The key result linking the additive and multiplicative structure of a finite field is the Primitive Normal Basis Theorem; this was established by Lenstra and Schoof in 1987 in a proof which was heavily computational in nature. In this thesis, a new, theoretical proof of the theorem is given, and new estimates (in some cases, exact values) are given for the number of primitive free elements. A natural extension of the Primitive Normal Basis Theorem is to impose additional conditions on the primitive free elements; in particular, we may wish to specify the norm and trace of a primitive free element. The existence of at least one primitive free element of GF(qn) with specified norm and trace was established for n ³ 5 by Cohen in 2000; in this thesis, the result is proved for the most delicate cases, n = 4 and n = 3, thereby completing the general existence theorem.

This thesis is concerned with the study of certain metabelian groups which can be viewed as split extensions of a free abelian group Q by a finitely generated Q-module A. In particular we want to know which cohomological finiteness properties these groups possess in order to shed further light on the Bieri-Groves conjecture.

Algebraic equations on complex numbers and functional equations on generating functions are often used to solve combinatorial problems. But the introduction of common arithmetic operators such as subtraction and division always causes panic in the world of objects which are generated from constants by applying products and coproducts. Over the years, researchers have been endeavouring to interpretate some absurd calculations on objects which lead to meaningful combinatorial results. This thesis investigates connections between algebraic equations on complex numbers and isomorphisms of recursively defined objects. We are attempting to work out conditions under which isomorphisms between recursively defined objects can be decided by equalities between polynomials on multi-variables with integers as coefficients.

Aquesta tesi doctoral tracta de l’estructura algebraica anomenada braça no commutativa per l’esquerra, i de les seves aplicacions a les solucions conjuntistes no degenerades de l’equació de Yang-Baxter. Concretament, estudiem els següents problemes: (1) Construïm totes les solucions conjuntistes no-degenerades associades a una braça per l’esquerra no commutativa donada. A més, demostrem que totes les solucions no-degenerades involutives es poden construir a partir de braces per l’esquerra amb una construcció similar. Aquests resultats estan continguts als Capítols 2 i 3. Això redueix el problema de la classificació de solucions conjuntistes no-degenerades de l’equació de Yang-Baxter a la classificació de braces per l’esquerra no commutatives, un problema que tractem d’estudiar a la resta de la tesi. (2) A la Secció 4.1, presentem dos mètodes nous per a construir braces per l’esquerra: extensions de braces per l’esquerra per ideals trivials, i matched product de braces. Aquestes construccions estan basades en les construccions anàlogues en grups. (3) Motivats per les extensions de braces, a la Secció 4.2 construïm la primera família de braces per l’esquerra simples no trivials. Per a aconseguir-ho, fem servir la construcció de matched products que havíem definit prèviament. (4) Responent a una pregunta de Cedó, Jespers i del Río, trobem el primer exemple de p-grup finit que no és grup multiplicatiu d’una braça per l’esquerra. Això demostra que no tot grup finit resoluble és grup multiplicatiu d’una braça per l’esquerra. Aquest resultat es troba a la Secció 4.3. (5) Finalment, classifiquem totes les braces per l’esquerra d’ordre p, p^2 i p^3, on p és un primer. Aquest resultat es troba contingut a la Secció 4.4 i al Capítol 5. / This PhD thesis deals with the algebraic structure called non-commutative left brace, and with its applications for the non-degenerate set-theoretic solutions of the Yang-Baxter equation. Concretely, we study the following problems: (1) We construct all the non-degenerate set-theoretic solutions of the Yang-Baxter equation associated with a fixed non-commutative left brace. Moreover, we prove all the non-degenerate involutive solutions can be obtained from left braces with an analogous construction. These results are contained in Chapters 2 and 3. This reduces the problem of classification of non-degenerate set-theoretic solutions of the Yang-Baxter equation to the classification of non-commutative left braces, a problem that we study for the rest of the memoir. (2) In Section 4.1, we present two new methods to construct new left braces: extensions of left braces by trivial ideals, and matched product of left braces. These constructions are based on the analogous constructions of group theory. (3) Motivated by the extensions of left braces, in Section 4.2 we construct the first family of non-trivial simple left braces. We use the matched product method that we have defined previously to obtain this family. (4) Answering a question of Cedó, Jespers and del Río, we find the first exemple of finite p-group which is not the multiplicative group of any left brace. This proves that not all finite solvable groups are multiplicative groups of left braces. This results is contained in Section 4.3. (5) Finally, we classify all the left braces of order p, p^2 and p^3, where p is a prime. This result is contained in Section 4.4 and Chapter 5.

Our main objects of study are Lie{Rinehart algebras, their enveloping algebras and their relation with other structures (Gerstenhaber algebras, Hopf algebroids, Leibniz algebras and algebroids). In particular we focus on two aspects: 1. In the same way that the universal enveloping algebra of a Lie algebra carries a Hopf algebra structure, the universal enveloping algebra of a Lie-Rinehart algebra is one of the richest class of examples of Hopf algebroids (a generalisation of Hopf algebras). We prove that, unlike in the classical Lie algebra case, the universal enveloping algebra of Lie-Rinehart algebras may or may not admit an antipode. We use the characterisation due to Kowalzig and Posthuma [KP11] of the antipode on the Hopf algebroid structure on the enveloping algebra of a Lie-Rinehart algebra in terms of left (and right) modules over its enveloping algebra [Hue98] and give examples of Lie-Rinehart algebras that do not admit these right modules structures and hence no antipode on the universal enveloping algebra of a Lie-Rinehart algebra. Moreover, we prove that some Lie-Rinehart algebras admit a structure weaker than right modules over its enveloping algebra which yields a generator of the corresponding Gerstenhaber algebra while not a square-zero one, hence not a differential. Our examples of these algebras arise when considering Jacobi algebras [Kir76, Lic78], a certain generalisation of Poisson algebras. 2. Following the work of Loday and Pirashvili [LP98] in which they analyse the functorial relation between Lie algebras in the category LM of linear maps (which they define) and Leibniz algebras, we study the relation between Lie-Rinehart algebras and Leibniz algebroids [IdLMP99]: After describing Lie-Rinehart algebra objects in the category LM of linear maps, we construct a functor from Lie-Rinehart algebra objects in LM to Leibniz algebroids.

Automorphic Lie Algebras arise in the context of reduction groups introduced in the late 1970s [35] in the field of integrable systems. They are subalgebras of Lie algebras over a ring of rational functions, denied by invariance under the action of a finite group, the reduction group. Since their introduction in 2005 [29, 31], mathematicians aimed to classify Automorphic Lie Algebras. Past work shows remarkable uniformity between the Lie algebras associated to different reduction groups. That is, many Automorphic Lie Algebras with nonisomorphic reduction groups are isomorphic [4, 30]. In this thesis we set out to find the origin of these observations by searching for properties that are independent of the reduction group, called invariants of Automorphic Lie Algebras. The uniformity of Automorphic Lie Algebras with nonisomorphic reduction groups starts at the Riemann sphere containing the spectral parameter, restricting the finite groups to the polyhedral groups. Through the use of classical invariant theory and the properties of this class of groups it is shown that Automorphic Lie Algebras are freely generated modules over the polynomial ring in one variable. Moreover, the number of generators equals the dimension of the base Lie algebra, yielding an invariant. This allows the definition of the determinant of invariant vectors which will turn out to be another invariant. A surprisingly simple formula is given expressing this determinant as a monomial in ground forms. All invariants are used to set up a structure theory for Automorphic Lie Algebras. This naturally leads to a cohomology theory for root systems. A first exploration of this structure theory narrows down the search for Automorphic Lie Algebras signicantly. Various particular cases are fully determined by their invariants, including most of the previously studied Automorphic Lie Algebras, thereby providing an explanation for their uniformity. In addition, the structure theory advances the classification project. For example, it clarifies the effect of a change in pole orbit resulting in various new Cartan-Weyl normal form generators for Automorphic Lie Algebras. From a more general perspective, the success of the structure theory and root system cohomology in absence of a field promises interesting theoretical developments for Lie algebras over a graded ring.

The Iwahori-Hecke algebras of finite Coxeter groups play an important role in many areas of mathematics. In this thesis we study the representation theory of the Iwahori-Hecke algebras of the Coxeter groups of type Bn and F4, in the unequal parameter case. We denote these algebras HQ and KQ respectively. This follows on from work done by Dipper, James, Murphy and Norton. We are interested in the Iwahori-Hecke algebras of type Bn and F4 since these are the only cases, apart from the dihedral groups, where the Coxeter generators lie in different conjugacy classes, and therefore the Iwahori-Hecke algebra can have unequal parameters. There are two parameters associated with these algebras, Q and q. Norton dealt with the case Q = q = 0, whilst Dipper, James and Murphy addressed the case q 6= 0 in type Bn. In this thesis we look at the case Q 6= 0; q = 0. We begin by constructing the simple modules for HQ, then compute the Ext quiver and find the blocks of HQ. We continue by observing that there is a natural embedding of the algebra of type n 1 in the algebra of type n, and this gives rise to the notion of an induced module. We look at the structure of the induced module associated with a given simple HQ-module. Here we are able to construct a composition series for the induced module and show that in a particular case the induced modules are self-dual. Finally, we look at KQ and find that the representation theory is related to representation theory of the Iwahori-Hecke algebra of type B3. Using this relationship we are able to construct the simple modules for KQ and begin the analysis of the Ext quiver.

Consider a finite group G and subgroups H;K of G. We say that H and K permute if HK = KH and call H a permutable subgroup if H permutes with every subgroup of G. A group G is called quasi-Dedekind if all subgroups of G are permutable. We can define, for every finite group G, an arithmetic quantity that measures the probability that two subgroups (chosen uniformly at random with replacement) permute and we call this measure the subgroup permutability degree of G. This measure quantifies, among others, how close a finite group is to being quasi-Dedekind, or, equivalently, nilpotent with modular subgroup lattice. The main body of this thesis is concerned with the behaviour of the subgroup permutability degree of the two families of finite simple groups PSL2(2n), and Sz(q). In both cases the subgroups of the two families of simple groups are completely known and we shall use this fact to establish that the subgroup permutability degree in each case vanishes asymptotically as n or q respectively tends to infinity. The final chapter of the thesis deviates from the main line to examine groups, called F-groups, which behave like nilpotent groups with respect to the Frattini subgroup of quotients. Finally, we present in the Appendix joint research on the distribution of the density of maximal order elements in general linear groups and offer code for computations in GAP related to permutability.

No description available.

Let \(G\) be a reductive algebraic group over an algebraically closed field \(k\) of good characteristic, let \(g\)=Lie(\(G\)) be the Lie algebra of \(G\), and let \(P\) be a parabolic subgroup of \(G\) with \(p\)=Lie(\(P\)). We consider the commuting variety \(C\)(\(p\)) of \(p\) and obtain two criteria for \(C\)(\(p\)) to be irreducible. In particular we classify all cases when the commuting variety \(C\)(\(b\)) is irreducible, for \(b\) a Borel subalgebra of \(g\). We then let \(G\) be a classical group and let \(O\)\(_1\) and \(O\)\(_2\) be nilpotent orbits of \(G\) in \(g\). We say that \(O\)\(_1\) and \(O\)\(_2\) commute if there exists a pair (\(X\), \(Y\)) ∈ \(O\)\(_1\)×\(O\)\(_2\) such that [\(X\),\(Y\)]=0. For \(g\)=\(s\)\(p\)\(_2\)\(_m\)(\(k\)) or \(g\)=\(s\)\(o\)\(_n\)(\(k\)), we describe the orbits that commute with the regular orbit, and classify (with one exception) the orbits that commute with all other orbits in \(g\). This extends previously-known results for \(g\)=\(g\)\(l\)\(_n\)(\(k\)). Finally let φ be a Springer isomorphism, that is, a \(G\)-equivariant isomorphism from the unipotent variety \(U\) of \(G\) to the nilpotent variety \(N\) of \(g\). We show that polynomial Springer isomorphisms exist when \(G\) is of type G\(_2\), but do not exist for types E\(_6\) and E\(_7\) for \(k\) of small characteristic.