Dissertations / Theses on the topic 'Finite soluble groups'

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.

In this thesis we begin the study of finite groups possessing a model subgroup, where a model subgroup H of a finite group G is defined to be a subgroup satisfying 〖1H〗^(↑G)=∑_(x∊∕π(G))▒X We show that a finite nilpotent group possesses a model subgroup if and only if it is abelian and that a Frobenius group with Frobenius complement C and Frobenius kernel N possesses a model subgroup if and only if (a) N is elementary abelian of order r". (b) C is cyclic of order (r" — 1 )/(rd — 1), for some d dividing n. (c) The finite field F=Frn has an additive abelian subgroup HF of order rd satisfying NormF/K(HF) =K, where K=Frd. We then go on to conjecture that a finite soluble group G possessing a model subgroup is either metabelian or has a normal subgroup N such that G/N is a Frobenius group with cyclic Frobenius complement of order 2" +1 and elementary abelian Frobenius kernel of order 22". We consider a series of cases that need to be excluded in order to prove the conjecture and present some examples that shed light on the problems still to be overcome.

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.

This thesis is an investigation into some of the lattice properties of the strong containment lattice (H, «) of Schunck classes and also of its important sublattice (D, «). The general aim is to characterise lattice properties of Schunck classes by avoidance class properties. Our main result, Theorem 8.5, is an avoidance class characterisation of those D-classes all of whose maximal ascending proper chains of Q-classes to S have the,same length. The problem extended to H is much more difficult but in Corollary 4.3 we describe an avoidance class condition for a Schunck class only to have chains of finite length to S. The lack of duality in H shows up clearly in section 3. The fascinating problem of deciding whether or not H is atomic is considered in section 9. Our results suggest that it probably is since any counterexample must be very complicated.

The main topic of this thesis is the discovery and study of a cohomological property of the subgroups called F-normalizers in finite soluble groups; namely, the property that with certain coefficient modules the restriction map in cohomology from a soluble group to its F-normalizers vanishes in non-zero degrees. Chapter 3 is devoted to a proof of this fact It turns out that in some classes of soluble groups the F-normalizers are characterized by this property, and the study of these classes occupies Chapters 4 and 5. Various connections with cohomology and group theory are found; the approach seems to offer some unification of disparate results from the theory of soluble groups. The relation between F-normalizers and cohomology was discovered through study of the work of Jacques Thevenaz on the action of a soluble group on its lattice of subgroups. Chapter 1 is a summary of this work and its background, and is included to provide motivation. A link with the rest of the thesis arises through a new result, in which certain subgroups crucial to Thevenaz's analysis of soluble groups are shown to coincide with their system normalizers. A proof of this is given in Chapter 2, which also contains some miscellaneous results on soluble groups from the class considered by Thevenaz, comprising those groups whose lattices of subgroups are complemented. The problem of characterizing F-normalizers in soluble groups by the results of Chapter 3 is proposed in Chapter 4, and in Chapters 4 and 5 two essentially different approaches to this problem are taken, which lead to partial solutions in different sets of circumstances. In Chapter 4, the first cohomology groups of soluble groups are considered, and an application is given to a proof of a recent theorem of Volkmar Welker described in Chapter 1 on the homotopy type of the partially ordered set of conjugacy classes of subgroups of a soluble group. Another application is to the study of local conjugacy of subgroups of soluble groups, and these are combined in a result which shows that the set of conjugacy classes considered by Welker is homotopy equivalent to an analogous set obtained from local conjugacy classes. In Chapter 5 some known results on the local conjugacy of F-normalizers are exhibited, as evidence for a cohomological characterization of these subgroups. The results are used to study groups of p-length one by a 'local' analysis, whereby the problem of characterizing F-normalizers is translated into a question concerning the action of automorphisms on the cohomology rings of p-groups. In the study of this question a natural place to start is the case of abelian groups, whose cohomology rings are known; calculations in this case lead to results on the F-normalizers of A-groups. The question is then considered for other p-groups, revealing an elegant relationship between the cohomology of p-groups, the theory of varieties, and some well-known results on automorphisms of p-groups.

Orientador: Dessislava Hristova Kochloukova
Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica
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Mestrado
Matematica
Mestre em Matemática

The work in this thesis was carried out in the area of computational group theory. The latter is concerned with designing algorithm s and developing their practical implementations for investigating problem s regarding groups. An important class of groups are finite soluble groups. These can be described in a computationally convenient way by power conjugate presentations. In practice, however, they are usually supplied differently. The aim of this thesis is to propose algorithm s for computing power conjugate presentations for finite soluble groups. This is achieved in two different ways. One of the ways in which a finite soluble group is often supplied is as a quotient of a finitely presented group. T he first p art of the thesis is concerned with designing an algorithm to compute a power conjugate presentation for a finite soluble group given in this way. T he theoretical background for the algorithm is provided and its practicality is investigated on an implementation. T he second p a rt of the thesis describes the theoretical aspects of an algorithm to compute all pow er conjugate presentations for a certain class of finite soluble groups of a given order.

There is a large collection of effective algorithms for computing information about finite soluble groups. The success in computation with these groups is primarily due to a computationally convenient representation of them by means of (special forms of) power conjugate presentations. A notable omission from this collection of algorithms is an effective algorithm for computing the automorphism group of a finite soluble group. An algorithm designed for finite groups in general provides only a partial answer to this deficiency. In this thesis an effective algorithm for computing the automorphism group of a finite soluble group is described. An implementation of this algorithm has proved to be a substantial improvement over existing techniques available for finite soluble groups.

Let G be a finite group with normal subgroup N. Let K be a subgroup of G. We say that K is a partial complement of N in G if N and K intersect trivially. There are two main results in this work. The first main result arises from analysing when each partial complement of N in G is contained in a complement of N in G when G is a finite soluble group, N is the product of minimal normal subgroups, N is complemented and all the complements of N in G are conjugate. We show that each partial complement of N in G is contained in a complement of N in G if and only if N is projective. The next natural question is: if N is non-projective, which partial complements are contained in a complement of N in G? We say that a cyclic p-partial complement is a partial complement that is cyclic and its order is a power of p. We establish exactly which cyclic p-partial complements are contained in a conjugate of H. So the next question is: if the partial complement is a non-cyclic p-partial complement, how do we know that is contained in a conjugate of H? This is a difficult question and because of restriction on representation theory, we have needed to restrict H to be in the class of groups that are nilpotent p'-groups by p-groups. To answer this question, we use the first cohomology group. The first cohomology group is the number of conjugacy classes of complements to N in G. That is, if we have a partial complement K such that the first cohomology group vanishes then we know there is only one conjugacy class of complements of N in NK and therefore K is in a conjugate of H. The second main result finds exactly when the first cohomology group vanishes. This is a sufficient condition for a partial complement to be contained in a complement of N in G. -- provided by Candidate.

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.