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One of the main algorithms of computational group theory is the Todd-Coxeter coset enumeration algorithm, which provides a systematic method for finding the index of a subgroup of a finitely presented group. This has been extended in various ways to provide not only the index of a subgroup, but also a presentation for the subgroup. These methods tie in with a technique introduced by Reidemeister in the 1920's and later improved by Schreier, now known as the Reidemeister-Schreier algorithm. In this thesis we discuss some of these variants of the Todd-Coxeter algorithm and their inter-relation, and also look at existing computer implementations of these different techniques. We then go on to describe a new package for coset methods which incorporates various types of coset enumeration, including modified Todd- Coxeter methods and the Reidemeister-Schreier process. This also has the capability of carrying out Tietze transformation simplification. Statistics obtained from running the new package on a collection of test examples are given, and the various techniques compared. Finally, we use these algorithms, both theoretically and as computer implementations, to investigate a particular class of finitely presented groups defined by the presentation: < a, b | an = b2 = (ab-1) ß =1, ab2 = ba2 >. Some interesting results have been discovered about these groups for various values of β and n. For example, if n is odd, the groups turn out to be finite and metabelian, and if β= 3 or β= 4 the derived group has an order which is dependent on the values of n (mod 8) and n (mod 12) respectively.

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In this thesis we consider the following two fundamental problems for semigroup presentations: 1. Given a semigroup find a presentation defining it. 2. Given a presentation describe the semigroup defined by it. We also establish other related results. After an introduction in Chapter 1, we consider the first problem in Chapter 2, and establish a presentation for the commutative semigroup of integers Zpt. Dually, in Chapter 3 we consider the second problem and study presentations of semigroups related to the direct product of cyclic groups. In Chapter 4 we study presentations of semigroups related to dihedral groups and establish their V-classes structure in Chapter 5. In Chapter 6 we establish some results related to the Schutzenberger group which were suggested by our studies of the semigroup presentations in Chapters 3 and 4. Finally, in Chapter 7 we define and study new classes of semigroups which we call R, L-semi-commutative and semi-commutative semigroups and they were also suggested by our studies of the semigroup presentations in Chapters 3 and 4.

A descent algebra is a subalgebra of the group algebra of a Coxeter group. They were first defined over a field of characteristic zero. In this thesis, the main areas of research to be addressed are; 1. The formulation of a rule for multiplying two elements of descent algebra of the Coxeter groups of type D. 2. The identification of properties exhibited by descent algebras over a field of prime characteristic. In addressing the first, a framework which exploits the specific properties of Coxeter groups is set up. With this framework, a new justification is given for existing rules for multiplying together two elements in the descent algebras of the Coxeter groups of type A and B. This framework is then used to derive a new multiplication rule for the descent algebra of the Coxeter groups of type D. To address the second, a descent algebra over a field of prime characteristic, p, is defined. A homomorphism into the algebra of generalised p-modular characters is then described. This homomorphism is then used to obtain the radical, and allows the irreducible modules of the descent algebra to be determined. Results from the two areas addressed are then exploited to give an explicit description of the radical of the descent algebra of the symmetric groups, over a finite field. In this instance, the nilpotency index of the radical and the irreducible representations are also described. Similarly, the descent algebra of the hyper-octahedral groups, over a finite field, has its radical, nilpotency index, and irreducible representations explicitly determined.

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For the non-abelian simple groups G of order up to 106 , excluding the groups PSL(2,q), q > 9, the presentations in terms of an involution a and an element b of minimal order (with respect to a) such that G= < a,b > are well known. The presentations are complete in the sense that any pair (x,y) of generators of G satisfying x2=ym=1, with m minimal, will satisfy the defining relations of just one presentation in the list. There are 106 such presentations. Using a computer, we give generators for each maximal subgroup of the groups G. For each presentation of G, the generators of maximal subgroups are given as words in the group generators. Similarly generators for a Sylow p-subgroup of G, for each p, are given. For each group G, we give a representative for each conjugacy class of the group as a word in the group generators. Minimal presentations for each Sylow p-subgroup of the groups G, and for most of the maximal subgroups of G are constructed. To obtain such presentations, the Schur multipliers of the underlying groups are calculated. The same tasks are carried out for those groups PSL(2,q) of order less than 106 which are included in the "ATLAS of finite groups". For these groups we consider a presentation on two generators x, y with x2=y3=1. A finite group G is said to be efficient if it has a presentation on d generators and d+rank(M(G)) relations (for some d) where M(G) is the Schur multiplier of G. We show that the simple groups J1, PSU(3,5) and M22 are efficient. We also give efficient presentations for the direct products A5xA6, A5xA6,A6xA7 where Ĥ denotes the covering group of H.

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In Chapter 2 of this thesis we look at methods for finding efficient presentations of the transitive permutation groups of degree ≤ 12. Chapter 3 gives efficient presentations for certain direct products of groups including PSL(2, P)2 SL(2, p) X SL(2, 8), PSL(2, p) x C2, for prime p ≥ 5 and PSL(2, 25)3. Chapter 4 introduces a new class of inefficient groups and Chapter 5 gives a brief survey of some of the open problems relating to the efficiency of finite groups.

We introduce in this thesis a new family of semigroup congruences, and we set out to prove that it is worth studying them for the following very important reasons: (a) that it provides an alternative way of studying algebraic structures of semigroups, thus shedding new light over semigroup structures already known, and it also provides new information about other structures not formerly understood; (b) that it is useful for constructing new semigroups, hence producing new and interesting classes of semigroups from known classes; and (c) that it is useful for classifying semigroups, particularly in describing lattices formed by semigroup species such as varieties, pseudovarieties, existence varieties etc. This interesting family of congruences is described as follows: for any semigroup S, and any ordered pair (n,m) of non-negative integers, define ⦵(n,m) = {(a,b): uav = ubv, for all ⋿Sn and v ⋿Sm}, and we make the convention that S1 = S and that S0 denotes the set containing only the empty word. The particular cases ⦵(0,1), ⦵(1,0) and ⦵(0,0) were considered by the author in his M.Sc. thesis (1991). In fact, one can recognise ⦵(1,0) to be the well known kernel of the right regular representation of S. It turns out that if S is reductive (for example, if S is a monoid), then ⦵(i,j) is equal to ⦵(0,0) - the identity relation on S, for every (i,j). After developing the tools required for the latter part of the thesis in Chapters 0-2, in Chapter 3 we introduce a new class of semigroups - the class of all structurally regular semigroups. Making use of a new Mal'tsev-type product, in Chapters 4,5,6 and 7, we describe the lattices formed by certain varieties of structurally regular semigroups. Many interesting open problems are posed throughout the thesis, and brief literature reviews are inserted in the text where appropriate.