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Dissertations / Theses on the topic 'Finite groups'

Author: Grafiati
Published: 4 June 2021
Last updated: 1 August 2024

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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>&nbsp<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>
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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, w
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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.
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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
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5

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 nécessai
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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 proper
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7

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, an
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8

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 commutat
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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
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10

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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11

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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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-bl
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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
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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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16

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 nontrivia
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17

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 'num
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18

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 facto
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19

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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20

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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21

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 t
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22

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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23

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 in
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24

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 \(H\)\(^i\)\(_r\)\
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25

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 s
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26

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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27

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
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28

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 s
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29

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 ca
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30

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 degre
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31

Nenciu, Adriana. "Character tables of finite groups." [Gainesville, Fla.] : University of Florida, 2006. http://purl.fcla.edu/fcla/etd/UFE0014824.

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32

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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33

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
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34

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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35

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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36

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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38

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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39

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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40

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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41

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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42

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 so
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43

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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44

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
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45

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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46

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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47

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 dimen
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48

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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49

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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50

Sinanan, Shavak. "Algorithms for polycyclic-by-finite groups." Thesis, University of Warwick, 2011. http://wrap.warwick.ac.uk/49186/.

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Abstract:
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 fo
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