This thesis constitutes a contribution to applied stability theory. We consider the classification problem of the stable simple locally finite groups. First the classification of the finite simple groups is used to reduce the problem to an identification problem for the simple locally finite groups of Lie type and an interpretation problem in model theoretic algebra. In chapter three, the identification problem is solved. It is shown that the union of a chain of groups of the same Lie type over finite fields is a group of Lie type over a locally finite field. This result, together with the classification of the finite simple groups, implies that an infinite simple periodic linear group is a group of Lie type over a locally finite field. The next two chapters solve the interpretation problem, and complete the proof that a stable simple locally finite group is a Chevalley group over an algebraically closed field. We also show that the class of Chevalley groups of a fixed Lie type is finitely axiomatisable. Chapter six contains a partial classification of the nonsoluble locally finite groups of finite Morley rank. In the final chapter, we show that a simple constructible group over an algebraically closed field is a Chevalley group. The proof is model theoretic, and makes no use of algebraic geometry or Lie algebras. This result can be regarded as a nonstandard corollary of the classification of the finite simple groups.

In this dissertation, Cohn's methods for constructing skew fields from firs are extended to methods for constructing simple artinian rings from hereditary rings; the flexibility this gives is used in order to prove results about the skew field coproduct, and other skew fields originally investigated by Cohn. Part I is devoted to developing the techniques needed; this has two main themes; the first is a detailed study of the finitely generated projective modules over an hereditary ring; the second is an investigation of the ring construction, universal localisation. The construction of universal homomorphisms from suitable hereditary rings to simple artinian rings leads to the simple artinian coproduct with amalgamation of simple artinian rings, which is the natural generalisation of the skew field coproduct of Cohn. Part II is a detailed study of the skew fields and simple artinian rings that are constructed in this way from firs or more generally from hereditary rings. The finite dimensional division subalgebras are classified (apart from the case of the skew field coproduct of two quadratic extensions of a skew field), the transcendence degree of the commutative subfields is bounded, and the centralisers of transcendental elements is studied in special cases. In addition, there are isolated results on the isomorphism classes of skew field coproducts. Part III is distinct from the rest of the dissertation; it consists of a description of generic solutions to the. partial splitting for finite dimensional central simple algebras. These are twisted forms of grassmannian varieties; this is used to show that over number fields, these varieties satisfy the Hasse principle.

Let R be an associative ring with a 1 . Denote by GLn(R) the group of invertible nxn matrices over R, and by GEn(R) the subgroup of GLp(r) generated by the elementary and invertible diagonal matrices. Certain specified relations between these generators hold universally, that is, for any ring R. We call a ring R universal for GEn if GEn(R)'has these relations as defining relations, and we shew that if R is a local ring (i.e. a ring in which the set of all non-units is an ideal) or the ring of rational integers, then R is universal for GEn, for all n. This both generalizes known results for n=2, and includes the classical case where R is a field, possibly skew. By adding further relations to those already, considered we obtain in a similar way the concept 'quasi-universal for GE? ', giving a class of rings which strictly includes the class of all rings universal for GEp, but which is better behaved than the latter under certain ring constructions. We shew that every semi-local ring (i.e. every ring R such that R modulo its Jacobson radical has the minimum condition on right ideals) is quasi-universal for GEn , for all n. Finally we shew how to obtain a presentation of GEn(R) for any R. This is unwieldy, but simplifies greatly for a certain class of rings called GE2-reducible rings, which includes all Euclidean rings. We shew that for such rings R a set of defining relations for GEn(R), for n > 3, is obtained by taking the universal relations together with certain relations in GEa(R).

This thesis is mainly concerned with the following sort of question. Let A and C be Abelian groups with certain model-theoretic properties. What can be deduced about the model-theoretic properties of F(A,C) where F is some operation on the class of Abelian groups? Since we never consider any other sort of group, we use the term group to mean Abelian group throughout. We give the following results: in chapter 1, a characterisation of groups A such that for any group A', A ≡ A' if and only if T(A) ≡ T(A') and A/T(A) ≡ A'/T(A'): in chapter 2 we shew that the above characterisation also characterises groups A and C such that for any groups A' and C', A ≡ A' and C ≡ C' implies A ⊗ C ≡ A' ⊗ C': in chapter 3 we shew that for any groups A, A', C, C' A ≡ _kw A' and C ≡ _kw C' implies Tor(A,C) ≡ _kw Tor(A', C'): in chapter 4 we obtain some results about groups without elements of infinite height: in chapters 5 and 6 we extend our investigations to Hom and Ext. Finally in the appendix, we shew how to extend most of these results to modules over Dedekind domains.

The rings considered in this thesis are the free algebras k⟨X⟩ (k a commutative field) and the more general rings K_k⟨X⟩ (K a skew field and k a subfield of the centre of K) given by the coproduct of K and k⟨X⟩ over k. The results fall into two distinct sections. The first deals with normal forms; using a process of linearization we establish a normal form for full matrices over K_k⟨X⟩ under stable association. We also give a criterion for a square matrix A over a skew field K to be cyclic - that is, for xI - A to be stably associated to an element of K_k⟨X⟩ (here k = centre(K)). The second section deals with factorizations and eigenrings in free algebras. Let k be a commutative field, E/k a finite algebraic extension and P a matrix atom over k⟨X⟩. We show that if E/k is Galois then the factorization of P over E⟨X⟩ is fully reducible; if E/k is purely inseparable then the factorization is rigid. In the course of proving this we prove a version of Hilbert's Theorem 90 for matrices over a ring R that is a fir and a k-algebra; namely that H¹(Gal (E/k),GL_n (R⊗_kE)) is trivial for any Galois extension E/k. We show that the normal closure F of the eigenring of an atom p of k⟨X⟩ provides a splitting field for p (in the sense that p factorizes into absolute atoms in F⟨X⟩). We also show that if k is any commutative field and D a finite dimensional skew field over k then there exists a matrix atom over k⟨X⟩ with eigenring isomorphic to D.

This thesis deals with two topics. In Part I it is shown that if ZFC is consistent, then so is ZF + the order extension principle + there is an abelian group without a divisible hull. The proof is by forcing. In Part II a technique is developed which, in many varieties of algebras, enables the construction for each positive integer not a non-free X_alpha+n-free algebra of cardinality X_alpha+n from a suitable non-free X_alpha-free algebra, when is regular. The algebras constructed turn out to be elementarily equivalent in the language L_{infinityXalpha+n} to free algebras in the variety. As applications of the technique, it is shown that for any positive integer n there are 2^Xn X_n-free algebras which are generated X_n elements, cannot be generated by fewer than this number and are L_infinityXn-equvalent to free algebras in each of the following varieties: any torsion-free variety of groups, all rings with a 1, all commutative rings with a 1, all K-algebras (with K a not-necessarily commutative integral domain), all Lie algebras over a given field. By a different analysis it is shown too that in any variety of nilpotent groups, a lambda-free group of uncountable cardinality lambda is free (respectively, equivalent in L_{infinitylambda} to a free group) if and only if its abelianisation is, in the abelian part of the variety. Finally, sufficient conditions are given for a X-free group in a variety of groups to be also para free in the variety. The results imply that in the varieties of all groups soluble of length at most k and of all groups polynil potent of given class, if lambda is singular or weakly compact, then a lambda-free group of cardinality lambda is parafree, while if lambda is strongly compact, then a lambda-free group of any cardinality is parafree.

The object of this thesis is to solve, in integers X and Y, various equations of the form [equation] where d and N are given square free integers. The work stems from two papers by J.H.E. Cohn in which the equations [equation] are solved for certain values of d. It is well known that the solutions of X2-dY2=4, and those of X2-dY2 = -4 where such solutions exist, may be expressed in terms of the least positive solutions of these equations. Solutions of the equations [equations] may now be sought among those of x2-dY2 = +1,+4. Extensive work in this direction has been done by W. Ljunggren working in the quadratic field R(d2) and other allied algebraic fields. His methods are powerful but deep and complicated. It is possible to show that the solutions of the equations X2 for sequences which satisfy a three-term recurrence relation. By applying the elementary theory of quadratic residues to these equations Cohn has solved the equations and for those d for which either of the equations has solutions (X,Y) for which X and Y are both odd. This thesis extends Cohn's work, Using similar methods, to solve the equations and for the same values of d as above and any given integer N. A few limited results are given for other values of d. L.J. Mordell has given simple conditions under which the equation can have no solutions. A theorem of a similar type concerning the equations x4-dY2 = 1,4 is proved. Finally the results proved in this thesis are compared with those of Cohn, Ljunggren and Mordell.

The dichromatic polynomial of a graph (or the Tutte polynomial) is a polynomial function of two variables from which a large amount of important information about the graph may be obtained , including the chromatic polynomial and the complexity of the graph. Some properties of the Tutte polynomial and an algorithm for its computation are given.a recursive family of graphs is defined to be a family of graphs whose Tutte polynomials satisfy a homogeneous linear recurrence relation. The smallest possible order of such a recurrence relation is called the recursiveness of the family. The existence of such a recurrence relation enables us to consider the Tutte polynomials of large graphs. Some elementary properties of recursive families are found and two large classes of recursive families of graphs are defined. The proof that the families in these classes are recursive is constructive and the methods used are applied to some families from the two classes with small recursiveness. The problem of the location of the chromatic roots of a graph is considered in the light of the information thus gained and several conjectures are made. The most important of these is a generalisation of Brooks' theorem [6] and states that for a graph whose greatest valency is k the chromatic roots all have modulus not greater than k + 1. Much of the work may be generalised immediately to matroid theory and where this is so the appropriate results are stated.

We are concerned with the idealizer S of a principal right ideal rR in a free associative algebra R. This comes up naturally in the theory of the corresponding one-relator associative algebra R/RrR, and seems to be a reasonable route to the investigation of such algebras. In many ways, S is well-behaved. It is always a pure (in the sense of Kosevoi) subalgebra. The case where r is homogeneous seems to produce a free idealizer - we are able to demonstrate freeness in such disparate cases as r being a monomial, and r being such that R/RrR has no zero divisors. The latter result is due to Jacques and Tekla Lewin, and we will prove a more general form of it. If r is not homogeneous we know that S need not be free - indeed, not even a 2-fir. But we can give examples of cases where S is free, and one of these provides the first known example of a non-regularly embedded free subalgebra of a free algebra. For certain types of one-relator algebras we can prove analogues of Magnus' theorems on one-relator groups, and in particular we prove a result of Sirsov's. We also investigate the Golod-Safarevic power series associated to these and other algebras. With machinery we develop, we derive the non-homogeneous generalization of the Golod-Safarevic criterion for infinite dimensionality, and extend the Schreier-Lewin dimension formula to rings with weak algorithm. Finally, we append our work with Bergman extending Lewin's upper triangular matrix representations.

We are concerned with two problems. Firstly, given a ring R and an epic R-field K, under what conditions can K be fully ordered? Epic R-fields can be constructed in terms of matrices over R; this makes it natural, in describing full orders on K, to consider matrix cones over R rather then ordinary cones of elements of K. Essentially, a matrix cone over R, associated with a given ordering of K consists of all square matrices which either become singular or have positive Dieudonne determinant over K. We give necessary and sufficient conditions in terms of matrix cones for (i) an epic R-field to be orderable, (ii) a full order on R to be extendible to a field of fractions of R and (iii) for such an extension to be unique. The second problem is finding K₁(U(R)), where R is is a Sylvester domain and U(R) denotes its universal field of fractions. Let R be a Sylvester domain and let Sigma be the monoid of full matrices over R. We show that K₁(U(R)) is naturally isomorphic to alpha(Sigma), the universal abelian group of Sigma. The inclusion R ⊆ U(R) induces a map K₁(R) → K₁(U(R)); we also prove that if R is a fully atomic semifir (e.g. if R is a fir) then K₁(U(R)) = K₁(R) X D(R), where K₁(R) denotes the image of K₁(R) in K₁(U(R)) and D(R) is the free abelian group on the set of equivalence classes of stably associated matrix atoms over R.