Abstract not found

Abstract. Let G be a finite permutation group acting on a set Γ. A base for G is a finite sequence of elements of Γ whose pointwise stabiliser in G is trivial. Most (families of) finite permutation groups admit a base that grows very slowly as the degree of the group increases. Such groups are known as small base and very efficient algorithms exist for dealing with them. However, some families of permutation groups, such as the symmetric groups, do not admit a small base. Dealing with these so-called large base groups is a fascinating area of current research. This thesis explores two closely interrelated strands of modern group theory. Initially, the focus is on identifying the large base primitive permutation groups, which can be achieved by making use of two landmark results in finite group theory: The Classification of Finite Simple Groups and the O’Nan-Scott Theorem for primitive permutation groups. Focus then shifts to algorithmic aspects of large base groups, in particular to the family known as the giants. We cover details such as recognition of large base Galois groups, generation of random elements of finite groups and give details of the very new paradigm of algorithms for black box groups. We conclude with an investigation into the constructive recognition problem for large base black box groups.

We describe an algorithm for computing a chief series, the soluble radical, and two other characteristic subgroups of a matrix group over a finite field, which is intended for matrix groups that are too large for the use of base and strong generating set methods. The algorithm has been implemented in MAGMA by the second author.