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Symmetric generation has provided concise ways of constructing many classical and sporadic groups, in fact every non-abelian finite simple groups arises in this manner. A symmetric presentation for a group is a homomorphism from a progenitor, p*m : N. onto the group. We give details of our program which constructs all permutation images of a given progenitor. In a monomial progenitor, p*m :m N, the control subgroup N has a monomial action on the symmetric generators of order p > 3. We study monomial progenitors in which the control subgroup has an irreducible monomial representation, as several such progenitors have beell found to map onto sporadic groups. We classify all irreducible monomial representations of the alternating, symmetric and sporadic groups and their covering groups. We use the irreducible monomial representations of the covers of the alternating groups to construct monomial progenitors and we obtain sporadic images of several of these progenitors.

In this thesis we study at two rather different problems within the arithmetic of abelian varieties. In the first case, we consider the problem of getting uniform bounds for the number of integer points which an elliptic curve defined over Q can obtain within a square box with sides of length N. In particular, our aim is to break the bounds of Bombieri and Pila which in this case give Oe.(N1/3+e). We accomplish this for a large family of elliptic curves using a variety of techniques including repulsion of integer points via Gap Principles, the theory of heights and the Large Sieve. As an application, we prove a result concerning the number of rational points of bounded height on a del Pezzo surface of degree L The second part of the project considers the behaviour of ranks of abelian varieties which arise as Jacobians of curves defined over a number field K. In particular, we ask for which values d are there infinitely many degree d extensions L/K such that the rank of the Jacobian increases as we base change from K to L. In the case of elliptic curves we show that this occurs for every d> 1 and for general Jacobians we show that this holds for all sufficiently large d.

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This thesis is centered on the following question: Given an abstract group action by an Abelian group G on a set X, when is there a compact Hausdorff topology on X such that the group action is continuous? If such a topology exists, we call the group action compact-realizable. We show that if G is a locally-compact group, a necessary condition for a G-action to be compact-realizable, is that the image of X under the stabilizer map must be a compact subspace of the collection of closed subgroups of G equipped with the co-compact topology. We apply this result to give a complete characterization for the case when G is a compact Abelian group in terms of the existence of continuous compact Hausdorff pre-images of a certain topological space associated with the group action. If G is not compact, we will show that the necessary condition is not sufficient. Together with various examples, we then present a general two-stage method of construction for compact Hausdorff topologies for R-actions. For discrete groups, the necessary condition above turns out to be not very strong. In the case of G = Z² we will see that the two cases |X| < c and |X| ≥ c must be treated very differently. We derive necessary conditions for a group action with |X| < c to be compact-realizable by constructing particularly nice open partitions of the space X. We then use symbolic dynamics together with some generic constructions to obtain a partial converse in this case. If |X| ≥ c we give further constructions of compact Hausdorff topologies for which the group action is continuous.

This thesis concerns M-finite groups and a notion of discrete measure in models of Peano Arithmetic. First we look at a measure construction for arbitrary non-M-finite sets via suprema and infima of appropriate M-finite sets. The basic properties of the measures are covered, along with non-measurable sets and the use of end-extensions. Next we look at nonstandard finite permutations, introducing nonstandard symmetric and alternating groups. We show that the standard cut being strong is necessary and sufficient for coding of the cycle shape in the standard system to be equivalent to the cycle being contained within the external normal closure of the nonstandard symmetric group. Subsequently the normal subgroup structure of nonstandard symmetric and alternating groups is given as a result analogous to the result of Baer, Schreier and Ulam for infinite symmetric groups. The external structure of nonstandard cyclic groups of prime order is identified as that of infinite dimensional rational vector spaces and the normal subgroup structure of nonstandard projective special linear groups is given for models elementarily extending the standard model. Finally we discuss some applications of our measure to nonstandard finite groups.

This thesis studies non-simple Jacobians and non-simple abelian varieties. The moti- vation of the study is a construction which gives a distinguished genus 4 curve in the linear system of a (1, 3)-polarised surface. The main theorem characterises such curves as hyperelliptic genus 4 curves whose Jacobian contains a (1, 3)-polarised surface. This leads to investigating the locus of non-simple principally polarised abelian g- folds. The main theorem of this part shows that the irreducible components of this locus are Is~, defined as the locus of principally polarised g-folds having an abelian subvariety with induced polarisation of type d. = (d1, ... , dk), where k ≤ g/2 Moreover, there are theorems which characterise the Jacobians of curves that are etale double covers or double covers branched in two points. There is also a detailed computation showing that, for p > 1 an odd number, the hyperelliptic locus meets IS4(l,p) transversely in the Siegel upper half space