Title:

Some topics in the theory of finite groups

A regular 2graph consists of a set andOmega; together with a (nonempty) set t of threeelement subsets of andOmega; such that any twoelement subset of andOmega; is contained in the same number of elements of t, any fourelement subset of andOmega; contains an even number of elements of t and not every threeelement subset of andOmega; is in t. These objects were introduced by G. Higman who used a regular 2graph with 276 points to provide a combinatorial setting for the doubly transitive representation of Conway's sporadic simple group C_{3}. In this thesis it is shown that regular 2graphs are in oneone correspondence with equivalence classes of strong graphs (as defined by J.J. Seidel). Moreover, for each point of a regular 2graph there is a natural way of defining a strongly regular graph on the remaining points. These graphical representations are used to obtain restrictions on the structure and on the parameters of a regular 2graph. It is also possible, via the strong graphs, to represent a regular 2graph as a configuration of equiangular lines in Euclidean space. Conversely, results about regular 2graphs obtained in this thesis extend the results of J.J. Seidel on equiangular lines. Regular 2graphs are constructed which admit the PSL(2,q) , qandequiv;l (mod 4), Sp(2m,2), in both doubly transitive representations; PSU(3,q^{2}), q odd; all groups of Ree type together with ^{2}G_{2}(3) = Aut(PSL(2,8)); the sporadic simple groups C_{3} and HiS; the group V.Sp(2m,2) which is the semidirect product of the group V of translations of a vector space of dimension 2m over the field GF(2) by Sp(2m,2). By studying the centraliser ring of a monomial representation associated with the doubly transitive representation it is shown that (with the possible exception of some groups with a regular normal subgroup) the above groups are the only known groups which can act as doubly transitive groups of automorphisms of a regular 2graph.
