Primitive bicirculant association schemes and a generalization of wielandt's theorem

Bannai and Ito defined association scheme theory as doing "group theory without groups", thus raising a basic question as to which results about permutation groups are, in fact, results about association schemes. By considering transitive permutation groups in a wider setting of association schemes, it is shown in this paper that one such result is the classical theorem of Wielandt about primitive permutation groups of degree 2p, p a prime, being of rank at most 3 (see Math. Z. 63 (1956), 478-485). More precisely, it is proved here that if X is a primitive bicirculant association scheme of order 2p^e, p > 2 is a prime, then X is of class at most 2, and if it is of class exactly 2, then 2p^e = (2s + 1)² + 1 for some natural number s, with the valencies of X being 1, s(2s + 1), (s + 1)(2s + 1), and the multiplicities of X being 1, p^e, p^e - 1. Consequently, translated into permutation group theory language, a primitive permutation group G of degree 2p^e, p a prime and e ≥ 1, containing a cyclic subgroup with two orbits of size p^e, is either doubly transitive or of rank 3, in which case 2p^e = (2s+1)²+1 for some natural number s, the sizes of suborbits of G are 1, s(2s + 1) and (s + 1)(2s + 1), and the degrees of the irreducible constituents of G are 1, p ^e and p^e - 1.