Salih Haval M. "Finite groups of small genus." Thesis, University of Birmingham, 2015. http://etheses.bham.ac.uk//id/eprint/5574/.
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For a finite group \(G\), the Hurwitz space \(H\)\(^i\)\(_r\)\(_,\)\(^n\)\(_g\) (\(G\)) is the space of genus \(g\) covers of the Riemann sphere with \(r\) branch points and the monodromy group \(G\). Let ε\(_r\)(\(G\)) = {(\(x\)\(_1\),...,\(x\)\(_r\)) : \(G\) = \(\langle\)\(x\)\(_1\),...,\(x\)\(_r\)\(\rangle\), Π\(^r\)\(_i\)\(_=\)\(_1\) \(x\)\(_i\) = 1, \(x\)\(_i\) ϵ \(G\)#, \(i\) = 1,...,\(r\)}. The connected components of \(H\)\(^i\)\(_r\)\(_,\)\(^n\)\(_g\)(\(G\)) are in bijection with braid orbits on ε\(_r\)(\(G\)). In this thesis we enumerate the connected components of \(H\)\(^i\)\(_r\)\(_,\)\(^n\)\(_g\)(\(G\)) in the cases where \(g\) \(\leq\) 2 and \(G\) is a primitive affine group. Our approach uses a combination of theoretical and computational tools. To handle the most computationally challenging cases we develop a new algorithm which we call the Projection-Fiber algorithm.
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Sannella, Stefano. "Broué's conjecture for finite groups." Thesis, University of Birmingham, 2018. http://etheses.bham.ac.uk//id/eprint/8462/.
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This research project consists of using the theory of perverse equivalences to study Broue's abelian defect group conjecture for the principal block of some finite groups when the defect group is elementary abelian of rank 2. We will look at G=\Omega^{ +} 8(2} and prove the conjecture in characteristic 5, the only open case for this group. We will also look at which result the application of our algorithm leads when G= { }^2F 4(2}'.2, {}^3D_ 4(2}, Sp_8(2}; for those groups, it seems that a slight modification of our method is required to complete the proof of the conjecture. Finally, we will see what happens when we apply our method -which is mainly used for groups G of Lie type- to some sporadic groups, namely G=j_2, He, Suz, Fi_{22}.
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Morris, Thomas Bembridge Slater. "Nilpotent injectors in finite groups." Thesis, University of Birmingham, 2011. http://etheses.bham.ac.uk//id/eprint/3066/.
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We prove that the odd nilpotent injectors (a certain type of maximal nilpotent subgroup) f a minimal simple group are all conjugate, extending the result from soluble groups. We also prove conjugacy in GU(\_3\)(q) and SU(\_3\)(q). In a minimal counterexample to the onjecture that the odd nilpotent injectors of an arbitrary ¯nite group are all conjugate we show that there must be a component, which cannot be of type A\(_n\) except possibly 3 ¢ A(\_6\) or 3 ¢ A(\_7\). Finally, we produce a partial result on minimal simple groups for a more general type of nilpotent injector.
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Levy, Matthew. "Word values in finite groups." Thesis, Imperial College London, 2013. http://hdl.handle.net/10044/1/23929.
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Let w denote a group word in d variables, that is, an element of the free group of rank d. For a finite group G we may define a word map that sends a d-tuple of elements of G, to its w-value by substituting variables and evaluating the word in G by performing all relevant group operations. In this thesis we study a number of problems to do with the behaviour of word maps over various classes of groups. The first problem we look at concerns the distribution of word values in nilpotent groups. We obtain a lower bound for the probability that a random d-tuple of elements from any nilpotent group of class 2 evaluates at the identity for any fixed word in d variables answering a special case of a question of Alon Amit. Another problem we look at deals with the question of which possible subsets of a group can be obtained as the image of a word map. This was first studied by Kassabov & Nikolov and later by Lubotzky who gave a complete description in the case of simple groups. We obtain a partial classification for the almost simple groups and quasisimple groups and completely describe what happens in the case of symmetric groups. Finally, we study twisted commutator maps over the alternating groups, special linear groups and special unitary groups. Twisted commutators are similar to commutators but are twisted by group automorphisms. These have been studied by Nikolov & Segal in where they obtain bounds on the width of twisted commutator words over the finite quasisimple groups. Our goal is to improve these bounds. Throughout this thesis we will also look at variations of the above problems as well as related questions.
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Bradford, Henry. "Spectral properties of finite groups." Thesis, University of Oxford, 2015. https://ora.ox.ac.uk/objects/uuid:10babab2-8d11-4d53-aea8-7479b868a57d.
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This thesis concerns the diameter and spectral gap of finite groups. Our focus shall be on the asymptotic behaviour of these quantities for sequences of finite groups arising as quotients of a fixed infinite group. In Chapter 3 we give new upper bounds for the diameters of finite groups which do not depend on a choice of generating set. Our method exploits the commutator structure of certain profinite groups, in a fashion analogous to the Solovay-Kitaev procedure from quantum computation. We obtain polylogarithmic upper bounds for the diameters of finite quotients of: groups with an analytic structure over a pro-p domain (with exponent depending on the dimension); Chevalley groups over a pro-p domain (with exponent independent of the dimension) and the Nottingham group of a finite field. We also discuss some consequences of our results for random walks on groups. In Chapter 4 we construct new examples of expander Cayley graphs of finite groups, arising as congruence quotients of non-elementary subgroups of SL2(Fp[t]) modulo certain square-free ideals. We describe some applications of our results to simple random walks on such subgroups, specifically giving bounds on the rate of escape from algebraic subvarieties, the set of squares and the set of elements with reducible characteristic polynomial in SL2(Fp[t]) Finally, in Chapter 5 we produce new expander congruence quotients of SL2 (Zp), generalising work of Bourgain and Gamburd. The proof combines the Solovay-Kitaev procedure with a quantitative analysis of the algebraic geometry of these groups, which in turn relies on previously known examples of expanders.
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McHugh, John. "Monomial Characters of Finite Groups." ScholarWorks @ UVM, 2016. http://scholarworks.uvm.edu/graddis/572.
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An abundance of information regarding the structure of a finite group can be obtained by studying its irreducible characters. Of particular interest are monomial characters – those induced from a linear character of some subgroup – since Brauer has shown that any irreducible character of a group can be written as an integral linear combination of monomial characters. Our primary focus is the class of M-groups, those groups all of whose irreducible characters are monomial. A classical theorem of Taketa asserts that an M-group is necessarily solvable, and Dade proved that every solvable group can be embedded as a subgroup of an M-group. After discussing results related to M-groups, we will construct explicit families of solvable groups that cannot be embedded as subnormal subgroups of any M-group. We also discuss groups possessing a unique non-monomial irreducible character, and prove that such a group cannot be simple.
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Taylor, Paul Anthony. "Computational investigation into finite groups." Thesis, University of Manchester, 2011. https://www.research.manchester.ac.uk/portal/en/theses/computational-investigation-into-finite-groups(8fe69098-a2d0-4717-b8d3-c91785add68c).html.
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We briefly discuss the algorithm given in [Bates, Bundy, Perkins, Rowley, J. Algebra, 316(2):849-868, 2007] for determining the distance between two vertices in a commuting involution graph of a symmetric group.We develop the algorithm in [Bates, Rowley, Arch. Math. (Basel), 85(6):485-489, 2005] for computing a subgroup of the normalizer of a 2-subgroup X in a finite group G, examining in particular the issue of when to terminate the randomized procedure. The resultant algorithm is capable of handling subgroups X of order up to 512 and is suitable, for example, for matrix groups of large degree (an example calculation is given using 112x112 matrices over GF(2)).We also determine the suborbits of conjugacy classes of involutions in several of the sporadic simple groups?namely Janko's group J4, the Fischer sporadic groups, and the Thompson and Harada-Norton groups. We use our results to determine the structure of some graphs related to this data.We include implementations of the algorithms discussed in the computer algebra package MAGMA, as well as representative elements for the involution suborbits.
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Nenciu, Adriana. "Character tables of finite groups." [Gainesville, Fla.] : University of Florida, 2006. http://purl.fcla.edu/fcla/etd/UFE0014824.
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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.
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Ersoy, Kivanc. "Centralizers Of Finite Subgroups In Simple Locally Finite Groups." Phd thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/3/12610850/index.pdf.
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A group G is called locally finite if every finitely generated subgroup of G is finite. In this thesis we study the centralizers of subgroups in simple locally finite groups. Hartley proved that in a linear simple locally finite group, the fixed point of every semisimple automorphism contains infinitely many elements of
distinct prime orders. In the first part of this thesis, centralizers of finite abelian subgroups of linear simple locally finite groups are studied and the following result is proved: If G is a linear simple locally finite group and A is a finite d-abelian
subgroup consisting of semisimple elements of G, then C_G(A) has an infinite abelian subgroup isomorphic to the direct product of cyclic groups of order p_i for infinitely many distinct primes pi.
Hartley asked the following question: Let G be a non-linear simple locally finite group and F be any subgroup of G. Is CG(F) necessarily infinite? In the second part of this thesis, the following problem is studied: Determine the nonlinear
simple locally finite groups G and their finite subgroups F such that C_G(F) contains an infinite abelian subgroup which is isomorphic to the direct product of cyclic groups of order pi for infinitely many distinct primes p_i. We prove the following: Let G be a non-linear simple locally finite group with a split Kegel cover K and F be any finite subgroup consisting of K-semisimple elements of G. Then the centralizer C_G(F) contains an infinite abelian subgroup isomorphic to the direct product of cyclic groups of order p_i for infinitely many distinct primes
p_i.
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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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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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Semikina, Iuliia [Verfasser]. "G-theory of group rings for finite groups / Iuliia Semikina." Bonn : Universitäts- und Landesbibliothek Bonn, 2018. http://d-nb.info/1173789642/34.
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Crawley-Boevey, W. W. "Polycyclic-by-finite affine group schemes and infinite soluble groups." Thesis, University of Cambridge, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.372868.
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Peterson, Aaron. "Pipe diagrams for Thompson's Group F /." Diss., CLICK HERE for online access, 2007. http://contentdm.lib.byu.edu/ETD/image/etd1959.pdf.
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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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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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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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Wetherell, Chris, and chrisw@wintermute anu edu au. "Subnormal Structure of Finite Soluble Groups." The Australian National University. Faculty of Science, 2001. http://thesis.anu.edu.au./public/adt-ANU20020607.121248.
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The Wielandt subgroup, the intersection of normalizers of subnormal subgroups, is
non-trivial in any finite group and thus gives rise to a series whose length is a measure
of the complexity of a group's subnormal structure. Another measure, akin to the
nilpotency class of nilpotent groups, arises from the strong Wielandt subgroup, the
intersection of centralizers of nilpotent subnormal sections. This thesis begins an
investigation into how these two invariants relate in finite soluble groups.
¶
Complete results are obtained for metabelian groups of odd order: the strong
Wielandt length of such a group is at most one more than its Wielandt length, and
this bound is best possible. Some progress is made in the wider class of groups with
p-length 1 for all primes p. A conjecture for all finite soluble groups, which may be
regarded as a subnormal analogue of the embedding of the Kern, is also considered.
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Szechtman, Fernando. "Weil representations of finite symplectic groups." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1999. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape9/PQDD_0006/NQ39598.pdf.
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Quinlan, Rachel. "Irreducible projective representations of finite groups." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp02/NQ59658.pdf.
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Duncan, Alexander Rhys. "Finite groups of low essential dimension." Thesis, University of British Columbia, 2011. http://hdl.handle.net/2429/35116.
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Informally, essential dimension is the minimal number of parameters required to define an algebraic object. This invariant has numerous connections to Galois cohomology, linear algebraic groups and birational geometry. In particular, the essential dimension of finite groups has connections to the Noether problem, inverse Galois theory and the simplification of polynomials via Tschirnhaus transformations.
This thesis studies finite groups of low essential dimension using methods from birational geometry. Specifically, the main results are a classification of finite groups of essential dimension 2, and a proof that the alternating and symmetric groups on 7 letters have essential dimension 4.
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Martino, Ivan. "The Ekedahl Invariants for finite groups." Licentiate thesis, Stockholms universitet, Matematiska institutionen, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-94950.
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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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Sinanan, Shavak. "Algorithms for polycyclic-by-finite groups." Thesis, University of Warwick, 2011. http://wrap.warwick.ac.uk/49186/.
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A set of fundamental algorithms for computing with polycyclic-by-finite groups is presented here. Polycyclic-by-finite groups arise naturally in a number of contexts; for example, as automorphism groups of large finite soluble groups, as quotients of finitely presented groups, and as extensions of modules by groups. No existing mode of representation is suitable for these groups, since they will typically not have a convenient faithful permutation representation. A mixed mode is used to represent elements of such a group; utilising a polycyclic presentation or a power-conjugate presentation for the elements of the normal subgroup, and a permutation representation for the elements of the quotient.