Title:

Radicals of group algebras and permutation representations of symplectic groups

In part A we consider three separate problems concerned with the radical of the group algebra of a finite group over a field of characteristic p dividing the order of the group. In Section I we characterise grouptheoretically those soluble groups for which the radical of the centre of the group algebra is an ideal of the group algebra. In Section II we find a canonical basis for the radical of the centre of the group algebra of a finite group. In Section III we give an algorithm for determining the radical of the group algebra of a psoluble group. We evaluate the result for groups of pIength one and prove that the exponent of the radical in this case is the same as for a Sylow psubgroup. We show by examples that no similar result holds in the general case. In part B we quote a conjecture of J. A. Green's on characters of Chevalley groups and prove Theorem A (i) If the conjecture holds then, excepting for each r at most a finite number of values of q, the group PSp(2r+1,q) has no multiply transitive permutation representations for r > 1. (ii) PSp (4,q) has no multiply transitive permutation representations for q > 2, regardless of the conjecture.
