Some topics in the theory of finite groups

 A regular 2-graph consists of a set andOmega; together with a (non-empty) set t of three-element subsets of andOmega; such that any two-element subset of andOmega; is contained in the same number of elements of t , any four-element subset of andOmega; contains an even number of elements of t and not every three-element subset of andOmega; is in t . These objects were introduced by G.andnbsp;Higman who used a regular 2-graph 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 2-graphs are in one-one correspondence with equivalence classes of strong graphs (as defined by J.J.andnbsp;Seidel). Moreover, for each point of a regular 2-graph 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 2-graph. It is also possible, via the strong graphs, to represent a regular 2-graph as a configuration of equiangular lines in Euclidean space. Conversely, results about regular 2-graphs obtained in this thesis extend the results of J.J.andnbsp;Seidel on equiangular lines. Regular 2-graphs 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 semi-direct 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 2-graph.