Computing with finite groups

The character table of a finite group G is constructed by computing the eigenvectors of matrix equations determined by the centre of the group algebra. The numerical character values are expressed in algebraic form. A variant using a certain sub-algebra of the centre of the group algebra is used to ease problems associated with determining the conjugacy classes of elements of G. The simple group of order 50,232,960 and its subgroups PSL(2,17) and PSL(2,19) are constructed using general techniques. A combination of hand and machine calculation gives the character tables of the known simple groups of order < 106 excepting Sp(4,4) and PSL(2,q). The characters of the non- Abelian 2-groups of order < 2 6 are computed. Miscellaneous computations involving the symmetric group Sn are given

Design and implementation of a language for manipulating algebraic formulae

This thesis explores the possibilities of doing mathematical problems involving algebra on a computer. A language is designed which allows names to occur as unknown quantities. This language has all the facilities of a general purpose language such as IMP, but is designed to be used inter-actively by a user at a console. The language also includes instructions which cause the usual algebraic operations to be applied to expressions. These operators include simplification, differentiation, but not integration. A brief survey is given of other languages in the field, with comments on their capabilities and restrictions. The second part of the thesis describes how the language is implemented. An interpreter is used. Statements of the language are analysed syntactically and then obeyed. Algebraic expressions are stored in byte arrays, using a type of prefix Polish notation. Finally the language is reviewed in the light of recent work done in the field, and suggestions are made for a further version